跨方法流场学习几何泛化调研：表示机制、验证证据与有限观测重建

Generalizing neural operators across varying geometries remains a central challenge in scientific machine learning. While most operator-learning frameworks assume fixed geometries and parametric variability in boundary conditions or coefficients, many fluid-dynamics problems require prediction across distinct domains. In this work, we introduce the Geometry-informed transformer neural operator (GINTO), a physics-informed architecture that reformulates the physics-informed neural transformer operator (PINTO) paradigm for domain-geometry generalization. The proposed model uses a spatial representation of obstacle geometry encoded as point clouds and propagates geometric information to interior evaluation points through stacked linear cross-attention blocks. The approach is based on a point-wise collocation representation and does not require a structured or unstructured body-fitted mesh, geometric parameterization, or domain alignment. Training requires neither data nor reference solutions, physical consistency is enforced through residual minimization of the steady incompressible Navier–Stokes equations. To systematically study geometric generalization, we introduce a composite geometric distance metric and construct training subsets with controlled coverage of the geometric space. Numerical experiments demonstrate that extrapolation accuracy strongly depends on geometric diversity in the training set. Increasing coverage yields substantial improvements up to a saturation regime, beyond which gains become marginal. Compared with the original PINTO architecture, the proposed model achieves comparable accuracy while delivering a 7–10-fold reduction in training time on the benchmark problems considered in this study, thereby enabling large-scale experiments with diverse geometry sets. The results establish geometry-conditioned transformer operators as a scalable and physically consistent surrogate modeling framework for fluid flows on arbitrary domains, with relevance to shape optimization and geometry-driven simulation tasks. The code and experiments are available in the open-source framework MultiPINN at https://github.com/labadt/multipinn.

No abstract available

Repeatedly solving flow around structures with varying parameters using computational fluid dynamics (CFD) is often essential for structural design. This study proposes a boundary-assimilation Fourier neural operator (BAFNO) method to address the challenges of manually setting initial conditions for CFD. The focus of the BAFNO is on the generalization ability to predict initial flow fields without relying on observational data. BAFNO addresses the boundary constraint requirements of the existing physics-informed neural operator models in parametric geometries. Inspired by the ghost node method, the domain boundary conditions are assimilated into the loss function instead of adding penalty terms. Meanwhile, the structure boundaries are assimilated into a damping source term using a level set function. BAFNO can flexibly handle parametric geometries with different shapes and quantities. Subsequently, a series of numerical experiments for flow-around structures are conducted to confirm the performance of the BAFNO. The results indicate that the BAFNO has strong generalization capability, and the BAFNO + CFD can obtain dynamic stable fields faster than the direct CFD.

Fluid flow exhibits strong nonlinearity, and traditional solutions based on computational fluid dynamics (CFD) are limited by high computational costs. To efficiently model complex fluid dynamics, this study proposes a Fourier neural operator network (FNONet) to approximate solutions for 2D incompressible non-uniform steady laminar flows around obstacles. FNONet transforms flow geometry into Fourier space via discrete Fourier transform (DFT), parameterizes integral kernels using a neural network, and reconstructs velocity and pressure fields through hybrid frequency-spatial computations. A pressure gradient-constrained data-driven loss function is introduced to incorporate spatial-domain errors and frequency-domain pressure gradient constraints, thereby enhancing prediction accuracy near fluid-solid interfaces. Experiments demonstrate that the total mean squared error (MSE) of FNONet is over 20 times lower than that of convolutional neural network (CNN) based framework and 1.5 times lower than that of CNN-Transformer based framework. Compared with traditional CFD solvers, FNONet achieves a prediction speedup of over 3 orders of magnitude on central processing units (CPUs) and 4 orders of magnitude on graphics processing units (GPUs). The robustness and generalization of the model to unseen geometries are validated through qualitative visual analysis and quantitative error metrics on test datasets, providing a promising data-driven approach for CFD.

Amid the global transition to carbon neutrality, tidal current energy has become a strategic sustainable energy resource due to its high predictability, power density, and environmental compatibility. Horizontal-axis turbines show great potential for marine energy harvesting, yet the large-scale commercialization of tidal turbines is severely hindered by complex wake dynamics and the lack of reliable, efficient prediction tools for out-of-distribution (OOD) operating conditions. Traditional high-fidelity CFD methods are computationally prohibitive for engineering optimization, while conventional data-driven surrogate models suffer from poor extrapolation performance, extrapolation collapse near training parameter boundaries, and the absence of uncertainty quantification. To address these bottlenecks, this study focuses on the OOD extrapolation of wake flow prediction across tip speed ratio (TSR) distributions for a single horizontal-axis tidal turbine. A CFD-generated spatiotemporal benchmark dataset is constructed for comparative OOD evaluation across various TSR conditions with 9504 total samples. A novel physics-constrained Fourier neural operator framework named TSR-FNO is proposed to improve OOD generalization. The model integrates TSR–Lipschitz regularization to suppress extrapolation collapse and Monte Carlo Dropout to provide reliable uncertainty estimation. Extensive experiments demonstrate that the proposed method effectively reduces prediction error in unseen TSR regimes, mitigates performance degradation in far-field extrapolation, and produces well-calibrated uncertainty estimates consistent with actual prediction confidence. This work provides a data-driven surrogate modeling strategy for fast and reliable wake prediction on a common CFD-generated benchmark, supporting the efficient design, array layout optimization, and engineering deployment of tidal current energy systems.

Flow field information within cascades is crucial for refined turbomachinery design. Currently, this information is primarily obtained through experimental methods or numerical simulations, both of which are complex and time-consuming. Data-driven deep learning approaches offer a potential solution for rapid flow field evaluation. However, existing deep learning-based flow field prediction models exhibit certain limitations in accuracy and generalization, particularly in regions with high gradients, which are often the primary sources of aerodynamic losses. To address these issues, this study develops a high-precision cascade flow field prediction model, A-FNO, based on a Galerkin-type self-attention mechanism and Fourier Neural Operator (FNO). A-FNO is designed based on the newly proposed FNO, which has demonstrated excellent performance in solving partial differential equations. This study extends its application to cascade flow field prediction problems. To mitigate the limitations of FNO in predicting areas with steep gradient changes, we incorporate the self-attention mechanism to capture dependencies between different regions of the flow field, thereby enhancing FNO's ability to express flow field details. Experimental results demonstrate that A-FNO significantly improves prediction accuracy in regions surrounding the boundary layer. The maximum relative error for velocity field predictions is within 5%, for pressure field predictions within 2%, and for temperature field predictions within 1%.

Accurate and efficient prediction of fluid dynamics plays a critical role in aerospace engineering, energy systems, and environmental applications, where reliable flow modeling supports design optimization and reduces computational costs. Traditional computational fluid dynamics (CFD) solvers, while highly accurate, become prohibitively expensive for unsteady and multi-scale flows. Recent neural operator approaches provide an attractive alternative by learning mappings between flow states and solutions, yet their current formulations struggle to preserve physical consistency under complex geometries and often exhibit strong low-frequency bias, which limits their capability to capture essential multi-scale features in fluid dynamics. To overcome these issues, we propose a physics-aware neural operator that unifies global flow coherence modeling and local geometric consistency under a spectral balancing mechanism. Specifically, our approach captures long-range dependencies through an efficient attention-based integral representation while refining local gradients via convolutional operators, and employs adaptive normalization to mitigate low-frequency dominance. This design ensures physical consistency and accurate multi-scale feature representation without compromising efficiency. We validate the proposed approach on multiple fluid dynamics benchmarks, including canonical flow problems and complex geometry scenarios. Compared to state-of-the-art operator learning models, our method significantly reduces prediction error while maintaining strong generalization under limited training data. These results demonstrate the potential of physics-aware neural operators as an efficient and accurate alternative to traditional CFD solvers for complex fluid dynamics applications.

No abstract available

Global warming has increased ocean wave heights and the frequency of extreme wave events. As a result, ocean waves have contributed to more coastal erosion, flooding, and fatigue damage of ocean-based structures. Researchers interested in understanding the effects of ocean waves must rely on slow numerical methods such as particle-in-cell (PIC), which can take hours. Therefore, researchers have good occasion to find a faster alternative for modeling ocean waves. Here, we present a Fourier Neural Operator (FNO) for predicting ocean wave behavior on grids of arbitrary geometry. An FNO is designed to map an input image to an output image in infinite dimensions. This allows the FNO to act as a solution operator, which decreases computational complexity. To create data for our FNO, we used the PIC method to create 290 simulations of an ocean wave crashing against a random obstruction, i.e., some arbitrary geometry. Each simulation modeled 38,000 water particles for 600 time steps, which required 30 minutes. From these simulations, we created training and testing datasets of 274 and 16 image pairs, respectively. We trained the FNO for 75 epochs, which took 900 seconds. After training, we validated the FNO against our 16 test points. At a resolution of $256 \times 256$, our FNO had an inference time of only 0.073 second. Our FNO results were good, especially since only 274 training points were used. Using our approach, FNOs can be designed to predict levee overtopping, flooding, blast waves, and the effect of water drops over wildfires.

The introduction of deep learning has resolved the high-cost issues associated with traditional methods in handling complex aerodynamics problems and is commonly used for simulating fluid behavior and optimizing aircraft design. However, flow field prediction based on deep learning typically encodes the freestream conditions and geometric information into the neural network model concurrently. This encoding scheme makes it difficult for the model to distinguish and deal with the intrinsic differences between these two types of information. As a result, the ability of the model to capture complex flow field features decreases and the difficulty of model fitting increases, which in turn reduces the effectiveness of the model. To solve these problems, this paper proposes the Operator-Convolution MultiModal Fusion Network (OCMMFNet), a new neural network architecture to predict the flow fields of airfoils with various geometries and freestream conditions. The proposed network architecture uses a freestream generalization network to encode the input freestream conditions. The resulting approximate flow field information is combined with the airfoil geometry information and fed into a shape feature compensation network to improve the prediction accuracy. We compare the performance of OCMMFNet with those of a deep operator network(DeepONet) and a vision transformer(ViT) model. When generalizing both freestream conditions and airfoil shapes, OCMMFNet reduces the prediction error in the pressure field by 9.71% and 3.76% compared to DeepONet and ViT, respectively. In tests involving extrapolation of Reynolds numbers, OCMMFNet significantly reduces the prediction error in the pressure field by 13.73% and 11.84% compared to DeepONet and ViT, respectively. The results show that OCMMFNet achieves better prediction accuracy than both DeepONet and ViT and displays superior robustness and generalization ability.

Particle Image Velocimetry (PIV) aims to infer underlying velocity fields from time-separated particle images, forming a PDE-constrained inverse problem governed by advection dynamics. Traditional cross-correlation methods and deep learning-based feature matching approaches often struggle with ambiguity, limited resolution, and generalization to real-world conditions. To address these challenges, we propose a PIV Neural Operator (PIVNO) framework that directly approximates the inverse mapping from paired particle images to flow fields within a function space. Leveraging a position informed Galerkin-style attention operator, PIVNO captures global flow structures while supporting resolution-adaptive inference across arbitrary subdomains. Moreover, to enhance real-world adaptability, we introduce a self-supervised fine-tuning scheme based on physical divergence constraints, enabling the model to generalize from synthetic to real experiments without requiring labeled data. Extensive evaluations demonstrate the accuracy, flexibility, and robustness of our approach across both simulated and experimental PIV datasets. Our code is at https://github.com/ZXS-Labs/PIVNO.

Electrical impedance tomographic (EIT) tactile sensing holds great promise for whole-body coverage of contact-rich robotic systems, offering extensive flexibility in sensor geometry. However, low spatial resolution restricts its practical use, despite the existing deep-learning-based reconstruction methods. This study introduces EIT-GNN, a graph-structured data-driven EIT reconstruction framework that achieves super-resolution in large-area tactile perception on unbounded form factors of robots. EIT-GNN represents the arbitrary sensor shape into mesh connections, then employs a twofold architecture of transformer encoder and graph convolutional neural network to best manage such the geometrical prior knowledge, resulting in the accurate, generalized, and parameter-efficient reconstruction procedure. As a proof-of-concept, we demonstrate its application using large-area face-shaped sensor hardware, which represents one of the most complex geometries in human/humanoid anatomy. An extensive set of experiments, including simulation study, ablation analysis, single-touch indentation test, and latent feature analysis, confirm its superiority over alternative models. The beneficial features of the approach are demonstrated through its application in active tactile-servo control of humanoid head motion, paving the new way for integrating tactile sensors with intricate designs into robotic systems.

Structured mesh generation serves as a crucial preprocessing step in numerical simulations and can be formulated as a mapping problem from geometry to structured mesh. Existing approaches typically establish an isolated mapping for each geometry. This geometry-specific paradigm fails to capture and leverage commonalities across geometries, inevitably requiring recomputation or costly retraining for new geometries. To overcome this limitation, we propose ICL-Mesh, a meta-learning framework based on in-context learning (ICL) for structured mesh generation. It treats learning one mapping as one task and trains a single neural network to extract commonalities across tasks and learn from in-context examples within each task, enabling rapid generalization to unseen tasks without parameter updates. Experimental results demonstrate that ICL-Mesh effectively generalizes to diverse geometries with only a few context examples, and even without examples. It also exhibits robustness to in-context example order sensitivity and can be extended to various mesh generation scenarios, including mesh refinement and coarsening.

In this work, we study the generalization capabilities of graph neural networks (GNNs) over geometric graphs sampled from manifolds. Specifically, we focus on graphs constructed from randomly sampled points over an embedded manifold with underlying geometric information captured. The graph is relatively sparse under a practical model assumption. We derive a generalization gap between the empirical and statistical risks of this GNN. We observe that the gap increases with the dimension of the underlying manifold and decreases with the number of sampled points from the manifold. This result demonstrates that a GNN trained on a graph constructed from a finite set of sampled points can effectively process unseen graphs generated from the same underlying manifold. Our theoretical analysis is grounded in the non-asymptotic convergence of a GNN on the sampled geometric graph to the underlying manifold neural network (MNN). We verify this theoretical result with experiments on a citation network.

