实验流体力学PIV/热线流场超分：传统到深度学习发展脉络、分类优势与局限、共性待解问题、ST-GNN真实增量与适用边界

Abstract High-resolution particle image velocimetry (PIV) particle-to-velocity analyses using small interrogation areas (IAs) often require substantial processing time. To overcome this limitation, a generative adversarial network (GAN)-based model is proposed to achieve spatio-temporal super-resolution (SR) reconstruction from low-resolution PIV data with large IAs, thereby significantly reducing post-processing time. Time-resolved PIV measurements of plasma-induced vortex flows, covering vortex formation, growth, transition and breakdown stages, are employed to train and evaluate the model with multi-scale vortical structures. By sequentially constructing spatial and temporal datasets, the GAN-based model enables reliable SR reconstruction at different scaling factors. Reconstruction accuracy is systematically assessed using time-averaged, instantaneous and phase-averaged velocity fields. At SR factors of $\times$ 4 and $\times$ 8, the reconstructed fields closely match high-resolution references, effectively capturing both fluctuating velocities and small-scale vortical structures. In contrast, $\times$ 16 reconstructions exhibit diminished accuracy due to the loss of fine-scale information from highly downsampled inputs. For time-averaged fields, high-resolution reconstructions reliably capture plasma jet characteristics at all SR factors. To enhance generalisation, transfer learning is introduced to fine tune the parameters of SR-related layers in the generator, enabling accurate reconstructions under varying vortex dynamics. In addition, the efficiency gains in PIV particle-to-velocity analysis and the fundamental limitations on achievable SR factors imposed by spatio-temporal data correlations are discussed. This study demonstrates that GAN-based spatio-temporal SR models offer a promising approach to accelerate PIV analyses while maintaining high reconstruction fidelity with diverse flow conditions.

Research on three-dimensional (3D) flow velocity fields holds significant importance in aerodynamic performance design, energy power, and biomedicine. Nevertheless, current techniques for measuring three-dimensional flow velocity fields, such as tomographic particle image velocimetry (Tomo-PIV), have challenges in achieving accurate and high-resolution measurements of tiny structures in flow fields. Consequently, a 3D flow field super-resolution (SR) reconstruction method based on Swin-Transformer framework (SWINFlow-3D) has been proposed in this paper. SWINFlow-3D comprises stacked residual channel attention Swin-transformer blocks, each containing multiple Swin-Transformer standard layers, incorporating a hybrid attention mechanism that allows for integrating relevant information from several channels and gives greater importance to critical information. Second, a loss function for SR reconstruction of the flow field has been introduced, taking into account the physical constraints such as divergence and curl. Furthermore, the characteristics obtained by interpolation downsampling methods are different from those of real experiments. To address this limitation, we construct a dataset based on cross correlation downsampling. Simulation experiments are carried out on Johns Hopkins Turbulence Database isotropic turbulence data and cylindrical wake data. The results are subsequently compared with those of the interpolation approach and 3D flow field SR reconstruction method, and our model yields the best results for all the metrics. Ultimately, to ascertain the accuracy and practical applicability of the model in practical tests, we conduct experiments on jet data and cylindrical wake recorded by Tomo-PIV. The experimental results demonstrate that SWINFlow-3D with the loss function presented in this study can be used to effectively reconstruct the 3D flow field and flow features, exhibiting strong generalizability.

"A novel full meshless super-resolution method for image velocimetry is proposed. This method builds upon the combination, presented in Tirelli et al. (2023a), of K-nearest neighbour Particle Tracking Velocimetry (Tirelli et al., 2023b, KNN-PTV) and constrained Radial Basis Functions regression (Sperotto et al., 2022, c-RBFs). The main novelty is that the algorithm is implemented here in a fully meshless version, i.e. it does not require the definition of a common Eulerian grid at any step of the process. KNN-PTV enhances the spatial resolution of vector fields by exploiting data coherence in space and time, even for non-time-resolved measurements. On the other and, c-RBFs provide an analytical representation of the field from scattered data while allowing the introduction of physical constraints. In this new version, the dictionary to blend data from different snapshots is computed with a meshless POD approach developed by the authors and presented in another contribution to this conference. This approach removes the last constraint of Eulerian grids and paves the way for a fully meshless algorithm. The algorithm is validated on a challenging 2D synthetic case, such as turbulent channel flow, and an experimental case involving the separation bubble around the frontal portion of a Ground Transportation System (GTS)."