Unstructured meshes offer the flexibility to capture complex details in flow fields, and graph neural networks (GNNs) are regarded as a natural tool for processing such mesh data. However, in practical flow field simulations, the topology of computational meshes changes with variations in aircraft geometry and flight conditions. When confronted with such fundamental topological variations, the predictive performance of existing GNN methods declines significantly, severely limiting their ability to generalize to unseen mesh topologies. To address this challenge, we propose a graph convolution method based on a geometric attention mechanism, which integrates a learnable local flow direction encoding and a cosine similarity weighting strategy to actively learn geometric relational rules that are independent of specific mesh connectivity. This approach enhances feature extraction and representation in flow fields, maintains stable performance across different mesh structures, and improves the capture of high-gradient features such as shock waves. As a result, the model's robustness to mesh topological changes and its predictive accuracy are significantly improved. Numerical experiments on several typical supersonic flow cases demonstrate that, compared to existing GNN methods, the proposed approach achieves superior performance in both accuracy and generalization capability, effectively adapting to topological variations in unstructured meshes.

No abstract available

No abstract available

Many applications in computational and experimental fluid mechanics require effective methods for reconstructing the flow fields from limited sensor data. However, this task remains a significant challenge because the measurement operator that provides the punctual sensor measurement for a given state of the flow field is often ill-conditioned and non-invertible. This issue impedes the feasibility of identifying the forward map, which is, theoretically, the inverse of the measurement operator, for field reconstruction purposes. While data-driven methods are available, their generalizability across different flow conditions (e.g., different Reynold numbers) remains questioned. Moreover, they frequently face the problem of spectral bias, which leads to smooth and blurry reconstructed fields, thereby decreasing the accuracy of reconstruction. We introduce FLRNet, a deep learning method for flow field reconstruction from sparse sensor measurements. FLRNet employs a variational autoencoder with Fourier feature layers and incorporates an extra perceptual loss term during training to learn a rich, low-dimensional latent representation of the flow field. The learned latent representation is then correlated to the sensor measurement using an attention-based network. We validated the reconstruction capability and the generalizability of FLRNet under various fluid flow conditions and sensor configurations, including different sensor counts and sensor layouts. Numerical experiments show that in all tested scenarios, FLRNet consistently outperformed other baselines, delivering the most accurate reconstructed flow field and being the most robust to noise.

The design of thermal protection modules (such as film cooling) for combustion chambers requires a high-fidelity swirling flow field. Although numerical methods provide insights into three-dimensional mechanisms of swirling flow, their predictions of key features such as recirculation zones and swirling jets are often unsatisfactory due to inherent anisotropy and the isotropic nature of the Boussinesq hypothesis. Experimental methods, such as hot wire, laser Doppler velocimetry, and planar particle image velocimetry (PIV), offer more accurate reference data but are limited by sparse or planar observations. In this study, considering the outperformed capability of solving inverse problems, physics-informed neural network (PINN) was adopted to reconstruct the mean swirling flow field based on limited experimental observations from two-dimensional and two-component (2D2C) results. It was found that adding partial information characterizing the swirling flow, such as the swirling jet, could significantly improve the reconstruction of flow field. In addition, film cooling effectiveness was the key variable to evaluate the film cooling performance, which was relatively measurable in the scalar field. To further improve the accuracy of the reconstruction, the multi-source strategy was adopted into the neural network, where the film cooling effectiveness (FCE) of the effusion plate was imported as the scalar source. It was found that the prediction of the flow field near the target plate was improved, where the highest error reduction could reach 76.5%. Finally, through the reconstructed three-dimensional vortex distribution, it was found that swirling flow vortex structures near the swirler exit had a significant impact on cooling effectiveness, causing a non-uniform cooling distribution. This study aims to diagnose the three-dimensional swirling flow field with deep learning by leveraging limited experimental data and deepen the understanding of effusion cooling under swirling flow condition so that obtains a more accurate reference in the design of thermal protection modules.

Urban lakes face increasing pollution pressures, yet accurate prediction of contaminant transport and source identification remains challenging due to the high computational cost and sparse monitoring data. This study develops a hybrid framework based on proper orthogonal decomposition (POD) and radial basis function (RBF) interpolation for three tasks: rapid prediction of pollutant fields under new boundary conditions, full flow field reconstruction from sparse observations, and inversion of key boundary condition parameters. The POD-RBF fast prediction model achieves excellent accuracy (coefficient of determination R2 > 0.97, mean absolute error MAE =0.35) while reducing computation time minutes to seconds. With only 10 monitoring points, flow fields reconstruction from sparse data yields MAE = 0.042 and mean squared error MSE = 0.004. Using 20 monitoring points and the Nelder-Mead optimizer, boundary inversion recovers COD concentration with a relative error of 1.6% and achieves an excellent fit at monitoring points (R2 = 0.98). SHAP (SHapley Additive exPlanations) analysis identifies discharge concentration as the most influential factor on pollutant dispersion, followed by diffusion time and rainfall intensity. The proposed framework provides an accurate, computationally efficient, and data-sparse solution for real-time water quality forecasting and source identification in urban lakes, particularly valuable for emergency response in data-scarce environments.

Acquiring high-resolution flow field data in reacting flow systems is essential for understanding their underlying mechanisms. However, it still remains challenging due to experimental limitations in capturing full-dimensional high-resolution measurements. This study introduces a novel observation-guided generative diffusion model and takes the rotating detonation combustor as a case study to reconstruct detailed flow fields from sparse or incomplete observations. Compared to discriminative models, the generative model demonstrates robust generalization across different observation configurations without the need for retraining. Results also indicate that the proposed observation-gradient-guided strategy surpasses existing flow field reconstruction studies in accuracy and consistency. The model operates effectively in scenarios including incomplete velocity fields and sparse pressure fields. Furthermore, the model can generate higher-resolution flow fields through a cascaded adaptive refinement reconstruction. This study provides a model with strong generative performance and generalization capability for flow field reconstruction research. As a surrogate model, it has the potential for flow field visualization and analysis in scientific research and engineering applications.

Reconstructing flow fields from sparse observations constitutes a fundamental challenge in aerodynamic analysis, since existing methods often fail to reconcile accuracy with computational efficiency under severe data constraints. We introduce a Graph Transformer Reconstruction Network (GTREN), a novel model designed to infer latent data patterns from extremely sparse measurements and to reconstruct high-fidelity, full-field flow solutions. The approach represents discrete grid points as nodes within a graph data structure, enabling a principled encoding of spatial relationships. We integrate a Transformer-style attention mechanism into the graph network's message-passing operations, allowing the model to selectively emphasize salient neighbor interactions while attenuating irrelevant signals; this facilitates efficient and stable information propagation. After training, GTREN uncovers implicit correlations among sparse sensor measurements and successfully generalizes these relationships to reconstruct the entire flow field. Results indicate that, with only ten measurement points, the model accurately reconstructs both pressure and velocity distributions across the field. When compared to Computational Fluid Dynamics simulations, the mean squared error (MSE) of the reconstruction is as low as 0.05%. By contrast, a conventional graph neural network yields an MSE of 18.31%. We also provide a systematic analysis of how both the number and spatial arrangement of measurement points affect reconstruction accuracy. Our analysis reveals that, rather than merely increasing sensor count, strategically locating sensors within the core vortex-shedding region of the wake substantially improves reconstruction accuracy—offering practical guidance for flexible sensor deployment. Overall, GTREN achieves high-precision full-field reconstruction with very few pressure sensors, enabling real-time sparse-sensing flow-field monitoring.

Ultrasound localization microscopy (ULM) has improved microvascular imaging by tracking microbubble (MB) trajectories to reconstruct vessel networks. However, the ensemble of trajectories produced by this technique is usually difficult to interpret, due to the underlying complexity of vascular geometries, MB motion, and unavoidable localization and tracking errors. In this work, we introduce a physics-informed channel modeling approach that reconstructs both vessel geometry and flow field exclusively from MB velocity measurements. By leveraging the inherent flow dynamics, each MB refines the vessel model, thus creating a unique, physically coherent reconstruction. We validate our method on an in vitro dataset, showing its robustness under sparse measurement conditions. This approach might allow for recovery of vessel information in tubes with low track density, while facilitating the extraction of clinically relevant global features.

Real-time perception of global flow field evolution is critical for ensuring the operational safety of deep-sea mining hydraulic lift systems. However, existing monitoring techniques are constrained by sparse observations and lack predictive capabilities. To address this, this paper proposes a novel spatiotemporal prediction and reconstruction framework tailored for solid-liquid two-phase flows. We introduce a cascaded deep learning architecture that integrates a long short-term memory network for temporal extrapolation with a pre-trained generative network for spatial reconstruction. This framework establishes a non-linear mapping from sparse historical cross-sectional observations to future three-dimensional global particle volume fraction fields. The proposed model achieves high prediction accuracy R2 > 0.94 with an optimal look-back window of 5 frames. By enabling forward-looking visualization of flow instabilities, this method provides a foundational digital twin technology for intelligent early warning and proactive control in deep-sea engineering.

Traditional experimental methods usually measure the aerodynamic load characteristics of an object by deploying a large number of pressure sensors on its surface, which are often challenging to economically and efficiently obtain accurate surface pressure distribution due to limitation imposed by experimental space, the complexity of geometry, and the cost of measurement instruments. To address this, a compressed sensing (CS) based framework has been proposed in this paper to investigate the reconstruction of the original surface pressure field from extremely sparse measurement data. The proposed framework integrates the generalized proper orthogonal decomposition method for flow field dimensionality reduction, the CS technique for accurate reconstruction of the original signal, and the improved particle swarm optimization algorithm for optimizing sensor placement strategies. Unlike image and unsteady flow field reconstructions, the method presented in this paper has been successful in reconstructing surface pressure fields of high-speed train under various conditions. Based on the accurate reconstruction of surface pressure fields, further aerodynamic load data can be obtained. Additionally, this paper optimizes the traditional pressure sensor layout using a particle swarm optimization method, which not only improves reconstruction accuracy but also significantly reduces the deployment of redundant sensors. Moreover, traditional point selection strategies based on experience can still be incorporated into the pressure sensor layout scheme and effectively reduce reconstruction errors under crosswind conditions. Comparison of the results showed that the proposed framework can accurately and efficiently reconstruct the surface flow field of three-dimensional complex-shaped objects from sparse measurement.

The design of thermal protection modules (such as film cooling) for combustion chambers requires a high-fidelity swirling flow field. Although numerical methods provide insights into three-dimensional mechanisms of swirling flow, their predictions of key features such as recirculation zones and swirling jets are often unsatisfactory due to inherent anisotropy and the isotropic nature of the Boussinesq hypothesis. Experimental methods, such as hot wire, laser Doppler velocimetry, and planar particle image velocimetry (PIV), offer more accurate reference data but are limited by sparse or planar observations. In this study, considering the outperformed capability of solving inverse problems, physics-informed neural network (PINN) was adopted to reconstruct the mean swirling flow field based on limited experimental observations from two-dimensional and two-component (2D2C) results. It was found that adding partial information characterizing the swirling flow, such as the swirling jet, could significantly improve the reconstruction of flow field. In addition, film cooling effectiveness was the key variable to evaluate the film cooling performance, which was relatively measurable in the scalar field. To further improve the accuracy of the reconstruction, the multi-source strategy was adopted into the neural network, where the film cooling effectiveness (FCE) of the effusion plate was imported as the scalar source. It was found that the prediction of the flow field near the target plate was improved, where the highest error reduction could reach 76.5%. This study aims to diagnose the three-dimensional swirling flow field with deep learning by leveraging limited experimental data so that obtains a more accurate reference in the design of thermal protection modules.

Reconstructing physical fields from sensor measurements is a challenging task in scientific and engineering applications, especially when sensors are sparse and unevenly distributed. Existing generative models directly incorporate point observations into the generation process to guide the generation of the full field. However, they did not provide equitable guidance for relatively sparsely observed regions, resulting in an imbalanced reconstruction error. To mitigate this imbalance, we propose proper orthogonal decomposition conditioned flow matching (POD-FM), a framework that synergistically integrates gappy POD's global physical consistency propagation to provide reconstruction guidance for poorly observed regions. Specifically, we take the POD modal coefficients inverted from observations as physically interpretable global feature guidance, then jointly encode them with the point observations to guide the generative model to reconstruct the full field. These coefficients provide a low-dimensional global prior that anchors the probability flow to the dominant modes of the field, which supplements implicit physical consistency for sparsely observed areas, and thus alleviates the accuracy imbalance caused by uneven sensor distribution. The efficacy of POD-FM has been verified on three progressive POD modal complexity datasets. Experiments have shown that POD-FM achieved higher accuracy and could effectively suppress the influence of sensor noise. The code will be available at https://github.com/XinWang199/POD-FM.

Gas turbine combustors commonly feature swirling flows. The swirl is usually characterized by the swirler geometry. The development of swirl-stabilized burners includes the experimental assessment of the resulting flow field and the quantification of the swirl, e.g. through Laser Doppler Anemometry (LDA). LDA accurately acquires flow velocity components in a small probing volume. Nevertheless, the measurement quality represents a trade-off between invested measurement time and spatial resolution. In this work, the potential of Physics-Informed Neural Networks (PINNs) to assimilate flow fields from sparse LDA measurements is investigated. A novel burner is employed in which the swirl is fluidically adjustable from a non-swirled jet to a fully swirled flow through a secondary air flow injection. Data is acquired through LDA within spatial measurement grids at multiple axial distances from the swirler. Assuming symmetry, axial, tangential, and radial velocity components are determined. A PINN is subsequently trained with the acquired data, creating a continuous and differentiable flow field representation by evaluating RANS equations. By systematically reducing the training data while evaluating the physical validity of the reconstructed field, a minimum training data requirement is identified. As a result, for three operating conditions, the flow field is adequately characterized by a minimum of measurement points.

We present a dual-guided framework for reconstructing unsteady incompressible flow fields using sparse observations. The approach combines optimized sensor placement with a physics-informed guided generative model. Sensor locations are selected using mutual information theory applied to a reduced-order model of the flow, enabling efficient identification of high-information observation points with minimal computational cost. These sensors, once selected, provide targeted observations that guide a denoising diffusion probabilistic model conditioned by physical constraints. Extensive experiments on laminar cylinder wake flows demonstrate that under sparse sensing conditions, the structured sensor layouts fail to capture key flow dynamics, yielding high reconstruction errors. In contrast, our optimized sensor placement strategy achieves accurate reconstructions, with L2 (Euclidean) errors (computed on normalized velocity fields) as low as 0.05, even with a limited number of sensors, confirming the effectiveness of the proposed approach in data-limited regimes. When the number of sensors is higher than a threshold, however, both methods perform comparably. Our dual-guided approach bridges reduced-order model-based sensor position optimization with modern generative modeling, providing accurate, physics-consistent reconstruction from sparse data for scientific machine-learning problems.

Accurate prediction of flow field in centrifugal compressors using the present turbulence model remains challenging due to intense nonlinearity and strong adverse pressure gradients inherent in the flow. An ensemble-Kalman-filter-based data assimilation (DA) framework is developed to optimize the RNG (renormalization group) k–ε turbulence models for enhancing prediction accuracy. The DA-optimized RNG k–ε model eliminates anomalous static pressure peaks within high-pressure gradient regions, leading to a more physical pressure distribution. Compared with the original results, the maximum absolute percentage error of static pressure is reduced from 19.98% to 7.24%, and the mean absolute percentage error from 5.12% to 2.77%, achieving a significant improvement in prediction accuracy. Furthermore, constrained by compressor geometry and experimental costs, reconstruction from sparse measurements is critically needed. A sparse-sensing shallow neural network (SS-SNN) is therefore proposed to reconstruct the static pressure field using limited sensing data. The SS-SNN results achieve the maximum absolute percentage error of 7.80% and the mean absolute percentage error of 2.18%. These results indicate that DA-corrected turbulence model and SS-SNN reconstruction offer can rapidly and accurately determine the static pressure distribution in centrifugal compressors.

Abstract We investigate the ability of two data assimilation (DA) methods – ensemble Kalman filter (EnKF) and variational smoother (4D-Var) – to reconstruct the Kuramoto–Sivashinsky and complex Ginzburg–Landau systems from sparse observations. While turbulent systems can reconstruct small scales below a critical resolution by substituting larger scales into governing equations, we examine whether DA methods can accurately recover the full field with observations sparser than this threshold, and consider which method performs better. Our findings show that both methods can accurately reconstruct the full field even with observations much sparser than the substitution-based critical resolution. However, likelihood of successful reconstruction within a fixed assimilation time decreases as sparsity increases. The EnKF method needs smaller assimilation time and lower temporal sampling rate than 4D-Var, but needs ad hoc stabilisation (e.g. inflation) and higher memory for ensemble storage. We validate these results by applying EnKF to turbulent two-dimensional Kolmogorov flows at Reynolds numbers from 200 to 2000, and forcing at wavenumbers 4 and 5. For these flows, we achieve full field reconstruction from observations as sparse as $4\times 4$ , outperforming existing 4D-Var and machine learning results where denser observations are required for reconstruction. These findings highlight the strengths and trade-offs of DA methods, offer guidance for reconstructing turbulent flows, and establish benchmarks for evaluating alternative methods.

The internal flow of nozzles governs the performance and stability of aerospace and high-speed systems, making accurate and efficient flow prediction essential. However, traditional computational fluid dynamics (CFD) approaches are still limited by prohibitive computational costs and inflexibility under variable geometric configurations. To address these challenges, this study proposes a dual-end convolutional squeeze-and-excitation (DE-ConvSE) adapter U-Net for rapid nozzle flow prediction, in which a U-Net backbone is augmented with DE-ConvSE adapters. The network integrates geometric topology encoding and boundary condition mapping to accurately reconstruct velocity, pressure, and temperature distributions. To enable efficient adaptation to previously unseen nozzle geometries, an adapter-based incremental learning strategy is introduced, where the pretrained backbone is frozen and only a small set of adapter parameters is updated. This strategy substantially improves the generalization capability of the model while maintaining high computational efficiency and mitigating catastrophic forgetting. Validation on a convergent–divergent nozzle benchmark demonstrates that the proposed approach achieves high reconstruction accuracy, with maximum symmetric relative errors below 9.14% and mean absolute errors below 5.21%. Compared with CFD solvers, the proposed method is mesh-free and provides orders-of-magnitude acceleration, highlighting its potential for real-time flow prediction and design optimization in advanced fluid dynamic systems.

In practical computational fluid dynamics simulations around industrial products with complex surface shapes, the robustness of preprocessing to “dirty” geometry is an important issue. The dirty STL (Standard Triangle Language) data contains errors such as gaps be-tween facets, overlapping of facets, and flipping of normal vectors. These errors in the STL data are difficult to avoid in 3D modeling of complex geometry. Using a Cartesian grid is advantageous to the boundary-fitted grid in the aspect of preprocessing for dirty STL files. In this study, a robust and automatic generation method of a signed distance function for the preprocessing of Cartesian grid solvers is proposed. To ensure robustness to the complex and dirty STL data, the proposed method uses information of all STL facets to determine each grid point. The proposed preprocessing method is verified by numerical simulation of the flow around the NASA common research model.

The super-resolution reconstruction based on sparse measurement information is an important issue in the field of experimental fluid dynamics, and using a small amount of information for high-precision turbulent flow field reconstruction has high research value. Traditionally, interpolation and reconstruction methods are used to reconstruct the flow field. In recent years, artificial intelligence technologies represented by deep neural networks have been increasingly applied to flow field reconstruction problems due to their advantages in handling high-dimensional and nonlinear function fitting problems. However, these methods do not fully consider the physical mechanisms followed by flow field evolution and require a large amount of high-precision data, which may result in high experimental costs. Physical Information Neural Network (PINN) is an unsupervised learning method that utilizes neural networks to fit solutions to partial differential equations of flow field evolution, thereby overcoming the strict requirements of traditional methods on data attributes. However, due to the complexity of the Navier-Stokes equations, the accuracy of traditional PINN methods cannot meet the modeling requirements. This article proposes a PINN based method that effectively combines test data and physical information, and applies this method to the problem of super-resolution turbulent flow field reconstruction. Using high-precision measurement data of turbulence in a square channel as the experimental dataset, the method was used to reconstruct a high-resolution velocity field based on downsampling information of turbulent flow field, and compared with traditional bicubic interpolation methods and flow field reconstruction methods based on Convolutional Neural Networks(CNN). The experimental results show that the PINN method proposed in this paper can quickly complete super-resolution flow field reconstruction using sparse flow field measurement information, and is insensitive to noise in the data. It has significant advantages in accuracy and robustness when the sampling ratio is large.

No abstract available

High-frequency flow field super-resolution remains challenging in computational fluid dynamics due to spectral bias in physics-informed neural networks, which limits the accurate reconstruction of fine-scale structures. This work introduces a hierarchical adaptive Fourier feature network for physics-informed super-resolution (HAFFN–PISR) framework that systematically addresses spectral bias through innovative frequency-scale decomposition. The framework employs a three-tier hierarchical encoding strategy that decomposes spatial–temporal features across distinct frequency scales, including low-frequency fundamental structures, characteristic frequencies aligned with external forcing, and high-frequency fine-scale features via learnable adaptive Fourier basis functions. Additionally, a prior knowledge-based hierarchical sampling algorithm leverages governing equations, external forcing frequencies, and Nyquist sampling theory for frequency parameter initialization. Neural tangent kernel theoretical analysis reveals that HAFFN–PISR achieves superior eigenvalue distributions, enabling accelerated convergence for high-frequency modes while maintaining consistency with the Navier–Stokes equations. Comprehensive validation demonstrates exceptional performance in reconstructing high-resolution flow fields from sparse observations with only 0.098% data utilization. Under varying forcing frequencies and Reynolds numbers, HAFFN–PISR achieves 50%–80% performance improvements over baseline methods while exhibiting robust noise resilience and superior computational efficiency compared to state-of-the-art approaches, establishing HAFFN–PISR as a powerful tool for high-fidelity flow field reconstruction. The source code for HAFFN–PISR will be made publicly available upon acceptance of this manuscript at https://github.com/xgxgnpu/HAFFN-PISR.

High-fidelity flow field data are essential for accurate evaluation of the aerodynamic performance of airfoils, as they directly affect the prediction accuracy of lift, drag, and flow separation characteristics. However, traditional methods are either computationally expensive, e.g., computational fluid dynamics (CFD) solvers, or struggle to achieve sufficient reconstruction fidelity and physical consistency, e.g., data-driven super-resolution approaches. In this paper, we propose a novel super-resolution reconstruction method based on the denoising diffusion probabilistic model (DDPM). It incorporates deformable convolutional layers and attention modules into the U-Net denoising network to enhance the extraction of airfoil features. Meanwhile, residual networks are also integrated to improve data reconstruction performance. To embed physical information, we introduce pointwise physics-informed loss functions based on the continuity and momentum residuals to enforce physical consistency. We select various airfoils from the public University of Illinois at Urbana-Champaign (UIUC) airfoil database, generate computational grids using Gmsh, and obtain high-fidelity flow field data through OpenFOAM simulations for model training and validation. Experimental results demonstrate that our proposed method can accurately predict characteristics of the flow field data around the airfoils, with computation time being only 23% of that required by traditional computational fluid dynamics software. Moreover, our method outperforms other deep learning models in terms of generation accuracy and generalization ability. This indicates that the proposed method offers significant advantages in computational efficiency and generation accuracy, providing a new solution for airfoil flow field data reconstruction.

In vehicle shape design from an aerodynamic performance perspective, deep learning methods enable us to estimate the flow field in a short period. However, the estimated flow fields are generally coarse and of low resolution. Therefore, a super-resolution model is required to enrich them. In this study, we propose a novel super-resolution model to enrich the flow fields around the vehicle to a higher resolution. To deal with the complex flow fields of vehicles, we apply the residual-in-residual dense block (RRDB) as the basic network-building unit in the generator without batch normalization. We then apply the relativistic discriminator to provide better feedback regarding the lack of high-frequency components. In addition, we propose a distance-weighted loss to obtain better estimation in wake regions and regions near the vehicle surface. Physics-informed loss is used to help the model generate data that satisfies the physical governing equations. We also propose a new training strategy to improve learning effectiveness and avoid instability during training for our enrichment task. Experimental results demonstrate that the proposed method out-performs the previous study in vehicle flow field enrichment tasks by a significant margin.

Particle image velocimetry (PIV) techniques have a limited field of view of the flow field and can only capture high-resolution flow fields in localized areas. To obtain a larger measurement range, multiple cameras must be used to capture the flow field simultaneously and then stitch the parts together. However, this method can be expensive. We propose the local-global fusion convolutional neural network (LGF-CNN) for reconstructing large-field flow fields with high spatial resolution based only on two flow data types: local small-field high spatial resolution wake velocity fields and global large-field low spatial resolution velocity fields. The core of the model consists of convolutional neural network (CNN) architecture to learn the mapping relationship between the small field of view with high spatial resolution and the large field of view with low spatial resolution. Using the effectively trained LGF-CNN model, we demonstrate its ability to reconstruct high-resolution velocity fields around the circular cylinder. The LGF-CNN is rigorously validated on a number of representative datasets, including simulated data for Reynolds numbers of 200 and 500, as well as experimental data for a Reynolds number of 3.3 × 104 with the steady jet at the rear stagnation point of the cylinder. The results demonstrate the ability of LGF-CNN to generate accurate velocity fields with high spatial resolution, including reliable detection of high-frequency components. The proposed method could reduce the number of cameras required for large-field, high spatial resolution PIV measurements, thereby reducing experimental costs.

Flow field super-resolution (FFSR) aims at recovering high-resolution turbulent velocity fields from low-resolution flow fields. Existing FFSR methods mainly process the flow fields in natural image patterns, while the critical and distinct fluid visual properties are rarely considered. This negligence would cause the significant domain gap between fluid and natural images to severely hamper the accurate perception of turbulent flows, thereby undermining super-resolution performance in a wrong perception pattern. To tackle this dilemma, we rethink the FFSR task with the fluid visual properties, including the unique fluid imaging principle and morphological information, and propose the first fluid visual property-informed FFSR algorithm. Particularly, different from natural images that are constructed by independent red, green, and blue channels in the light field, flow fields build on the orthogonal streamwise, spanwise, and vertical (UVW) velocities in the fluid field. To empower the FFSR network with an awareness of the fluid imaging principle, we propose quaternion spatial modeling to model this orthogonal spatial relationship for improved FFSR. Moreover, due to viscosity and surface tension characteristics, fluids often exhibit a droplet-like morphology in flow fields. Inspired by this morphological property, we design the dynamic fluid convolution to effectively mine the morphological information to enhance FFSR. Extensive experiments on the newly acquired fluid field datasets demonstrate the state-of-the-art performance of our method.

The high-resolution (HR) spatio-temporal flow field plays a decisive role in describing the details of the flow field. In the acquisition of the HR flow field, traditional direct numerical simulation (DNS) and other methods face a seriously high computational burden. To address this deficiency, we propose a novel multi-scale temporal path UNet (MST-UNet) model to reconstruct temporal and spatial HR flow fields from low-resolution (LR) flow field data. Different from the previous super-resolution (SR) model, which only takes advantage of LR flow field data at instantaneous (SLR) or in a time-series (MTLR), MST-UNet introduces multi-scale information in both time and space. MST-UNet takes the LR data at the current frame and the predicted HR result at the previous moment as the model input to complete the spatial SR reconstruction. On this basis, a temporal model is introduced as the inbetweening model to obtain HR flow field data in space and time to complete spatio-temporal SR reconstruction. Finally, the proposed model is validated by the spatio-temporal SR task of the flow field around two-dimensional cylinders. Experimental results show that the outcome of the MST-UNet model in spatial SR tasks is much better than those of SLR and MTLR, which can greatly improve prediction accuracy. In addition, for the spatio-temporal SR task, the spatio-temporal HR flow field predicted by the MST-UNet model has higher accuracy either.

PURPOSE We present an unsupervised deep learning method to perform flow denoising and super-resolution without high resolution labels. We demonstrate the ability of this model to reconstruct 3D stenosis and aneurysm flows, with varying geometries, orientations, and boundary conditions. METHODS Ground truth data was generated using computational fluid dynamics, and then corrupted with multiplicative Gaussian noise. Autoencoders were used to compress the representations of the flow domain geometry and the (possibly noisy and low-resolution) flow field. These representations were used to condition a physics-informed neural network. A physics-based loss was implemented to train the model to recover lost information from the noisy input by transforming the flow to a solution of the Navier-Stokes equations. RESULTS Our experiments achieved mean squared errors in the true flow reconstruction of order 1.0e-4, and root mean squared residuals of order 1.0e-2 for the momentum and continuity equations. Our method yielded correlation coefficients of 0.971 for the pressure field and 0.82 for the wall shear stress magnitude field. CONCLUSION By performing point-wise predictions of the flow, the model was able to robustly denoise and super-resolve the field to 20x the input resolution.