Light field particle image velocimetry (LF-PIV) enables single-camera volumetric flow measurement, but its performance is limited by low spatial resolution and large depth-wise velocity uncertainty. This study proposes a LF-aware physics-informed neural network (LF-PINN) for super-resolved flow reconstructions from sparse LF-PIV measurements. Building on standard PINNs, which enforce governing equations during learning, the proposed method further incorporates LF imaging characteristics into the physics-informed framework to improve the reconstruction accuracy. This is achieved by introducing a degradation model and a reliability-weighted learning strategy that account for the spatial averaging effect and the anisotropic velocity uncertainty inherent in LF-PIV measurements. Numerical tests on forced isotropic turbulence demonstrate that LF-PINN effectively corrects depth-wise velocity errors, reducing the relative error of the depth-wise velocity component to below 10% when initial measurement errors reach 20%. Incorporating pressure information further enhances flow reconstruction accuracy, reducing the root mean square divergence of the flow field by 22%. Compared with standard PINN formulations and a proper orthogonal decomposition-based interpolation method, LF-PINN achieves superior performance in both spatial and temporal super-resolution reconstruction from sparse measurements. Fine-scale vortical structures and turbulence statistics are recovered with higher fidelity. Experimental measurements of the wake behind a circular cylinder further show that LF-PINN produces clearer wake structures and velocity profiles that are in better consistence with planar-PIV references. These results demonstrate that embedding LF imaging characteristics into physics-informed learning provides an effective route toward super-resolved and physically consistent volumetric flow reconstruction.

Particle image velocimetry (PIV) is a key non-intrusive technique for velocity measurement, yet dense flow estimation remains difficult in cases with small-scale vortices or occlusions. Based on Super-Kernel Flow Network (SKFlow), a novel lightweight deep learning framework called PIV-LightSKFlow has been proposed, integrating a feature encoder, multi-scale correlation pyramid, superkernel module, and global motion aggregation (GMA). Paired PIV images are processed to predict dense velocity fields. For PIV tasks, the encoder resolution is improved from 1/8 to 1/4 for better particle representation, and the correlation pyramid is redesigned, reducing parameters by 29.3%. A new synthetic dataset is created to evaluate the algorithm's ability to predict small occlusions in the flow field. Evaluations on synthetic and experimental datasets show that PIV-LightSKFlow surpasses traditional algorithms, and previous deep learning methods. It achieves root mean square error reductions of 6.9%–68.4% in velocity prediction of flow fields such as vortex shedding, and direct numerical simulation turbulent motion, while maintaining high spatial resolution and stable performance at various Reynolds numbers (Re). The superkernel and GMA modules further enhance stability and accuracy in occluded regions. These results highlight PIV-LightSKFlow as an efficient and reliable solution for advanced PIV estimation, with promising potential in experimental and industrial fluid mechanics.

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This work presents our ‘SPICY’ (meshlesS Pressure from Image veloCimetrY) toolbox for meshless data assimilation in image velocimetry. The proposed approach allows for computing an analytic representation of velocity and pressure fields from noisy and sparse fields, regardless of whether these are acquired using cross-correlation or particle tracking algorithms. SPICY uses penalized and constrained Radial Basis Functions (RBFs), allowing for enforcing physical priors (e.g., divergence-free in incompressible flows) or boundary conditions (e.g., no slip). The resulting analytic expression allows for super-resolution on arbitrary points and accurate computation of derivatives. These could be used to compute derived quantities (e.g., vorticity) and to integrate the pressure Poisson equation to compute pressure fields for incompressible flows. A set of video tutorials on how to use SPICY is provided.

Particle image velocimetry (PIV) and particle tracking velocimetry (PTV) are important flow visualization technologies for measuring global velocity fields in a nonintrusive manner. However, they are limited by the spatial resolution of the measurement, and they require further postprocessing steps to refine the flow fields. To this end, we employ a deep-learning method, physics-informed neural networks (PINNs), which can integrate the sparse velocity measurements from PIV or PTV with the governing equations of the fluid flow by a neural network. A real experiment, where the tomographic PTV setup is applied to measure the 3-D turbulent jet flow, is considered to evaluate the proposed method. We perform a systematic study based on the experimental data, demonstrating that the PINN-enhanced velocimetry approach can yield super-resolution for the velocity vectors, hence demanding only the order of 100 vectors per snapshot compared to 16 500 vectors at full resolution. In addition, PINNs infer the pressure field without providing any pressure information. The proposed algorithm can be readily implemented with the existing PIV/PTV software, providing a standard method for greatly enhancing experimental data in fluid dynamics.