The flow field obtained through numerical simulations often exhibits distortion after data processing steps, such as super-resolution (SR) reconstruction or flow field prediction. This study presents a flow field reconstruction method based on deep learning. The physics-informed convolutional neural network (PICNN) model, combining the convolutional neural network (CNN) and the physics-informed neural network (PINN), is specifically designed to address distortion issues present in data processing. The study shows that in the SR reconstruction of the square cavity flow field, the PICNN model increases the resolution of the velocity field by 16, 36, 64, and even 256 times with an error range significantly superior to traditional interpolation methods. However, in regions where the velocity changes are particularly abrupt, the super-resolution reconstruction performance of the PICNN model is suboptimal. At the same time, combined with the sparsity promoting dynamic mode decomposition (SPDMD) algorithm, the PICNN model significantly optimizes the flow field prediction of the SPDMD algorithm, even in the case of a small number of retained modes.

Blood flow observation is of high interest in cardiovascular disease diagnosis and assessment. For this purpose, 2D Phase-Contrast MRI is widely used in the clinical routine. 4D flow MRI sequences, which dynamically image the anatomic shape and velocity vectors within a region of interest, are promising but rarely used due to their low resolution and signal-to-noise ratio (SNR). Computational fluid dynamics (CFD) simulation is considered as a reference solution for resolution enhancement. However, its precision relies on image segmentation and a clinical expertise for the definition of the vessel borders. The main contribution of this paper is a Segmentation-Free Super-Resolution (SFSR) algorithm. Based on inverse problem methodology, SFSR relies on minimizing a compound criterion involving: a data fidelity term, a fluid mechanics term, and a spatial velocity smoothing term. The proposed algorithm is evaluated with respect to state-of-the-art solutions, in terms of quantification error and computation time, on a synthetic 3D dataset with several noise levels, resulting in a 59% RMSE improvement and factor 2 super-resolution with a noise standard deviation of 5% of the Venc. Finally, its performance is demonstrated, with a scale factor of 2 and 3, on a pulsed flow phantom dataset with more complex patterns. The application on in-vivo were achievable within the 10 min. computation time.

No abstract available

We suggest an algorithm for generation of a moving adaptive mesh for a system of moving bodies defined by the distance-like implicit function. The objective is to improve accuracy of the immersed boundary conditions for viscous flow solver NOISETTE Time-dependent control metric is constructed on top of the implicit function, and minimization of quasi-isometric mesh quality functional [2] is used to make the mesh follow the metric. We expect that mesh compression zone would travel in the computational domain and would precisely follow the boundary of bodies. For efficient parallel execution of NOISETTE solver it is required that general unstructured 2d or 3d mesh is deformed without connectivity changes, which makes mesh adaptation problems quite hard.

Computation modeling for large thermoplastic deformation of plastic solids is critical for industrial applications like non-invasive assessment of engineering components. While deep learning-based methods have emerged as promising alternatives to traditional numerical simulations, they often suffer from systematic errors caused by geometric mismatches between predicted and ground truth meshes. To overcome this limitation, we propose a novel boundary geometry-constrained neural framework that establishes direct point-wise mappings between spatial coordinates and full-field physical quantities within the deformed domain. The key contributions of this work are as follows: (1) a two-stage strategy that separates geometric prediction from physics-field resolution by constructing direct, point-wise mappings between coordinates and physical quantities, inherently avoiding errors from mesh misalignment; (2) a boundary-condition-aware encoding mechanism that ensures physical consistency under complex loading conditions; and (3) a fully mesh-free approach that operates on point clouds without structured discretization. Experimental results demonstrate that our method achieves a 36–98% improvement in prediction accuracy over deep learning baselines, offering a efficient alternative for high-fidelity simulation of large thermoplastic deformations.

We study neural-operator surrogates for steady incompressible Navier-Stokes flow on variable two-dimensional coronary-like domains. Two finite-element datasets are constructed: stenosed channels and bifurcating vessels, each with 1000 geometries. We compare a reference-domain MIONet based on diffeomorphic registration with geometry-aware point-cloud operators, GNOT and PCNO. PCNO gives the strongest in-distribution accuracy and best extrapolation to unseen inflow velocities, whereas MIONet is most stable under the tested OOD shape shift and fastest per forward pass. Currents-based geometric dissimilarity helps characterize error trends and supports reliability-aware surrogate assessment.

. A convolution neural network (CNN)-based approach for the construction of reduced order surrogate models for computational fluid dynamics (CFD) simulations is introduced; it is inspired by the approach of Guo, Li, and Iori [X. Guo, W. Li, and F. Iorio, Convolutional neural networks for steady flow approximation, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, New York, USA, 2016, ACM, pp. 481–490]. In particular, the neural networks are trained in order to predict images of the flow field in a channel with varying obstacle based on an image of the geometry of the channel. A classical CNN with bottleneck structure and a U-Net are compared while varying the input format, the number of decoder paths, as well as the loss function used to train the networks. This approach yields very low prediction errors, in particular, when using the U-Net architecture. Furthermore, the models are also able to generalize to unseen geometries of the same type. A transfer learning approach enables the model to be trained to a new type of geometries with very low training cost. Finally, based on this transfer learning approach, a sequential learning strategy is introduced, which significantly reduces the amount of necessary training data.

Modeling the turbulent flow within different draft tube configurations in a cost-effective way is essential for efficient turbine optimization and exploring the underlying flow mechanics. In this study, a convolution neural network (CNN) based surrogate model was proposed to predict local flow parameters within different inclined Francis turbine draft tubes. Three symbolic representations denoting the complex geometry and boundary conditions were set as the input, and pressure and velocity were the output. The adopted CNN framework consists of the U-Net architecture with a contracting path and four expansive paths. Six representative hyperparameters were considered to analyze their influence on the performance and generalization ability of the CNN model. The results show that the predicting accuracy of the CNN model with a U-Net network is 7.53% higher than the traditional CNN model, as skip connections improve image segmentation accuracy. The CNN model with a larger convolution kernel can more comprehensively capture the main features of the flow field. The model with three input variables improves prediction accuracy by 2.4% as more geometrical features correlate with the key flow patterns. For the four different image resolutions, the model with a resolution of 200 × 400 performs exceptionally well. In addition, appropriately increasing the number of convolutional layers or blocks can significantly improve the prediction accuracy of the CNN model. The proposed innovative surrogate model is useful for facilitating the optimization of hydraulic turbine components.

For flow-related design optimization problems, computational fluid dynamics (CFD) simulations are commonly used to predict the flow fields. However, the computational expenses of CFD simulations limit the opportunities for design exploration. Motivated by this tricky issue, a convolutional neural network (CNN) based on U-Net architecture with attention mechanism (AM) is proposed to efficiently learn flow representations from CFD results to shorten the compressor blade design cycle. The proposed model converts the provided shape information and flow conditions into grayscale images to directly predict the expected flow field, saving computational time. An extensive hyper-parameter search is performed to determine the optimal model. Qualitative and quantitative analysis of the results are studied to evaluate the accuracy for the calculation of Mach number distributions. In particular, two new attention mechanisms is developed to preserve the physical consistency of the complex flow field with shock wave. Mach number flow fields under different working conditions are predicted using the proposed model, and the prediction is well consistent with CFD results. Over three orders of magnitude of speedup is achieved at all batch sizes compared to traditional CFD methods, while maintaining low prediction errors.

Despite advancements in high-performance computing and numerical algorithms, Computational Fluid Dynamics (CFD) remains challenging for practical real-time applications, particularly in analysis and design tasks such as digital twin implementations. While traditional Reduced-Order Methods offer efficient and accurate predictions of entire flow fields, autoencoder Convolutional Neural Networks (CNNs) have demonstrated success in reconstructing CFD solutions due to their exceptional local feature extraction capabilities and scalability. However, their applicability is constrained to simple geometries because of the reliance on Cartesian or pixel-like grid structures. In this study, we propose a novel Point-based U-Net (PointUNet) framework incorporating Local Point Encoding (LPE) as a mesh-independent autoencoder model. The key functionality of LPE lies in its ability to transform point cloud data into a standard input array for conventional CNNs using a Virtual Reference Grid. This approach avoids data loss typically associated with interpolation or extrapolation, enabling greater flexibility in mesh generation and complex geometry handling. Verification was conducted using airfoil flows at transonic speeds and cylinder flows at low Reynolds numbers with various cross-sectional shapes, achieving minimal verification errors. The results were compared directly with other point cloud methods, demonstrating superior accuracy and efficiency in predicting highly nonlinear flows involving separation and shock waves, showing better agreement with full-order CFD solutions.

We evaluate the capability of convolutional neural networks (CNNs) to predict a velocity field as it relates to fluid flow around various arrangements of obstacles within a two-dimensional, rectangular channel. We base our network architecture on a gated residual U-Net template and train it on velocity fields generated from computational fluid dynamics (CFD) simulations. We then assess the extent to which our model can accurately and efficiently predict steady flows in terms of velocity fields associated with inlet speeds and obstacle configurations not included in our training set. Real-world applications often require fluid-flow predictions in larger and more complex domains that contain more obstacles than used in model training. To address this problem, we propose a method that decomposes a domain into subdomains for which our model can individually and accurately predict the fluid flow, after which we apply smoothness and continuity constraints to reconstruct velocity fields across the whole of the original domain. This piecewise, semicontinuous approach is computationally more efficient than the alternative, which involves generation of CFD datasets required to retrain the model on larger and more spatially complex domains. We introduce a local orientational vector field entropy (LOVE) metric, which quantifies a decorrelation scale for velocity fields in geometric domains with one or more obstacles, and use it to devise a strategy for decomposing complex domains into weakly interacting subsets suitable for application of our modeling approach. We end with an assessment of error propagation across modeled domains of increasing size.

We propose a steady-state aerodynamic data-driven method to predict the incompressible flow around airfoils of NACA (National Advisory Committee for Aeronautics) 0012-series. Using the Signed Distance Function (SDF) to parameterize the geometric and flow condition setups, the prediction core of the method is constructed essentially by a consecutive framework of a convolutional neural network (CNN) and a deconvolutional neural network (DCNN). Impact of training parameters on the behavior of the proposed CNN-DCNN model is studied, so that appropriate learning rate, mini-batch size, and random deactivation rate are specified. Tested by “unseen” airfoil geometries and far-field velocities, it is found that the prediction process is three orders of magnitudes faster than a corresponding Computational Fluid Dynamics (CFD) simulation, while relative errors are maintained lower than 1% on most of the sample points. The proposed model manages to capture the essential dynamics of the flow field, as its predictions correspond reasonably with the reconstructed field by proper orthogonal decomposition (POD). The performance and accuracy of the proposed model indicate that the deep learning-based approach has great potential as a robust predictive tool for aerodynamic design and optimization.

An intelligent modelling method driven by flow field images for predicting steady and unsteady flow filed around aerofoils has been developed. Signed Distance Field (SDF) images achieve dimensionality enhancement of aerofoil geometric information, and ‘synthesised images’ achieve dimensionality enhancement of the angle of attack of the aerofoil and Mach number. An intelligent aerodynamic model for steady flow field of aerofoils is constructed based on the U-Net neural network architecture, and further incorporating a long short-term memory (LSTM) module to construct a U-Net-LSTM neural network architecture to extract the temporal features. Typical NACA aerofoils results show that, the prediction error for steady flow is less than 1.98%, while the prediction error for unsteady flow is less than 2.56%. Additionally, the model demonstrates good generalization capability, with a generalization error for steady flow less than 2.45% and a generalization error for unsteady flow less than 3.34%. This research provides a new method for intelligent aerodynamic modelling based on physical representations. Compared to existing methods, this method avoids the need for extracting aerofoil geometry information and eliminates the necessity of predicting the flow field point by point, making it more concise and efficient. Highlights 1. An aerodynamic model was constructed using U-Net to rapidly predict the steady flow field around airfoils.2. A Long Short-Term Memory (LSTM) module was incorporated to capture temporal information, enabling the rapid prediction of the unsteady flow field around airfoils. To address the problem of ‘dimension loss’ in the modelling datasets, effective data dimensionality enhancement was achieved using SDF images and ‘synthesized images’.

We propose a super-resolution framework for accelerating three-dimensional free-surface flow simulations, termed the free-surface flow super-resolution convolutional neural network (FSFlow-SRCNN), by integrating a 3D U-Net-based CNN with an advanced free-surface flow solver. The free-surface flow solver is based on the coupled level-set and volume-of-fluid (CLSVOF) method and the full-variable Cartesian grid (FVCG) method. Numerical experiments show substantial acceleration while maintaining close agreement with high-resolution reference solutions and satisfying mass conservation to a comparatively high standard; a speed-up factor of approximately 139 is achieved in one computational environment. FSFlow-SRCNN also demonstrates generalization to nearby unseen conditions, remaining stable and maintaining robust super-resolution performance. Among the three tested loss functions, Loss3 combining an L1 term with divergence and gradient terms provides the best overall balance: the divergence term improves mass conservation, whereas the gradient term enhances the reconstruction of the density field across gas–liquid density discontinuities. Finally, training on mixed one- and two-droplet datasets yields reliable performance in both cases, indicating the potential of FSFlow-SRCNN as a versatile model for a range of free-surface flows.

Nasal air resistance is a key indicator of respiratory health and is essential for understanding nasal physiology and functions. Accurately measuring this quantity, however, remains challenging both experimentally and computationally. Data-driven methods, particularly deep learning models, offer a promising avenue for the rapid and reliable prediction of flow features, but they require large and diverse training datasets to generalize effectively to unseen cases. This study has two primary objectives: first, to develop machine learning models for respiratory flow simulations capable of accurately predicting the air resistance; and second, to introduce a data augmentation strategy for generating large virtual populations from a limited number of real patient geometries. Due to the complex and unstructured nature of nasal cavity geometries, training samples are represented as graphs, allowing direct use of computational fluid dynamic simulations as model inputs. The model is implemented as a distributed graph convolutional neural network to efficiently handle large-scale datasets, demonstrated here with 8000 graphs and scalable to even larger populations. Results show that the model achieves an R2 score of 0.999 in predicting the pressure drop, and that the prediction error on unseen cases decreases substantially as the virtual population is expanded from a limited set of real geometries.

As the geometries of high-rise buildings become increasingly complex, accurately and efficiently predicting their wind field characteristics is essential for wind-resistant structural design. Traditional computational fluid dynamics (CFD) methods, although highly accurate, incur substantial computational costs and are thus impractical for rapid evaluation. This study proposes a CFD data-driven deep learning model for fast and accurate prediction of flow fields around common high-rise buildings under various wind directions. Based on the exposure category B wind field defined in the Load Standard Code of China, a flow field dataset was constructed using steady Reynolds-averaged Navier–Stokes for five building shapes under different wind directions. A residual U-shaped convolutional neural network with Fourier skip connection (FSC-ResUNet) was developed, and ablation experiments were conducted against U-shaped convolutional neural network (UNet), residual U-shaped convolutional neural network (ResUNet), and U-shaped convolutional neural network with Fourier skip connection (FSC-UNet). Results show that the predictions of FSC-ResUNet are highly consistent with CFD simulations, with coefficients of determination (R2) for velocity and pressure fields close to 0.99, minimal root mean squared error, and overall weighted absolute percentage errors generally below 10% and below 5% for most cases, significantly outperforming the comparison models. Flow field visualization and streamline characteristics analysis further confirm that the model successfully captures key flow features across different cross-sectional shapes, such as separated shear layers and recirculating vortices. Compared with the hour-level computation time of CFD, the trained model predicts new cases within seconds, achieving several orders of magnitude improvement in efficiency. This study demonstrates the significant potential of deep learning in wind engineering flow field prediction and provides an efficient and reliable data-driven tool for fast wind load evaluation and aerodynamic shape optimization of buildings.

Fluid dynamics is governed by partial differential equations (PDEs) which are solved numerically. The limitations of traditional methods in data assimilation hinder their effective engagement with experiments. Physics-informed neural network (PINN) has emerged as a hybrid data-physics-driven model for convective problems. However, the approach suffers from low accuracy and poor efficiency due to the way of incorporating PDEs. In this work, a novel convolutional neural network framework integrating finite volume method (FVM) is developed to address the challenge. The interface variables of the grid are predicted by the neural network for the first time, rather than a complex procedure in FVM. The physical law is then learned by minimizing the residual of the discretized conservative form of PDEs. A comparison between this model and the original PINN regarding prediction accuracy demonstrates the superiority of embedding PDEs through FVM. The effects of sampling strategies and quantities are studied. The result confirms the model’s capability to utilize sparsely available internal measurement data. Furthermore, the model performs well even in scenarios where partial initial and boundary conditions are absent.

We develop a convolutional neural network model to predict multi-resolution steady flow data. Based on the image-to-image translation model pix2pixHD, our model can predict high resolution flow fields from the set of patched signed distance functions. By patching the high resolution data, our model uses roughly the one third of memory used by pix2pixHD. The accuracy of our model is almost the same as the U-Net model using the unpatched high resolution data.

Flow field prediction of variable geometry aims for the precise prediction of varying flow fields with complex geometric shapes and their variations. By combining a novel deep convolutional neural network (CNN) model with a Computational Fluid Dynamics (CFD) method, the solution of the Navier-Stokes equation can be effectively approximated with a significant reduction in the corresponding computation time cost. This paper proposes an approach named AM UNet-3 for predicting the flow field of variable geometry, which can directly predict the expected flow field of new shapes after the training of deep CNN through extracting and identifying dual features of flow field regions: geometric information features and multi-class channel features. The AM UNet-3 is a convolutional neural network model that combines a U-Net variant structure with three independent decoders and attention mechanisms. The framework of U-Net can learn the complete numerical solutions of the N-S equations to velocity and pressure fields from ground-truth data. The three independent decoders are used to output the pressure field and the velocity components in the x-direction and y-direction respectively. The embedded channel attention and spatial attention mechanisms can enhance the ability of AM UNet-3 to extract fast-changing boundary layer characteristics. The performances of AM UNet-3, U-Net and UNet-3 are tested and compared on a dataset consisting of flow field data with 981 different geometric shapes. Experiment results show that AM UNet-3 gets the fastest inference and the lowest prediction error in the three methods. Compared with the standard CFD method, AM UNet-3 is able to speed up the computation more than three orders of magnitude.

Thermal convection is normally calculated by numerical solvers such as Nektar++. However, due to their high memory cost and time cost, their limitations promote the research to solve the complex fluid phenomena with alternative methods. In this research, the U- Net CNN was successfully constructed for learning the thermal convection problem and studying velocities, temperature, and pressure in a closed geometry with a periodic bottom heating source. Tests were conducted for its generalization for different bottom heating patterns and its performance regarding the time length as well as Raleigh’s number. The optimized model proved its high accuracy in prediction and was able to achieve a lower MSE and lower time cost compared with the result generated by Nektar++. Importantly, the model showed a stronger capability in recognizing the temperature distribution. The prediction quality of the model was decreased if the time skip length as well as the time length increased. Huge memories also occurred due to it pursuing high precision and studying four variables together.

No abstract available

We present a data-driven or non-intrusive reduced-order model (NIROM) which is capable of making predictions for a significantly larger domain than the one used to generate the snapshots or training data. This development relies on the combination of a novel way of sampling the training data (which frees the NIROM from its dependency on the original problem domain) and a domain decomposition approach (which partitions unseen geometries in a manner consistent with the sub-sampling approach). The method extends current capabilities of reduced-order models to generalise, i.e., to make predictions for unseen scenarios. The method is applied to a 2D test case which simulates the chaotic time-dependent flow of air past buildings at a moderate Reynolds number using a computational fluid dynamics (CFD) code. The procedure for 3D problems is similar, however, a 2D test case is considered sufficient here, as a proof-of-concept. The reduced-order model consists of a sampling technique to obtain the snapshots; a convolutional autoencoder for dimensionality reduction; an adversarial network for prediction; all set within a domain decomposition framework. The autoencoder is chosen for dimensionality reduction as it has been demonstrated in the literature that these networks can compress information more efficiently than traditional (linear) approaches based on singular value decomposition. In order to keep the predictions realistic, properties of adversarial networks are exploited. To demonstrate its ability to generalise, once trained, the method is applied to a larger domain which has a different arrangement of buildings. Statistical properties of the flows from the reduced-order model are compared with those from the CFD model in order to establish how realistic the predictions are.

Fluid dynamics computations for tube-like geometries are crucial in biomedical evaluations of vascular and airways fluid dynamics. Physics-Informed Neural Networks (PINNs) have emerged as a promising alternative to traditional computational fluid dynamics (CFD) methods. However, vanilla PINNs often demand longer training times than conventional CFD methods for each specific flow scenario, limiting their widespread use. To address this, multi-case PINN approach has been proposed, where varied geometry cases are parameterized and pre-trained on the PINN. This allows for quick generation of flow results in unseen geometries. In this study, we compare three network architectures to optimize the multi-case PINN through experiments on a series of idealized 2D stenotic tube flows. The evaluated architectures include the ‘Mixed Network’, treating case parameters as additional dimensions in the vanilla PINN architecture; the “Hypernetwork”, incorporating case parameters into a side network that computes weights in the main PINN network; and the “Modes” network, where case parameters input into a side network contribute to the final output via an inner product, similar to DeepONet. Results confirm the viability of the multi-case parametric PINN approach, with the Modes network exhibiting superior performance in terms of accuracy, convergence efficiency, and computational speed. To further enhance the multi-case PINN, we explored two strategies. First, incorporating coordinate parameters relevant to tube geometry, such as distance to wall and centerline distance, as inputs to PINN, significantly enhanced accuracy and reduced computational burden. Second, the addition of extra loss terms, enforcing zero derivatives of existing physics constraints in the PINN (similar to gPINN), improved the performance of the Mixed Network and Hypernetwork, but not that of the Modes network. In conclusion, our work identified strategies crucial for future scaling up to 3D, wider geometry ranges, and additional flow conditions, ultimately aiming towards clinical utility.

In recent years, machine learning techniques have emerged as pivotal tools across scientific and engineering disciplines. One notable application is in computational fluid dynamics (CFD), where there is a growing demand for cost-effective alternatives to traditional, resource-intensive simulations. This study explores a surrogate modeling approach known as Design Variable Coded Multilayer Perceptron (DV-MLP), aimed at predicting velocity on unseen geometries using CFD simulation data. The DV-MLP model integrates spatial and design variables directly, offering a mesh-independent, scalable solution for rapid flow prediction across various geometric configurations. This method seeks to replace CFD simulations for geometries resembling those used in training, thereby accelerating the design process. The dataset comprises CFD results from four distinct bump heights. The objective is to train the MLP using data from three bump heights and assess its predictive performance on a middle bump height. The results indicate that the MLP accurately predicts flow features for the middle bump height. This demonstrates its ability to provide precise predictions of near-wall flow behavior for similar geometries, making it a valuable tool for industries requiring swift and reliable CFD predictions for design refinement and optimization. The DV-MLP model is designed primarily for interpolation between geometries and flow conditions represented in the training data, providing accurate predictions for cases that fall within the range of the sampled design space.

In the field of data-driven turbulence modeling, the consistency of the a posteriori results and generalizability are the most critical aspects of new models. In this study, we combine a multi-case surrogate optimization technique with a progressive augmentation approach to enhance the performance of the popular k−ω shear stress transport (SST) turbulence model in the prediction of flow separation. We introduce a separation factor into the transport equation of a turbulent specific dissipation rate (ω) to correct the underestimation of the turbulent viscosity by the k−ω SST model in the case of flow separation for two-dimensional cases. The new model is optimized based on their performance on the training cases including periodic hills and curved backward-facing step flow. Simulation of the channel flow is likewise included in the optimization process to guarantee that the original performance of k−ω SST is preserved in the absence of separation. The new model is verified on multiple unseen cases with different Reynolds numbers and geometries. Results show a significant improvement in the prediction of the recirculation zone, velocity components, and distribution of the friction coefficient in both training and testing cases, where flow separation is expected. The performance of the new models on the test case with no separation shows that they preserve the successful performance of k−ω SST when flow separation is not expected.

The design of various biomedical, electronics cooling, and microfluidic devices relies on geometry-specific models and empirical correlations for flow and heat transfer through microscale pin fin geometries. Machine learning techniques are being used across many branches of science to develop more generalized surrogate models that can predict such transport processes. To collapse the simulation of flow and thermal properties across many different pin fin surfaces into a single predictive tool, the present study develops machine-learning-based surrogate models for the friction factor and Nusselt number (for constant wall temperature conditions) for fully developed low Reynolds number flow across pin fin geometries of differing cross-section shape (circular, square, triangular) in aligned or staggered arrangements, oriented at any angle to the incoming flow, and for a range of transverse and longitudinal pitches, with water as the working fluid. The model training data are generated using an automated workflow that allows thousands of numerical simulations to be carried out on across different geometric and flow configurations. A total of ~14,800 distinct simulation cases, for both friction factor and Nusselt number, are generated whilst varying the Reynolds number and aforementioned geometric parameters to train and test the machine learning models. The machine learning model architecture takes inputs of both image and vector data, and then outputs a scalar friction factor or Nusselt number. The trained models yield a goodness of fit (R2) value of 0.98 on unseen data.

Rapid and accurate simulations of fluid dynamics around complicated geometric bodies are critical in a variety of engineering and scientific applications. While scientific machine learning (SciML) has shown considerable promise, most studies in this field are limited to simple geometries. This paper addresses this gap by benchmarking diverse SciML models, including neural operators and vision transformer-based foundation models, for fluid flow prediction over intricate geometries. We evaluate the impact of geometric representations—Signed Distance Fields (SDF) and binary masks—on model accuracy, scalability, and generalization using a high-fidelity dataset of steady-state flow over complex geometries. We introduce a unified scoring framework that integrates metrics for global accuracy, boundary layer fidelity, and physical consistency. Our findings reveal that newer foundation models significantly outperform neural operators, particularly in data-limited scenarios. In addition, binary mask representation enhances the performance of vision transformer models by up to 10%, while SDF representations improve neural operator performance by up to 7%. Despite these promises, all models struggle with out-of-distribution generalization, highlighting a critical challenge for future SciML applications. Our work paves the way for robust and scalable ML solutions for fluid dynamics across complex geometries. Ali Rabeh, Ethan Herron and colleagues benchmark diverse scientific machine learning models, including neural operators and vision transformer-based foundation models, for fluid flow prediction over intricate geometries using a unified scoring framework. They show that geometry representations can have a measurable impact on the accuracy and generalization of different models.

With the rapid development of artificial intelligence technology, the physics-informed neural network (PINN) has gradually emerged as an effective and potential method for solving N-S equations. The treatment of constraints is vital to the PINN prediction accuracy. Compared to soft constraints, hard constraints are advantageous for the avoidance of difficulties in guaranteeing definite conditions and determining penalty coefficients. However, the principles on the formulation of hard constraints of PINN currently remain to be formed, which hinders the application of PINN in engineering fields. In this study, hard-constraint-based PINN models are constructed for Couette flow, plate shear flow and stenotic/aneurysmal flow with curved geometries. Particular efforts have been devoted to assessing the impact of the model parameters of hard constraints, i.e., degree and scaling factor, on the prediction accuracy of PINN at different Reynolds numbers. The results show that the degree is the most important factor that influences the prediction accuracy, followed by the scaling factor. As for the N-S equations, the degree of hard constraints should be at least two, while the scaling factor is recommended to be maintained around 1.0. The outcomes of the present work are of reference value for the development of PINN methods in fluid mechanics.