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In this work, we demonstrate and assess the performances of a novel super-resolution generative adversarial network (GAN) framework. The algorithm, recently introduced by the authors, leverages random spatial sampling in particle images to provide incomplete views of the high-resolution underlying fields. The main novelty is that the architecture, named Randomly Seeded GAN (RaSeedGAN), does not need full high-resolution training samples. The training is performed directly using the sparse sensors (e.g. particles) available in each snapshot, reduced on a regular grid by spatial averaging in bins. Bins without vectors are simply skipped during training. Provided that the particles randomly sample the space within the dataset, it is possible to reconstruct the mapping from low to high-resolution input with such incomplete ``gappy'' views. The proposed technique is tested on several synthetic datasets based on simulations, ocean surface temperature distribution measurements, and particle image velocimetry data of a zero-pressure-gradient turbulent boundary layer flow. While the applications in this work target fluid mechanics examples, the proposed method can be applied to more general frameworks where mapping is performed by moving sensors and/or with random on-off status. The results show increased accuracy compared to standard processing and direct cubic interpolation of the scattered velocity vectors. An analysis of the turbulent flow features, turbulence statistics, and spectra demonstrates an increase in spatial resolution of at least a factor of 3 in terms of cut-off frequency, with physically consistent estimated fields.

Recent attempts to employ deep learning technology for the super-resolution (SR) reconstruction of turbulence have focused chiefly on reconstructing two-dimensional (2D) slices of the three-dimensional (3D) flow fields. However, tomographic particle image velocimetry (Tomo-PIV) measurements yield 3D velocity fields. The resolutions of 3D velocity field data along three directions cannot be simultaneously enhanced using a 2D SR reconstruction model. Additionally, a 3D velocity flow field structure cannot be accurately reconstructed using a 2D model. Therefore, we present a 3D flow field SR reconstruction method (FSR-3D) based on a back-projection network. The FSR-3D model incorporates a multiscale convolutional residual block and an up-and-down projection module and establishes the mapping relationship between low-resolution features and high-resolution (HR) features through an iterative mechanism. The proposed model is validated using two experiments. First, the original velocity field is reconstructed from downsampled forced isotropic turbulence data. Second, the downsampled direct numerical simulation (DNS) data of a turbulent channel flow are used to reconstruct the HR flow field to verify the reconstruction capability of the FSR-3D model with respect to the flow field of the boundary layer. Third, Tomo-PIV measurements of the wake flow behind a circular cylinder are used to further verify the generalization ability of the model and the reconstruction effect of the measured data. The experimental results demonstrate that the proposed FSR-3D model can accurately reconstruct the HR 3D velocity field. At the same time, the properties of the reconstruction results, such as their vortex structure, kinetic energy spectrum, and Reynolds stress, are closer to the DNS results than those of the 2D model. The reconstruction results of the Tomo-PIV measurement data show that the FSR-3D model has good generalization ability.

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.

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.

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.

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Fluid mechanics’ simulations are widely incorporated in place of expensive and complex experiments, providing accurate property fields in a dense grid fashion. It is a fact that experimental measurements due to their discrete and, most of the times, sparse data representation, cannot fully define velocity, pressure, temperature, vorticity, which are common properties of interest, especially in turbulent flows. The need to upscale flow features from coarse-grained data has been successfully addressed during the past years with aid of novel artificial intelligence -based methods. This upscaling refers either to sparse experimental data or to coarse simulation data, bypassing computationally intensive classical direct numerical simulations. This study presents a deep learning approach that accepts low resolution vorticity fields, from an open channel flow simulation, and reconstructs the field to resemble a fine grid result. This super resolution approach shows that reconstruction is possible as long as the network training is performed using pairs of low/high resolution fields. The proposed model is based on the U-Net architecture and is investigated over the effect of the resolution scale factor of the input images.

Recently, advanced deep-learning techniques have been successfully applied as deconvolution operators to super-resolve the low-resolution data in large-eddy simulation (LES). The super-resolution (SR) operator provides an approximate inverse to the filter operators in LES such that the under-resolved and un-resolved sub-grid information can be reconstructed from the resolved scales. In this work, a particle-aware attention-based conditional super-resolution generative adversarial network (PACASRGAN) is proposed for the fourfold SR of gas field scalars which are generated by the pyrolysis process in a hot turbulent flow laden with pulverised biomass particles. Multiple carrier-phase direct numerical simulations (DNS) of two-way coupled particle-laden flows with heat and mass transfer, that mimic the near-burner field of pulverised biomass combustion (PBC) systems, are carried out to build the training/testing datasets. The model performance is assessed in an a priori manner by investigating statistical quantities of interest for the modelling in LES of PBC. The results show that the proposed model can super-resolve the temperature and mixture fraction fields to a good accuracy and outperforms unconditional GAN models. Particles create localised sources/sinks via two-way coupling which sharpen scalar gradients in the subgrid. The particle mask and feature vector encode this localisation to improve the predictions of the generator. The scalar spectra, the conditional average of unresolved scalar variances, the probability density function (PDF), and the conditional average of the square of the mixture fraction gradient from the reconstructed fields follow the DNS values well. Slight deviations are observed at rich conditions in conditional statistics and at the tail of the PDFs. Nonetheless, the results demonstrate that SR is applicable to two-way coupled particle-laden flows with heat and mass transfer, providing reasonably accurate high-resolution information for both the entire gas field and particle positions.

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.

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.

This paper proposes a super-resolution (SR) reconstruction method based on deep learning, which efficiently reconstructs the global high-resolution wake flow field from the low-resolution (LR) wake data of a propeller. The extensive wake data for the propeller under various operating conditions are generated using numerical simulations based on a delayed detached eddy simulation model. The proposed approach, propeller super-resolution convolutional neural networks (PSCNN), uses a dilated convolutional module to capture multi-scale spatial characteristics of wake flow fields. The performance of the proposed SR method is evaluated by improving the resolution of the wake flow field under different scaling factors, and its superiority is demonstrated by comparing the reconstruction accuracy with that of two other typical SR reconstruction methods. The results indicate that PSCNN can effectively improve the resolution of the propeller wake flow field by 32 times, with an overall mean relative error of the three velocity components being less than 4.0%, and the reconstructed global SR wake flow field agrees well with the ground truth in spatial distribution variation. Furthermore, PSCNN can reconstruct the SR wake flow field with reasonable accuracy under unseen operating conditions, further proving the generalizability of the proposed SR model in capturing spatial relationships of the propeller wake. Overall, the proposed SR reconstruction method has significant applications in obtaining high-resolution flow snapshots in fluid experiments.

With the rapid advancement of hypersonic flight technology, the high-precision measurement of hypersonic flow field parameters has become an urgent issue. The computational fluid dynamics (CFD) method for obtaining high-resolution flow field data requires high grid quality and involves complex numerical solving processes, leading to significant computational costs. There is a growing need to quickly obtain high-resolution flow field data with less effort. Leveraging the powerful nonlinear fitting capabilities of neural networks, it is possible to perform fine super-resolution reconstruction of low-resolution flow field data in a data-driven manner. In this study, we propose an improved Enhanced Super-resolution Convolutional Neutral Network (ESRCNN) model tailored for hypersonic target flow fields. This enhanced model is applied to the super-resolution reconstruction of low-resolution flow fields of blunt body aircraft targets and compared with interpolation methods and traditional deep learning methods. Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) are used as evaluation metrics. The results validate the accuracy and superiority of this model in reconstructing hypersonic target flow fields. This method provides an effective approach for the precise measurement and reconstruction of hypersonic flow fields, thereby contributing to the advancement of hypersonic flight technology.

Two-phase flow exists widely in the industrial field, and its research is of great significance for industrial production. In this brief, we design a wire-mesh sensor system with 256 measurement points and conduct vertical upward gas-liquid two-phase flow experiments to capture the flow information under different flow conditions. We find that flow structure can be characterized by the gas-liquid distribution imaging obtained through the wire-mesh sensor. To obtain more detailed imaging, we apply super-resolution methods based on deep learning to improve the imaging quality and choose No-Reference Structural Sharpness to evaluate the effects. The results show that compared with traditional interpolation method, the deep learning-based super resolution can restore more abundant details such as edges and textures, thereby characterizing the flow structure more finely. This provides an important support for further analysis of the flow characteristics and flow regularity, which can effectively optimize the industrial process.

The development of cerebrovascular disease is tightly coupled to changes in cerebrovascular hemodynamics, with altered flow and relative pressure indicative of the onset, development, and acute manifestation of pathology. Image-based monitoring of cerebrovascular hemodynamics is, however, complicated by the narrow and tortuous vasculature, where accurate output directly depends on sufficient spatial resolution. To address this, we present a method combining dedicated deep learning and state-of-the-art 4D Flow MRI to generate super-resolution full-field images with coupled quantification of relative pressure using a physics-driven image processing approach. The method is trained and validated in a patient-specific in-silico cohort, showing good accuracy in estimating velocity (relative error: 12.0 ± 0.1%, mean absolute error (MAE): 0.07 ± 0.06 m/s at peak velocity), flow (relative error: 6.6 ± 4.7%, root mean square error (RMSE): 0.5 ± 0.1 mL/s at peak flow), and with maintained recovery of relative pressure through the circle of Willis (relative error: 11.0 ± 7.3%, RMSE: 0.3 ± 0.2 mmHg). Furthermore, the method is applied to an in-vivo volunteer cohort, effectively generating data at <0.5mm resolution and showing potential in reducing low-resolution bias in relative pressure estimation. Our approach presents a promising method to non-invasively quantify cerebrovascular hemodynamics, applicable to dedicated clinical cohorts in the future.