Kolmogorov–Arnold Networks (KANs) have emerged as a promising alternative to traditional Multilayer Perceptrons (MLPs) in deep learning. KANs have already been integrated into various architectures, such as convolutional neural networks, graph neural networks, and transformers, and their potential has been assessed for predicting physical quantities. However, the combination of KANs with point-cloud-based neural networks (e.g., PointNet) for computational physics has not yet been explored. To address this, we present Kolmogorov–Arnold PointNet (KA-PointNet) as a novel supervised deep learning framework for the prediction of incompressible steady-state fluid flow fields in irregular domains, where the predicted fields are a function of the geometry of the domains. In KA-PointNet, we implement shared KANs in the segmentation branch of the PointNet architecture. We utilize Jacobi polynomials to construct shared KANs. As a benchmark test case, we consider incompressible laminar steady-state flow over a cylinder, where the geometry of its cross-section varies over the data set. We investigate the performance of Jacobi polynomials with different degrees as well as special cases of Jacobi polynomials such as Legendre polynomials, Chebyshev polynomials of the first and second kinds, and Gegenbauer polynomials, in terms of the computational cost of training and accuracy of prediction of the test set. Furthermore, we examine the robustness of KA-PointNet in the presence of noisy training data and missing points in the point clouds of the test set. Additionally, we compare the performance of PointNet with shared KANs (i.e., KA-PointNet) and PointNet with shared MLPs. It is observed that when the number of trainable parameters is approximately equal, PointNet with shared KANs (i.e., KA-PointNet) outperforms PointNet with shared MLPs. Moreover, KA-PointNet predicts the pressure and velocity distributions along the surface of cylinders more accurately, resulting in more precise computations of lift and drag.

In recent years, deep learning methods have been widely applied to airfoil flow field prediction in Computational Fluid Dynamics (CFD) simulations. Vision Transformer (ViT), known for its global modeling capabilities, has been introduced to geometric encoding tasks. However, existing ViT-based models still suffer from limited accuracy when handling complex airfoil geometries. In real-world flow fields, regions such as the leading edge, trailing edge, and near-wall zones often exhibit sharp flow variations and steep pressure gradients, playing a critical role in aerodynamic performance. Traditional CFD methods address this using mesh refinement, wall functions, or boundary layer modeling to enhance local prediction accuracy. Inspired by these strategies, we propose a novel model, CTF-DANet, for airfoil flow field prediction. Our model comprises a geometric encoder (CTF-GENet) and a Multilayer Perceptron (MLP) based prediction head, supporting end-to-end learning. CTF-GENet introduces a convolutional stem to enhance local feature extraction and employs a key-region guidance mechanism to strengthen attention to critical geometric structures using selected patch tokens. Moreover, a multi-scale fusion module aggregates intermediate-layer Transformer features to jointly encode local and global semantics. The encoded geometric representation is combined with other physical parameters and passed into the prediction head to estimate multiple flow variables. Experimental results on diverse airfoils and flow conditions show that CTF-DANet significantly outperforms baseline models in both prediction accuracy and generalization ability, particularly in critical regions.

Airfoil flow field data acquisition is pivotal to the study of aerodynamics, traditionally relying on time-consuming computational fluid dynamics simulations or expensive wind tunnel tests. Herein, we introduce a new methodology leveraging Transformer Neural Network (TNN), which differs from conventional methodologies by employing self-attention mechanisms, to effectively predict these critical flow field data using historical data. A comprehensive set of experiments demonstrates the TNN model’s exceptional predictive accuracy, achieving over 95% across various airfoils under various operating conditions. Beyond accuracy and efficiency, we introduce an attention principle in our TNN model enhancing its interpretability. By aligning the TNN model’s attention distribution with the aerodynamic principles of airfoils, we illustrate how it utilises these geometric attributes in its predictions, thereby offering theoretical backing to its predictive outcomes. Our TNN model’s commendable accuracy, efficiency and interpretability illuminate the pathway for continued exploration in the fusion of deep learning with computational fluid dynamics.

Accurate assessment of intracranial aneurysm rupture risk, particularly in Middle Cerebral Artery (MCA) aneurysms, relies on a detailed understanding of patient-specific hemodynamic behavior. In this study, we present an integrated framework that combines Computational Fluid Dynamics (CFD) with Proper Orthogonal Decomposition (POD) and machine learning (ML) to efficiently model pulsatile blood flow using a Casson non-Newtonian fluid model, without incorporating fluid-structure interaction (FSI). Patient-specific vascular geometries were reconstructed from DICOM imaging data and simulated using ANSYS Fluent to capture key hemodynamic factors, including velocity components, pressure, wall shear stress (WSS), and oscillatory shear index (OSI). POD was applied to reduce the dimensionality of the CFD data while retaining the dominant energetic flow structures. Results showed that fewer than 10 POD modes were sufficient to capture over 99% of the energy for pressure and WSS, while OSI required significantly more modes due to its inherent complexity. Machine learning models were trained on the reduced-order features to predict hemodynamic fields across time snapshots. The hybrid POD-ML approach yielded reasonable predictions for pressure and WSS in both training and test sets, while OSI prediction accuracy decreased in the test region, indicating the need for more advanced modeling strategies. The proposed method significantly reduces computational cost while preserving critical hemodynamic information, making it well-suited for real-time or near-real-time clinical decision support. This work demonstrates the potential of combining data-driven techniques with CFD for efficient, non-invasive risk assessment and treatment planning in cerebral aneurysm management.

The use of sonic nozzles as a high-precision standard for gas flow measurements in various engineering and metrological applications is well-established. Due to tolerances during manufacturing, the local curvature radius near the nozzle throat often deviates from the prescribed specifications. Consequently, the resulting non-ideal nozzle shapes can have a considerable influence on the flow rate. In this work, a computational fluid dynamics-based surrogate model is developed for flow prediction in non-ideally shaped sonic nozzles. A parameterisation to represent realistic nozzle shapes is proposed, wherein the curvature radius near the nozzle throat can be varied independently of that at the nozzle entry. A comprehensive numerical study of 121 parameterised non-ideal nozzle geometries is conducted in a wide Reynolds number range, providing the input data for the surrogate model. For the establishment of the surrogate model, five surrogate modelling methods are compared and assessed by means of a leave-one-out cross-validation. Flow results of the resulting surrogate model reveal a systematic impact on the flow behaviour depending on the parameter combination of throat curvature radius and axial transition length. Subsequently, a flow prediction approach using the developed surrogate model is demonstrated for twelve real test nozzles. For this purpose, an approximation procedure based on a weighted non-linear least squares method is conducted to obtain parameterised contours adequately representing the real nozzle shapes. The flow predictions of the surrogate model for the aforementioned parameterised contours are in much better agreement with experimental data for the respective test nozzle contour, exhibiting an average root mean square error of 0.17% over all test nozzles compared to the associated standard, with an average root mean square error of 0.35%. The presented flow prediction approach allows to assess the applicability and effectiveness of the developed surrogate model to each test nozzle individually, which is beneficial for further improvements.

Transformers have demonstrated effectiveness in solving partial differential equations (PDEs). However, extending them to solve PDEs on complex geometries remains a challenge. In this work, we propose SpiderSolver, a geometry-aware transformer that introduces spiderweb tokenization for handling complex domain geometry and irregularly discretized points. Our method partitions the irregular spatial domain into spiderweb-like patches, guided by the domain boundary geometry. SpiderSolver leverages a coarse-grained attention mechanism to capture global interactions across spiderweb tokens and a fine-grained attention mechanism to refine feature interactions between the domain boundary and its neighboring interior points. We evaluate SpiderSolver on PDEs with diverse domain geometries across seven datasets, including cars, airfoils, blood flow in the human thoracic aorta, as well as canonical cases governed by the Navier-Stokes, Darcy flow, elasticity, and plasticity equations. Experimental results demonstrate that SpiderSolver consistently achieves state-of-the-art performance across different datasets and metrics, with better generalization ability in the OOD setting. The code is available at https://github.com/Kai-Qi/SpiderSolver.

for flow-related design optimization problems, e.g., aircraft and automobile aerodynamic design, computational fluid dynamics (CFD) simulations are commonly used to predict flow fields and analyze performance. While important, CFD simulations are a resource-demanding and time-consuming iterative process. The expensive simulation overhead limits the opportunities for large design space exploration and prevents interactive design. In this paper, we propose FlowDNN, a novel deep neural network (DNN) to efficiently learn flow representations from CFD results. FlowDNN saves computational time by directly predicting the expected flow fields based on given flow conditions and geometry shapes. FlowDNN is the first DNN that incorporates the underlying physical conservation laws of fluid dynamics with a carefully designed attention mechanism for steady flow prediction. This approach not only improves the prediction accuracy, but also preserves the physical consistency of the predicted flow fields, which is essential for CFD. Various metrics are derived to evaluate FlowDNN with respect to the whole flow fields or regions of interest (RoIs) (e.g., boundary layers where flow quantities change rapidly). Experiments show that FlowDNN significantly outperforms alternative methods with faster inference and more accurate results. It speeds up a graphics processing unit (GPU) accelerated CFD solver by more than 14 000×, while keeping the prediction error under 5%.

Accurate reconstruction of flow fields from sparse sensor measurements remains challenging due to inherent limitations in sensor sparsity, irregular positioning, and computational inefficiency of existing methods. This study proposes a deep learning framework that integrates autoencoders with attention mechanisms through a two-stage training strategy. In the first stage, a convolutional autoencoder compresses high-resolution flow fields into low-dimensional latent representations. In the second stage, sensor data (including positions and measurements) are encoded via positional embeddings and processed by a self-attention module. These features are mapped to the latent space through a cross-attention regressor and subsequently decoded to reconstruct the complete flow field. The innovations of the proposed framework include the adaptability to arbitrary sensor numbers/positions and a significant improvement of computational efficiency. Case studies on the datasets of cylinder-wake flow and global sea surface temperature demonstrate that this framework achieves superior reconstruction accuracy, outperforming Voronoi convolutional neural network, and is comparable to the state-of-the-art Senseiver in terms of accuracy while reducing training time by 16 times. The framework also exhibits robustness against sensor failures and data noise, enabling real-time, high-precision flow reconstruction for engineering applications such as aerodynamic monitoring and climate modeling.

Real-time monitoring of the complex hydrodynamic states of underwater-launched vehicles during water-exit processes remains challenging due to highly nonlinear multiphase evolution, wake instability, and strong fluid–structure coupling. In this study, a physics-guided sparse sensing framework is developed to establish the relationship between limited wall-pressure measurements, global flow structures, and vehicle motion responses. The framework employs a proper orthogonal decomposition (POD) and Transformer-based multi-task learning strategy, where POD provides a physically interpretable latent representation of flow dynamics, while Transformer-based feature learning captures global correlations among modal features and sensor locations. By incorporating operating-condition information through feature modulation and attention-based fusion, the proposed framework simultaneously reconstructs pressure, velocity, and vorticity fields and predicts vehicle motion states. The results demonstrate that sparse wall-pressure measurements contain sufficient information to recover the dominant coherent structures and dynamic responses during the water-exit process. Under varying Froude number conditions, reconstruction errors are mainly concentrated in wake shear layers and vortex-dominated regions associated with highly unsteady flow structures, whereas the large-scale flow characteristics governing overall evolution remain accurately captured. Furthermore, sensor configuration analysis reveals that for the investigated vehicle geometry and flow conditions, a 12-sensor layout provides an effective balance between measurement sparsity and reconstruction accuracy, indicating the existence of an effective sensing threshold for capturing essential flow information. These findings provide new insights into the observability of complex water-exit flows, the encoding mechanism between sparse wall measurements and global flow dynamics, and the coupling between hydrodynamic evolution and vehicle motion responses.

Accurate reconstruction of velocity and pressure fields in fluid flows is essential for monitoring and control in industrial systems. Physics-Informed Neural Networks (PINNs) have emerged as powerful tools to infer hidden flow quantities by embedding physical laws directly into the deep learning process. However, reconstructing such fields remains particularly challenging due to their indirect nature and high sensitivity to sensor placement. This work introduces a gradient-weighted probabilistic sensor placement strategy within a sparse and decoupled supervision framework for PINNs. Instead of uniformly or deterministically selecting measurement points, the method assigns sampling probabilities based on the local gradient magnitudes of each target variable. This enables strategic allocation of sensors in physically informative regions while maintaining variable-specific sparse supervision. The approach is tested on a two-dimensional turbulent T-junction case, where it significantly improves reconstruction accuracy, particularly for the pressure field. Compared to the baseline strategy used in [1], the $\mathcal{L}_2$ error for pressure is reduced from 0.278 to 0.142, for transverse velocity from 0.144 to 0.094, and for streamwise velocity from 0.084 to 0.046. These results represent reductions of approximately 45–50% in the most challenging variables, highlighting the effectiveness of gradient-informed sensor strategies in enhancing PINN-based flow reconstruction.

In the realm of experimental fluid mechanics, accurately reconstructing high-resolution flow fields is notably challenging due to often sparse and incomplete data across time and space domains. This is exacerbated by the limitations of current experimental tools and methods, which leave critical areas without measurable data. This research suggests a feasible solution to this problem by employing an inverse physics-informed neural network (PINN) to merge available sparse data with physical laws. The method's efficacy is demonstrated using flow around a cylinder as a case study, with three distinct training sets. One was the sparse velocity data from a domain, and the other two datasets were limited velocity data obtained from the domain boundaries and sensors around the cylinder wall. The coefficient of determination (R2) coefficient and mean squared error (RMSE) metrics, indicative of model performance, have been determined for the velocity components of all models. For the 28 sensors model, the R2 value stands at 0.996 with an associated RMSE of 0.0251 for the u component, while for the v component, the R2 value registers at 0.969, accompanied by an RMSE of 0.0169. The outcomes indicate that the method can successfully recreate the actual velocity field with considerable precision with more than 28 sensors around the cylinder, highlighting PINN's potential as an effective data assimilation technique for experimental fluid mechanics.

No abstract available

This work proposes a novel approach that combines an optimal sparse sensor placement algorithm with a Convolutional Neural Network (CNN) to accurately reconstruct computational fluid dynamics (CFD) simulations. Fluid mechanics is a field with an ever-increasing amount of high-quality data, and machine learning offers new opportunities to extract essential flow features and enable real-time predictions and control. The key challenge addressed herein is to determine the optimal placement of a minimal number of sensors to capture the complexity of fluid flows while enabling fast CNN-based reconstruction. The proposed methodology was applied to a lid-driven cavity flow dataset comprising 100 steady laminar simulations of the vertical velocity component. The sparse sensor placement algorithm was used to determine the optimal sensor positions, which were then used to transform the original CFD dataset to feed into the CNN. The results demonstrate the approach’s effectiveness, achieving an accuracy of 95.50% using only 1000 sensors out of a possible set of 16384 and a single Proper Orthogonal Decomposition (POD) mode. This work has the potential to significantly reduce the computational burden associated with CFD simulations within Digital Twins, enabling real-time predictions and control for various fluid flow applications.