Details of flow field are highly relevant to understand the mechanism of turbulence, but obtaining high-resolution turbulence often requires enormous computing resources. Although the super-resolution reconstruction of turbulent flow fields is an efficient way to obtain the details, the traditional interpolation methods are difficult to reconstruct small-scale structures, and the results are too smooth. In this paper, based on the transformer backbone architecture, we present a super-resolution transformer for turbulence to reconstruct turbulent flow fields with high quality. It is supervised and has a broader perceptual field for better extraction of deep-level features. The model is applied to forced isotropic turbulence and turbulent channel flow dataset, and the reconstructed instantaneous flow fields are comprehensively compared and analyzed. The results show that SRTT can recover the turbulent flow fields with high spatial resolution and capture small-scale details. It can obtain either the isotropic or the anisotropic turbulent properties even in complex flow configurations.

In numerical simulations and experimental observations, flow field data are often limited by insufficient spatial resolution and incomplete time series, which in turn affect the accurate capture and modeling of flow structures and their evolution processes. To address this challenge, this paper proposes an end-to-end high-resolution flow field prediction framework that can rely solely on low-resolution input data to reconstruct the fine spatial structure at the current moment and predict the high-resolution evolution state at future moments. The framework consists of three types of deep neural network modules working together: the U-Net-based super-resolution reconstruction module (U-Net-SR) implements high-resolution spatial reconstruction at the current moment, the long short-term memory module (LSTM) models the temporal evolution trend of low-resolution sequences, and the U-Net-based multi-source fusion prediction module (U-Net-Pred) integrates the information from both to achieve accurate prediction of the highresolution flow field at future moments. This method is validated in three typical flow field tasks: laminar flow around a cylinder, laminar flow over a controlled pitch airfoil, and experimental turbulent flow field under cross-wind gusts passing through a flat plate airfoil. In multiple typical flow field tasks, the framework demonstrates excellent spatiotemporal modeling capabilities. Overall, the average Peak Signal-to-Noise Ratio(PSNR) of the prediction results generally remains in the range of 40–48 dB, and the Structural Similarity Index Measure(SSIM) index is mostly stable above 0.95, accurately capturing key structural features and maintaining good temporal evolution consistency. Compared with traditional single-path models that only handle spatial reconstruction or temporal prediction, the innovation of this framework lies in its dual U-Net collaborative mechanism: U-Net-SR provides the highresolution state at the current moment as a spatial benchmark, LSTM captures the overall evolution trend of the flow field, and U-Net-Pred completes cross-scale fusion and information collaboration in the feature space, thereby achieving better super-resolution prediction.

Single image super-resolution (SR) has become a promising research topic, with many deep learning-based models invented to reconstruct high-fidelity high-resolution (HR) images from low-resolution (LR) images. Motivated by a large amount of turbulent flow field data collected by experimental measurements and numerical simulation, researchers begin investigating the application of these data-driven deep learning models to conduct SR reconstruction of LR flow field data. Due to the limitations of experimental equipment and computing power, sometimes researchers can only obtain LR data. However, deep learning models can quickly reconstruct HR spatial-temporal turbulent data from LR data so that researchers can easily conduct further qualitative and quantitative analyses. This article reviews the development of flow field data SR reconstruction models and the problems encountered from the two aspects of network structure and loss function definition. Finally, we propose the research direction of applying the conditional generative adversarial network (cGAN) to turbulent flow SR reconstruction since seldom study has been conducted in this field.

This paper reports experiments on a hydrogen-fueled scramjet performed at different equivalence ratios in a ground pulse combustion wind tunnel with a Mach-2.5 incoming flow. In the non-chemical reaction flow before the fuel was ignited, the flow field was oscillatory, and from the pressure monitor, the oscillation period was 0.07 s and the oscillation amplitude was 0.035 MPa. Schlieren and flame self-luminescence images of the combustor flow were obtained synchronously, and the flow field structure was stable with the flame concentrated in the shear layer. Deep learning was used to subject the low-resolution combustion flow field to super-resolution analysis to reconstruct a high-resolution flow field. To improve the spatial resolution of the flow field during self-ignition of the hydrogen-fueled scramjet and study the flow mechanism and coupling rule of turbulent fluctuations in the ignition process, a multipath asymmetric residual network (MARN) is proposed based on a single-path super-resolution convolutional neural network (SRCNN) and a residual network model (ResNet_16). The experimental results show that compared with SRCNN and ResNet_16, MARN has the best precision and performance regarding the super-resolution flow field in the self-ignition of hydrogen fuel in terms of mean peak signal-to-noise ratio, mean structural similarity, and average correlation coefficient, as well as being the least complicated. The proposed method offers the possibility of developing lightweight super-resolution models for the flow fields in supersonic combustors, it shows enormous potential for revealing the physical flow of fuel and air mixture, and it offers accurate forecasts of self-ignition times.

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 hydrodynamic model, based on the strict conservation of momentum and continuity equations, can accurately simulate the distribution of a flow field. However, significant computing time and storage space requirements limit real-time prediction. Machine learning is well known for its fast computing speed and powerful learning ability, but its accuracy depends on an abundance of training data, hindering its wider use in locations without sufficient measurements. Application restrictions in data-deficient areas can be addressed through transfer learning, provided that two areas share common characteristics. In this study, a machine learning method based on a deep super-resolution convolutional neural network (DSRCNN) and transfer learning is proposed, validated, and applied to model two bend flows and one realistic test case. Firstly, the hydrodynamic model was established and validated against measured data. The validated model was considered to have the ability to generate real data and was used to generate a comprehensive data set for training and validating the machine learning model. Three different methods were compared and tested, with Realizable k-ε performing better than the others in predicting the outer bank flow distribution. DSRCNN was compared to a plain SRCNN (PSRCNN), as well as Bilinear, Nearest, and Bicubic methods, and the results showed that DSRCNN had the best performance. We compared Raw, RT, and TL methods, finding that the TL method performed the best overall. Therefore, the research results showed that the developed super-resolution convolutional neural network can provide more reliable predications and serve as an ideal tool for simulating flow field distribution in bends.

High-fidelity simulation and reconstruction of physical fields are essential in both scientific research and engineering, yet classical solvers can be prohibitively expensive at resolutions needed to capture fine-scale structures. We propose FNODiffSR, a data-driven super-resolution framework that couples a residual-guided diffusion model with an Adaptive Weighted Fourier Neural Operator (AWFNO). AWFNO models longrange spectral dependencies while selectively emphasizing highfrequency components, and the diffusion module employs a conditional probability-flow ODE instead of stochastic sampling to deterministically bridge low- and high-fidelity representations. Final reconstructions are obtained by integrating this ODE with an adaptive time-stepping solver. Experiments on quasigeostrophic turbulence across varied upsampling and sparsesampling regimes show that FNODiffSR consistently surpasses interpolation and learning-based baselines in reconstruction fidelity, structural similarity, and physical consistency (as assessed by a dimensionless equation-residual), while offering predictable runtime and scalability. These qualities make FNODiffSR a strong candidate for high-quality scientific data recovery and downstream analysis.

No abstract available

Super-resolution reconstruction (SR) of turbulent flow fields with high physical fidelity from low-resolution turbulence data is a novel and cost-effective way in a turbulence study. However, some naive image-to-image machine learning methods often produce nonphysical features inconsistent with the physical characteristics of turbulence. The present work proposes, respectively, convolutional neural network and generative adversarial network-based turbulence SR models using the kinetic energy spectra of turbulence flow as a physical constraint. The models have been validated in turbulence SR reconstruction for a Newtonian fluid under the flow condition of homogeneous isotropic turbulence at Reynolds number, Re=3140 and 4710 and viscoelastic fluid at the same Re numbers and Weisenberg number, Wi=0.796 and 1.194 (the elasticity number El=Wi/Re=0.000 254), respectively. The results show that with the energy spectra constraint (ESC), not only the nonphysical features occurred in the energy spectra of velocity field could be eliminated by the SR models, the errors of their reconstructed vorticity fields in comparison with the results of direct numerical simulation are also significantly smaller than those of the SR models without ESC. Therefore, incorporation of physical constraints is vital in preserving physical characteristics in SR of turbulent flow.

High-resolution turbulence data are essential for marine environment monitoring, climate change modeling, and aerospace engineering. However, numerical simulations and experimental measurements are often limited by high computational costs, insufficient sensor resolution, and sparse data acquisition, making it difficult to fully capture the multi scale details of turbulence. To address this challenge, a spatio-frequency fusion distillation network (SFDN) is developed to reconstruct finescale turbulent structures from low resolution inputs. The proposed model is built upon stacked Mamba-frequency distillation blocks, where each block integrates multiple Mamba-frequency fusion blocks (MFFBs) to refine and fuse the deep features of turbulence. By combining a Mamba-based state space model with a frequency attention mechanism, the MFFB is capable of capturing both long-range spatial dependencies and frequency-domain representations. Additionally, a physics-guided loss function is introduced to constrain the solution space and guide the learning process. To evaluate the performance of the proposed SFDN, comprehensive experiments were conducted on datasets of the forced isotropic turbulence and the turbulent channel flow, and comparisons were made with bicubic interpolation and several deep learning super-resolution models. The results demonstrate that SFDN achieves comparable or superior performance to state-of-the-art methods in terms of visual quality, quantitative accuracy, and preservation of physical characteristics, while using only 35.38% of the parameters of leading models. These findings highlight the effectiveness and efficiency of the proposed approach, as well as its strong generalization capability for reconstructing complex turbulent flows.