Accurate reconstruction of aerodynamic forces from sparse wall pressure sensor inputs presents significant challenges for monitoring supersonic nozzles, where the flow is characterized by intense nonlinearity and transient flow patterns under varying nozzle pressure ratios (NPR). Limited by the geometric constraints of nozzle contours and measurement cost, reconstructing the pressure distribution through sparse sensors has become critically important. This study presents a data-driven wall pressure reconstruction model based on a small number of pressure sensors on the surface of a supersonic nozzle wall. In this context, we advance data-driven methodologies by rigorously comparing two sparse sensor paradigms: discrete empirical interpolation method with proper orthogonal decomposition (DEIM-POD) and compressed sensing-POD (CS-POD). Through wind-tunnel experiments on an axisymmetric expansion nozzle, we demonstrate DEIM-POD’s superior performance: achieving 30% lower reconstruction errors than the vectorized CS-POD (Vec-CS-POD) at equivalent sensor densities while maintaining NPR adaptability with fixed sensor arrays. Additionally, this method can resolve shock-induced intense pressure gradients well and operates in practical configurations requiring 60% fewer sensors than conventionally uniformly spaced sensor layouts. Computational efficiency analyses reveal that DEIM-POD’s greedy sampling executes faster than CS-POD’s randomized approach. To address localized sensor failures, we also introduce a geometrically constrained DEIM variant, namely modified DEIM-POD, which improves missing-data robustness and a comparable error level compared to the DEIM-POD model. These findings establish DEIM-POD as a preferred and potential framework, balancing measurement economy with reconstruction fidelity in supersonic flow applications.

Oblique detonation engines (ODEs) hold significant promise for hypersonic propulsion, where accurate sensing of the internal flow field is critical for achieving optimal performance. However, reconstructing the full flow field from sparse, incomplete, and irregular wall temperature measurements remains a formidable challenge. To address this, we propose the diffusion-guided residual coordinate-attention U-shaped network (DRC-UNet), which is designed to recover the global combustion chamber temperature field. DRC-UNet integrates three key components: a score-based diffusion model (SDM) for robust data completion, residual connections to enhance deep feature propagation, and coordinate attention for improved spatial localization. Using only 36 uniformly distributed wall-temperature sensors, DRC-UNet outperforms the baseline U-shaped network (U-Net) by improving the average structural similarity index measure (SSIM) from 0.95 to 0.99 and increasing the average peak signal-to-noise ratio (PSNR) from 28.31 to 41.81 dB, while better recovering shock fronts. Furthermore, under extreme data sparsity—where only 12 randomly distributed wall sensors (approximately 2% of the total measurement points) are available—DRC-UNet still achieves high-fidelity reconstructions, maintaining a mean absolute percentage error (MAPE) below 1.5%. In contrast, conventional end-to-end networks often struggle under such conditions, as they rely on fixed sensor layouts and require retraining to adapt to varying data sparsity. These results demonstrate that the proposed framework provides a robust and generalizable solution for intelligent sensing, capable of accurately reconstructing complex flow fields from incomplete or irregular data. This study provides a foundation for advanced diagnostics and control in oblique detonation engines by enabling temperature field reconstruction from sparse sensors.

No abstract available

This study evaluates the reconstruction of three-dimensional steady and unsteady flow fields in a centrifugal impeller using Proper Orthogonal Decomposition (POD) and sparse sensing techniques. This approach is driven by the fundamental challenges in turbomachinery, where spatial and instrumentation limitations greatly restrict the range of measurable physical quantities. Using POD, this study seeks to efficiently extract the dominant flow structures, enabling their accurate reconstruction even from sparse sensor data. For time-averaged flow, the first two POD modes account for 99.4% of the energy, capturing deviations from the design flow rate and localized variations in secondary flow. Optimal sensor placement, algorithms, i.e., Determinant-based Greedy (DG) and Bayesian Determinant-based Greedy (BDG) methods, achieved accurate flow field estimation with a minimal number of sensors. For time-varying flow, the first two modes capture the interaction between the blade and tongue and the third and fourth modes represent higher-order harmonics of the first and second modes. The BDG method outperformed others in selecting sensor locations, leading to smaller estimation errors in both velocity and pressure fields. Incorporating Bayesian estimation into the reconstruction process further enhances robustness, ensuring reliable results even in the presence of noise. This research provides a deeper understanding of flow dynamics, even with limited measurement data, and holds the potential for controlling and optimizing turbomachinery in practical applications such as real-time monitoring, anomaly detection,and feed-back control systems.

ABSTRACT This work presents a combined methodology – referred to as SSPINNs – that integrates optimal sensor placement via sparse sensing (SS) with physics-informed neural networks (PINNs) to reconstruct temperature and velocity fields in axisymmetric flames. The SS procedure determines optimal positions for a limited number of point measurements taken in soot-free regions (e.g. via thermocouples), complementing widely used non-intrusive optical techniques based on soot spectral emission, which only provide data within the sooting region. A database of temperature and soot volume fraction fields is generated from detailed flame simulations, including comprehensive chemical kinetics and particle production mechanisms. This dataset is further augmented with randomized variations in spatial distribution, magnitude, and geometry, within physically expected ranges. Using this augmented dataset, a singular value decomposition (SVD) is employed to reduce dimensionality, while a pivoted QR decomposition is used to identify optimal sensor locations. This approach ensures a feasible experimental setup for obtaining complementary temperature information in soot-free regions. With these strategically placed measurements, temperature fields are reconstructed across the entire flow domain leveraging the low-rank approximation provided by the previous SVD. Subsequently, a PINN is trained on the reconstructed temperature field to solve the coupled mass, momentum, and energy transport equations, thereby inferring the corresponding velocity field. The method is demonstrated on a coflow flame from a Yale-type burner and validated on the well-known Santoro flame, where both experimental temperature and velocity data are available. Results show that SSPINNs can reliably estimate temperature from sparse measurements and accurately approximate velocity fields without the need for more sophisticated experimental techniques like particle image velocimetry. By combining sparse, physically feasible measurements with low-rank representations and governing equations, SSPINNs bridge the gap between soot-based diagnostics and full-field thermo-fluid reconstruction. In this sense, the burden of field reconstruction is partially shifted toward physically constrained data assimilation, rather than relying solely on direct measurements or purely data-driven regression.

No abstract available

The reconstruction of accurate and robust unsteady flow fields from sparse and noisy data in real-life engineering tasks is challenging, particularly when sensors are randomly placed. To address this challenge, a novel Autoencoder State Estimation (AE-SE) framework is introduced in this paper. The framework integrates sensor measurements into a machine learning-based reduced-order model (ROM) by leveraging the low-dimensional representation of flow fields. The proposed approach is tested on two direct numerical simulation benchmark examples, namely, circular and square cylinders and wake flow fields at Re = 100. The results demonstrate satisfactory performance in terms of accuracy and reconstruction efficiency. It achieves the same accuracy as traditional methods while improving reconstruction efficiency by 70%. Moreover, it preserves essential physical properties and flow characteristics even in the noisy data, indicating its practical applicability and robustness. Experimental data validation confirms a relative error below 5% even at a noise level of 12%. The flexibility of the model is further evaluated by testing it with a trained ROM under varying Reynolds numbers and benchmark cases, demonstrating its ability to accurately estimate and recognize previously unseen flow fields with appropriate training datasets. Overall, the proposed AE-SE flow reconstruction method efficiently and flexibly leverages ROM for the low-dimensional representation of complex flow fields from sparse measurements. This approach contributes significantly to the development of downstream applications such as design optimization and optimal control.

The proper orthogonal decomposition (POD) is commonly applied for the model order reduction (MOR) of parametric problems. The method generates a reduced basis from the left singular vectors of a matrix consisted by a set of snapshot solutions. The basis represents a lower dimensional subspace where the solutions lie on; thus, it can be utilized for order reduction of the original problem. However, the application of POD on high-dimensional parametric problems with varying geometries faces two essential difficulties. First, the snapshot solutions need to be with the same dimension, since the singular value decomposition (SVD) of the snapshot matrix is a key step to obtain the POD reduced basis and a prerequisite of SVD is that the snapshot solutions can form a rectangular matrix. Second, the “curse of dimensionality” problem needs to be tackled in high-dimensional parametric problems. This work engages to provide a solution that can resolve these two difficulties. To deal with the first difficulty, we propose to use an interface adaptive background mesh to assure the consistence of snapshot solution vector lengths with varying geometry. And considering the second difficulty, the work proposes and compares two adaptive sampling methods based on latin hypercubic sampling (LHS) and ANOVA expansion, respectively. The proposed approach is validated and applied on a parametric electrostatic problem for parasitic capacitance extraction of interconnects in integrated circuits.

Abstract A Non-Intrusive Reduced-Order Model (NIROM) based on Proper Orthogonal Decomposition (POD) has been proposed for predicting the flow fields of transonic airfoils with geometry parameters. To provide a better reduced-order subspace to approximate the real flow field, a domain decomposition method has been used to separate the hard-to-predict regions from the full field and POD has been adopted in the regions individually. An Artificial Neural Network (ANN) has replaced the Radial Basis Function (RBF) to interpolate the coefficients of the POD modes, aiming at improving the approximation accuracy of the NIROM for non-samples. When predicting the flow fields of transonic airfoils, the proposed NIROM has demonstrated a high performance.

In this paper, we present a computationally efficient reduced order model for obtaining blood perfusion profiles within parametric functional units of the liver called ‘lobules’. We consider Darcy’s equation in two-dimensional hexagonal lobule domains with six flow inlets and one outlet, whose positions are parameterized to represent varying lobule geometries. To avoid the meshing effort for every new lobule domain, we map the parametric domain onto a single reference domain. By making use of the contra-variant Piola mapping, we represent solutions of the parametric domains in the reference domain. We then construct a reduced order model via proper orthogonal decomposition (POD). Additionally, we employ the discrete empirical interpolation method (DEIM) to treat the non-affine parameter dependence that appears due to the geometric mapping. For sampling random shapes and sizes of lobules, we generate Voronoi diagrams (VD) from Delaunay triangulations and use an energy minimization problem to control the packing of the lobule structures. To reduce the dimension of the parameterized problem, we exploit the mesh symmetry of the full lobule domain to split the full domain into six rotationally symmetric subdomains. We then use the same set of reduced order basis (ROB) functions within each subdomain for the construction of the reduced order model. We close our study by a thorough investigation of the accuracy and computational efficiency of the resulting reduced order model.

In many-query applications such as design, optimization, and model-predictive control, reduced order models (ROMs) have the potential to significantly decrease the computational cost. In contrast to intrusive projection-based ROMs which require direct access to the high-fidelity model (HFM) operators, non-intrusive ROMs can be constructed in a data-driven fashion using the input-output data generated by the HFM. When used in the prediction of the system response in unseen parameter regimes, however, generalization capabilities have to be carefully assessed. In this study, we pursue a two-step approach to construct non-intrusive parametric ROMs: The first step involves the extraction of low-dimensional latent spaces using proper-orthogonal decomposition (POD) and convolutional neural network-based autoencoders (CNN-AE); and the second step uses regression for the latent variables in parameter space. We adapt a unique decomposition approach named Split-POD to enrich the low-dimensional subspace of the parametric ROM. The proposed methods are used to predict the flow field over a vehicle geometry as a function of vehicle speed and internal fan speed. Through orders of magnitude reduction in the degrees of freedom, the non-intrusive ROMs enable flow field prediction in real-time and bypass hours of computations by the HFM. The results show that the nonlinear encoding in CNN-AE can enhance ROM predictions given adequate data. POD-based ROMs are more robust and efficient, and in particular more accurate than CNN-AE in under-sampled regions of the parameter space. We further enhance the utility of non-intrusive ROMs by formulating a training process that minimizes the required number of high-fidelity simulations using Gaussian process regression to adaptively sample the input parameter space and optimize the estimated variance of the predictions.

Traditional low-fidelity, semi-empirical aerodynamic models are typically used to provide drag polars needed for conceptual-level mission analysis but are insufficient for unconventional vehicles. In such scenarios, higher-fidelity, computationally expensive models commonly used in preliminary design are required. Due to the associated high computational cost and longer model evaluation time, it is impractical to estimate aerodynamic performance for all the design perturbations explored in the conceptual sizing stage. This paper presents a novel approach for efficiently approximating parametric mission drag polars by integrating sensitivity analysis, multifidelity surrogate models, and parametric reduced-order modeling techniques. The standardized regression coefficients method is used to select the geometry design parameters that have the largest impact on key vehicle performance metrics. These parameters, combined with hierarchical kriging surrogate models, are employed to generate mission drag polars for fixed vehicle geometries. Then, the proper orthogonal decomposition reduced-order model with kriging interpolation is used to approximate the coherent structure of mission drag polars. The proposed method is demonstrated on a supersonic commercial aircraft. Results show that both the multifidelity surrogate model and the reduced-order model effectively emulate vehicle drag polars well for fixed and varying vehicle geometries, respectively.

Abstract Accurate and efficient modelling of cardiac blood flow is crucial for advancing data-driven tools in cardiovascular research and clinical applications. Recently, the accuracy and availability of computational fluid dynamics methodologies for simulating intraventricular flow have increased. However, these methods remain complex and computationally costly. This study presents a reduced-order model (ROM) based on higher-order dynamic mode decomposition (HODMD). The proposed approach enables accurate reconstruction and long-term prediction of left ventricle flow fields. The method is tested on two idealized ventricular geometries exhibiting distinct flow regimes to assess its robustness under different hemodynamic conditions. By leveraging a small number of training snapshots and focusing on the dominant periodic components representing the physics of the system, the HODMD-based model accurately reconstructs the flow field over entire cardiac cycles and provides reliable long-term predictions beyond the training window. The reconstruction and prediction errors remain below 5 % for the first geometry and below 10 % for the second, even when using as few as the first three cycles of simulated data, representing the transitory regime. Additionally, the approach reduces computational costs with a speed-up factor of at least $10^{5}$ compared with full-order simulations, enabling fast surrogate modelling of complex cardiac flows. These results highlight the potential of spectrally constrained HODMD as a robust and interpretable ROM for simulating intraventricular hemodynamics. This approach shows promise for integration in real-time analysis and patient specific models.