Abstract The super-temporal-resolution (STR) reconstruction of turbulent flows is an important data augmentation application for increasing the data reach in measurement techniques and understanding turbulence dynamics. This paper proposes a data assimilation (DA) strategy based on weak-constraint four-dimensional variation to conduct an STR reconstruction in a turbulent jet beyond the Nyquist limit from given low-sampling-rate observations. Highly resolved large-eddy simulation (LES) data are used to produce synthetic measurements, which are used as observations and for validation. A segregated assimilation procedure is realised to assimilate the initial condition, inflow boundary condition and model error separately. Different types of observational data are tested. The first type is down-sampled LES data containing many small-scale turbulence structures with or without synthetic noise. The DA results show that the temporal variation of the small-scale structures is well recovered even with noise in the observations. The spectra are resolved to a frequency approximately one order of magnitude higher than what can be captured within the Nyquist limit. The second type of observation is low-sampling-rate tomographic particle image velocimetry (tomo-PIV) data with or without the injection of small-scale structures. The modulation between the large-scale structures contained in the tomo-PIV fields and the small scales injected from the observations is improved. The resultant small scales in the STR reconstruction have the characteristics of authentic turbulence to a considerable extent. Additionally, DA yields much smaller errors in the prediction of particle positions when compared with the Wiener filter, demonstrating the great potential for Lagrangian particle tracking in measurement techniques.

Super-resolution (SR) reconstruction of turbulent flows, inspired by breakthroughs in image SR, aims to generate high-resolution flow fields from low-resolution data that are available from numerical simulations and experimental measurements. Deep convolutional neural networks (CNN) have long been the pillar in both image and turbulence SR. Recently, networks based on the Swin Transformer featuring an innovative attention mechanism have achieved great success in image SR. This study first applies a latest variant of the Swin Transformer, SwinFIR (Image Restoration with Swin Transformer using a fast Fourier convolution layer), to the super-resolution of forced homogeneous isotropic turbulence, and compares its performance with a well-established CNN-based model, the Residual-in-Residual Dense Block (RRDB). The results show that the SwinFIR slightly outperforms the RRDB after being trained with a Ln-norm-based pixel loss, but both models still significantly underestimate the high-frequency turbulent energy. Further improvement is hindered by the loss. To address this issue, a new fine-tuning strategy featuring a spectral loss is proposed in this work. The results show that the proposed fine-tuning strategy is highly effective in improving the reconstruction of high-frequency turbulence. The generalization tests on data with higher Reynolds numbers unseen in the training further demonstrate that the fine-tuned network generalizes well for 4× SR (i.e., enhancing resolution by a factor of four in each direction), and that good generalization performance can also be achieved for higher SR factors with a consistent ratio of grid size to Kolmogorov length scale. The underlying mechanisms dictating the generalization performance are discussed, providing clues for developing better training strategies that can endow models with unconditional generalization capability.

Abstract Super-resolution of turbulence is a term used to describe the prediction of high-resolution snapshots of a flow from coarse-grained observations. This is typically accomplished with a deep neural network and training usually requires a dataset of high-resolution images. An approach is presented here in which robust super-resolution can be performed without access to high-resolution reference data, as might be expected in an experiment. The training procedure is similar to data assimilation, wherein the model learns to predict an initial condition that leads to accurate coarse-grained predictions at later times, while only being shown coarse-grained observations. Implementation of the approach requires the use of a fully differentiable flow solver in the training loop to allow for time-marching of predictions. A range of models are trained on data generated from forced, two-dimensional turbulence. The networks have reconstruction errors which are similar to those obtained with ‘standard’ super-resolution approaches using high-resolution data. Furthermore, the methods are comparable to the performance of standard data assimilation for state estimation on individual trajectories, outperforming these variational approaches at initial time and remaining robust when unrolled in time where performance of the standard data-assimilation algorithm improves.

In a plethora of research endeavors concerning flow fields, acquiring high-resolution data is paramount. However, obtaining high-resolution turbulence data invariably requires substantial computational resources. Although super-resolution reconstruction of turbulent fields has emerged as a salient technique for detail extraction, conventional interpolation methods pose a significant challenge in reconstructing small-scale structures, often resulting in overly smooth outcomes. In this study, we propose a novel hybrid framework of spatially-adaptive feature attention (HSAFA) for the high-quality reconstruction of turbulent fields. This framework is characterized by the implementation of multidimensional feature fusion, which enhances the model's ability to capture details of turbulence. We rigorously applied the proposed model to datasets comprising laminar flow around a square cylinder and turbulent channel flows, with the reconstructed instantaneous velocity fields and statistics subjected to exhaustive and comparative analysis. Our findings demonstrate that HSAFA is capable of effectively reconstructing high-resolution turbulence fields from significantly low-resolution data, covering the range from laminar to turbulent flows.