This paper presents a nonlinear reduced-order modeling (ROM) framework that leverages deep learning and manifold learning to predict compressible flow fields with complex nonlinear features, including shock waves. The proposed DeepManifold (DM)-ROM methodology is computationally efficient, avoids pixelation or interpolation of flow field data, and is adaptable to various grids and geometries. The framework consists of four main steps: First, a convolutional neural network-based parameterization network extracts nonlinear shape modes directly from aerodynamic geometries. Next, manifold learning is applied to reduce the dimensionality of the high-fidelity output flow fields. A multilayer perceptron-based regression network is then trained to map the nonlinear input and output modes. Finally, a back-mapping process reconstructs the full flow field from the predicted low-dimensional output modes. DM-ROM is rigorously tested on a transonic RAE2822 airfoil test case, which includes shock waves of varying strengths and locations. Metrics are introduced to quantify the model's accuracy in predicting shock wave strength and location. The results demonstrate that DM-ROM achieves a field prediction error of approximately 3.5% and significantly outperforms reference ROM techniques, such as proper orthogonal decomposition (POD)-ROM and isometric mapping (ISOMAP)-ROM, for various training sample sizes.

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Engineering design workflows such as early‐stage vehicle development require controllable and interpretable parameterizations of complex 3D geometry, enabling designers to modify physically interpretable geometric attributes while keeping unrelated features stable. However, latent representations used in modern implicit shape models often exhibit interdependent coupling among multiple geometric factors, limiting their suitability as independent design variables. Motivated by reduced‐order modeling (ROM) and parametric shape design, we propose a geometrically structured latent subspace for signed distance function (SDF) representations that aligns latent directions with human‐interpretable geometric attributes (e.g., vehicle height, width, wheelbase) while keeping residual variation compact without prespecifying subspace dimension. The resulting formulation provides a numerically stable, geometry‐aware reduced‐order parameterization suitable for downstream engineering tasks. The model is trained end‐to‐end using lightweight global and local alignment losses together with a rank‐agnostic compactness regularizer. Anchor directions and normalized traversal step sizes are learned to enable monotonic, predictable edits with reduced cross‐attribute interference. Experiments on standard vehicle benchmarks demonstrate that the method preserves SDF reconstruction accuracy while producing attribute‐aligned and interpretable latent variations. These results indicate that embedding minimal geometric structure into SDF latent spaces substantially improves controllability, stability, and geometric consistency for vehicle‐centric 3D modeling and design exploration.

No abstract available

No abstract available

Predicting flows that occur both through and around porous bodies is challenging due to coupled physics across fluid and porous regions and the need to generalize across diverse geometries and boundary conditions. We address this problem using two Physics Informed learning approaches: Physics Informed PointNets (PIPN) and Physics Informed Geometry Aware Neural Operator (P-IGANO). We enforce the incompressible Navier Stokes equations in the free-flow region and a Darcy Forchheimer extension in the porous region within a unified loss and condition the networks on geometry and material parameters. Datasets are generated with OpenFOAM on 2D ducts containing porous obstacles and on 3D windbreak scenarios with tree canopies and buildings. We first verify the pipeline via the method of manufactured solutions, then assess generalization to unseen shapes, and for PI-GANO, to variable boundary conditions and parameter settings. The results show consistently low velocity and pressure errors in both seen and unseen cases, with accurate reproduction of the wake structures. Performance degrades primarily near sharp interfaces and in regions with large gradients. Overall, the study provides a first systematic evaluation of PIPN/PI-GANO for simultaneous through-and-around porous flows and shows their potential to accelerate design studies without retraining per geometry.

Accurate modeling of fluid dynamics around complex geometries is critical for applications such as aerodynamic optimization and biomedical device design. While advancements in numerical methods and high-performance computing have improved simulation capabilities, the computational cost of high-fidelity 3D flow simulations remains a significant challenge. Scientific machine learning (SciML) offers an efficient alternative, enabling rapid and reliable flow predictions. In this study, we evaluate Deep Operator Networks (DeepONet) and Geometric-DeepONet, a variant that incorporates geometry information via signed distance functions (SDFs), on steady-state 3D flow over complex objects. Our dataset consists of 1,000 high-fidelity simulations spanning Reynolds numbers from 10 to 1,000, enabling comprehensive training and evaluation across a range of flow regimes. To assess model generalization, we test our models on a random and extrapolatory train-test splitting. Additionally, we explore a derivative-informed training strategy that augments standard loss functions with velocity gradient penalties and incompressibility constraints, improving physics consistency in 3D flow prediction. Our results show that Geometric-DeepONet improves boundary-layer accuracy by up to 32% compared to standard DeepONet. Moreover, incorporating derivative constraints enhances gradient accuracy by 25% in interpolation tasks and up to 45% in extrapolatory test scenarios, suggesting significant improvement in generalization capabilities to unseen 3D Reynolds numbers.

Rapid and accurate simulations of fluid dynamics around complicated geometric bodies are critical in a variety of engineering and scientific applications, including aerodynamics and biomedical flows. However, while scientific machine learning (SciML) has shown considerable promise, most studies in this field are limited to simple geometries, and complex, real-world scenarios are underexplored. This paper addresses this gap by benchmarking diverse SciML models, including neural operators and vision transformer-based foundation models, for fluid flow prediction over intricate geometries. Using a high-fidelity dataset of steady-state flows across various geometries, we evaluate the impact of geometric representations -- Signed Distance Fields (SDF) and binary masks -- on model accuracy, scalability, and generalization. Central to this effort is the introduction of a novel, unified scoring framework that integrates metrics for global accuracy, boundary layer fidelity, and physical consistency to enable a robust, comparative evaluation of model performance. Our findings demonstrate that newer foundation models significantly outperform neural operators, particularly in data-limited scenarios, and that SDF representations yield superior results with sufficient training data. Despite these promises, all models struggle with out-of-distribution generalization, highlighting a critical challenge for future SciML applications. By advancing both evaluation models and modeling capabilities, our work paves the way for robust and scalable ML solutions for fluid dynamics across complex geometries.

Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries. Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks that encode node types (e.g., fixed boundary, free surface, hole edge), relative edge features (distance between neighbors), and global features (applied load). This architecture is inherently translation- and rotation-invariant, enabling generalization to unseen geometries without retraining. The MGN was trained on 11 plate geometries under 20 load conditions and evaluated on 7 unseen geometries and 3 unseen loads. In the most favorable case, the model achieves $R^2 \geq 0.97$ on an unseen geometry and unseen load, compared to $R^2 \approx 0.01$--$0.86$ for conventional models (Random Forest, Gradient Boosting , K-Nearest Neighbors) trained on identical data. However, even in less favorable cases, the MGN model still outperforms conventional models. This work extends the mesh-based simulation framework of Pfaff et al. (arXiv:2010.03409) to structural mechanics, demonstrating that graph neural networks can serve as efficient surrogates for finite element analysis across varying geometries.

Large-eddy simulations (LES) require closures for filtered production rates because the resolved fields do not contain all correlations that govern chemical source terms. We develop a graph neural network (GNN) that predicts filtered species production rates on non-uniform meshes from inputs of filtered mass fractions and temperature. Direct numerical simulations of turbulent premixed hydrogen-methane jet flames with hydrogen fractions of 10%, 50%, and 80% provide the dataset. All fields are Favre filtered with the filter width matched to the operating mesh, and learning is performed on subdomain graphs constructed from mesh-point connectivity. A compact set of reactants, intermediates, and products is used, and their filtered production rates form the targets. The model is trained on 10% and 80% blends and evaluated on the unseen 50% blend to test cross-composition generalization. The GNN is compared against an unclosed reference that evaluates rates at the filtered state, and a convolutional neural network baseline that requires remeshing. Across in-distribution and out-of-distribution cases, the GNN yields lower errors and closer statistical agreement with the reference data. Furthermore, the model demonstrates robust generalization across varying filter widths without retraining, maintaining bounded errors at coarser spatial resolutions. A backward facing step configuration further confirms prediction efficacy on a practically relevant geometry. These results highlight the capability of GNNs as robust data-driven closure models for LES on complex meshes.

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. Data-driven networks like GNN, Neural Operators have proved to be very effective in generalizing the model across unseen domain and resolutions. But one of the most critical issues in these data-based models is the computational cost of generating training datasets. Complex phenomena can only be captured accurately using deep networks with large training datasets. Furthermore, numerical error of training samples is propagated in the model errors, thus requiring the need for accurate data, i.e. FEM solutions on high-resolution meshes. Multi-fidelity methods offer a potential solution to reduce the training data requirements. To this end, we propose a novel GNN architecture, Multi-Fidelity U-Net, that utilizes the advantages of the multi-fidelity methods for enhancing the performance of the GNN model. The proposed architecture utilizes the capability of GNNs to manage complex geometries across different fidelity levels, while enabling flow of information between these levels for improved prediction accuracy for high-fidelity graphs. We show that the proposed approach performs significantly better in accuracy and data requirement and only requires training of a single network compared to other benchmark multi-fidelity approaches like transfer learning. We also present Multi-Fidelity U-Net Lite, a faster version of the proposed architecture, with 35% faster training, with 2 to 5% reduction in accuracy. We carry out extensive validation to show that the proposed models surpass traditional single-fidelity GNN models in their performance, thus providing feasible alternative for addressing computational and accuracy requirements where traditional high-fidelity simulations can be time-consuming.

This works investigates the generalization capabilities of MeshGraphNets (MGN) [Pfaff et al. Learning Mesh-Based Simulation with Graph Networks. ICML 2021] to unseen geometries for fluid dynamics, e.g. predicting the flow around a new obstacle that was not part of the training data. For this purpose, we create a new benchmark dataset for data-driven computational fluid dynamics (CFD) which extends DeepMind's flow around a cylinder dataset by including different shapes and multiple objects. We then use this new dataset to extend the generalization experiments conducted by DeepMind on MGNs by testing how well an MGN can generalize to different shapes. In our numerical tests, we show that MGNs can sometimes generalize well to various shapes by training on a dataset of one obstacle shape and testing on a dataset of another obstacle shape.

Despite the plethora of deep learning frameworks for flow field prediction, most of them deal with flow fields on regular domains, and although the best ones can cope with irregular domains, they mostly rely on graph networks, so that real industrial applications remain currently elusive. We present a deep learning framework for 3D flow field prediction applied to blades of aircraft engine turbines and compressors. Crucially, we view any 3D field as a function from coordinates that is modeled by a neural network we call the backbone-net. It inherits the property of coordinate-based MLPs, namely the discretization-agnostic representation of flow fields in domains of arbitrary topology at infinite resolution. First, we demonstrate the performance of the backbone-net solo in regressing 3D steady simulations of single blade rows in various flow regimes: it can accurately render important flow characteristics such as boundary layers, wakes and shock waves. Second, we introduce a hyper-net that maps the surface mesh of a blade to the parameters of the backbone-net. By doing so, the flow solution can be directly predicted from the blade geometry, irrespective of its parameterization. Together, backbone-net and hyper-net form a highly-accurate memory-efficient data-driven proxy to CFD solvers with good generalization on unseen geometries.

Reconstructing spatiotemporal fields from partial observations is fundamental to scientific inference, from inferring atmospheric states from satellite data to recovering fluid states from imaging. When observations are incomplete, the inverse problem is fundamentally ill-posed: even when the underlying PDE dynamics are Markovian in the full state, partial observation operators induce a non-Markovian posterior that cannot be resolved from a single timestep. We propose a history-bootstrapped autoregressive flow matching (HB-ARFM) for spatiotemporal inverse reconstruction under partial observability. Observation history bootstraps the initial reconstruction via conditional flow matching, reducing ambiguities. The same conditional transport model is then applied autoregressively, conditioning on both new observations and past predictions to propagate the reconstruction forward in time. We evaluate the method on boiling dynamics reconstruction, recovering full velocity and temperature fields from interface geometry and motion. Across two inverse tasks with varying observation sparsity, HB-ARFM produces physically and temporally valid reconstructions where other models fail.

Urban wind flow reconstruction is essential for assessing air quality, heat dispersion, and pedestrian comfort, yet remains challenging when only sparse sensor data are available. We propose GenDA, a generative data assimilation framework that reconstructs high-resolution wind fields on unstructured meshes from limited observations. The model employs a multiscale graph-based diffusion architecture trained on computational fluid dynamics (CFD) simulations and interprets classifier-free guidance as a learned posterior reconstruction mechanism: the unconditional branch learns a geometry-aware flow prior, while the sensor-conditioned branch injects observational constraints during sampling. This formulation enables obstacle-aware reconstruction and generalization to held-out mesh geometries, wind directions, and sensor configurations within the studied urban-flow setting, without retraining. We consider both sparse fixed sensors and trajectory-based observations using the same reconstruction procedure. When evaluated against supervised graph neural network (GNN) baselines and classical reduced-order data assimilation methods, GenDA reduces the relative root-mean-square error (RRMSE) by 25-57% and increases the structural similarity index (SSIM) by 23-33% across the tested meshes. Experiments are conducted on Reynolds-averaged Navier-Stokes (RANS) simulations of a real urban neighborhood in Bristol, United Kingdom, at a characteristic Reynolds number of $\mathrm{Re}\approx2\times10^{7}$, featuring complex building geometry and irregular terrain. The proposed framework provides a scalable path toward generative, geometry-aware data assimilation for environmental monitoring in complex domains.

This study proposes a universal flow field prediction framework based on knowledge transfer from large language model (LLM), addressing the high computational costs of traditional computational fluid dynamics (CFD) methods and the limited cross-condition transfer capability of existing deep learning models. The framework innovatively integrates Proper Orthogonal Decomposition (POD) dimensionality reduction with fine-tuning strategies for pretrained LLM, where POD facilitates compressed representation of flow field features while the fine-tuned model learns to encode system dynamics in state space. To enhance the model's adaptability to flow field data, we specifically designed fluid dynamics-oriented text templates that improve predictive performance through enriched contextual semantic information. Experimental results demonstrate that our framework outperforms conventional Transformer models in few-shot learning scenarios while exhibiting exceptional generalization across various inflow conditions and airfoil geometries. Ablation studies reveal the contributions of key components in the FlowBERT architecture. Compared to traditional Navier-Stokes equation solvers requiring hours of computation, our approach reduces prediction time to seconds while maintaining over 90% accuracy. The developed knowledge transfer paradigm establishes a new direction for rapid fluid dynamics prediction, with potential applications extending to aerodynamic optimization, flow control, and other engineering domains.