Super-resolution reconstruction of turbulent flows using deep learning has gained significant attention, yet challenges remain in accurately capturing physical small-scale structures. This study introduces the Conditional Enhanced Super-Resolution Generative Adversarial Network (CESRGAN) for reconstructing high-resolution turbulent velocity fields from low-resolution inputs. CESRGAN consists of a conditional discriminator and a conditional generator, the latter being called CoGEN. CoGEN incorporates subgrid-scale (SGS) turbulence kinetic energy as conditional information, improving the recovery of small-scale turbulent structures with the desired level of energy. By being aware of SGS turbulence kinetic energy, CoGEN is relatively insensitive to the degree of detail in the input. As shown in the paper, its advantages become more pronounced when the model is applied to heavily filtered input. We evaluate the model using direct numerical simulation (DNS) data of forced homogeneous isotropic turbulence. The analysis of Q-criterion isosurfaces, energy spectra, and probability density functions shows that the proposed CoGEN reconstructs fine-scale vortical structures more precisely and captures turbulent intermittency better compared to the traditional generator. Particle-pair dispersion simulations validate the physical fidelity of CoGEN-reconstructed fields, closely matching DNS results across various Stokes numbers and filtering levels. This paper demonstrates how incorporating available physical information enhances super-resolution models for turbulent flows.

The resolution of flow fields represents a significant factor influencing the accuracy of turbulent flow analysis. Nevertheless, the acquisition of high-resolution turbulence data remains a challenge due to the limitations imposed by computing resources. The interpolation method, while capable of achieving high-resolution turbulence at low cost, faces challenges in capturing details of turbulent flows. In this study, a local and global feature fusion network (LGFN) is designed for the reconstruction of high-resolution turbulent flows with high quality. First, dual parallel branches made of dense blocks are introduced into the LGFN to extract local features of turbulent flows. Moreover, the features after the first dense block of each branch are summarized into the self-attention block to obtain global features. The extracted local and global features are aggregated through learnable weight parameters to achieve feature fusion. Finally, the fused turbulence features are scaled to the same dimensional size as the high-resolution turbulence through the implementation of multilayer pixel shuffle layers and convolution layers. The effectiveness of the proposed network was evaluated using datasets of forced isotropic turbulence and channel turbulence. The results demonstrate that the reconstructed velocity fields of the LGFN exhibit the highest degree of similarity to the direct numerical simulation results, in comparison with bicubic interpolation, static convolutional neural network, and super-resolution dense connection network results. In addition, compared to alternative methods, the proposed network effectively captures the characteristics of isotropic or anisotropic turbulence even at larger scales.

High-resolution flowfield data are essential for turbulence research and engineering applications, yet their acquisition from numerical simulations and experiments remains prohibitively expensive. Super-resolution reconstruction offers a promising alternative, but conventional interpolation-based and generic image super-resolution methods often fail to recover turbulence-specific fine-scale features, such as coherent vortical structures, thin shear layers, and high-wavenumber fluctuations. To address this limitation, this study proposes a structure-aware deep-learning framework, termed the Coordinate-Attention Residual Super-Resolution Network (CARSR-Net), for reconstructing turbulent flows from severely underresolved inputs. Here, structure-aware means that the model is designed to preferentially preserve physically important flow structures and their multiscale statistical signatures. In particular, a turbulence-oriented composite loss function is introduced to emphasize high-gradient and vortical regions so that the reconstruction is constrained not only by pointwise field accuracy but also by the recovery of physically relevant small-scale structures. Lightweight attention and multiscale feature aggregation modules are further incorporated to improve the extraction of directional and multiscale flow information. The model is evaluated on wall-bounded turbulent channel flow and forced isotropic turbulence using both instantaneous fields and representative statistical quantities. The results show that CARSR-Net reconstructs high-resolution flowfields with lower relative [Formula: see text] error and more accurate energy spectra and vorticity probability density functions than baseline methods. In particular, it better preserves high-frequency content and the physical consistency of the reconstructed fields.

No abstract available

Turbulence is a complicated phenomenon because of its chaotic behavior with multiple spatio-temporal scales. Turbulence also has irregularity and diffusivity, making predicting and reconstructing turbulence more challenging. This study proposes a deep-learning approach to reconstruct three-dimensional (3D) high-resolution turbulent flows from spatially-limited data using a 3D enhanced super-resolution generative adversarial networks (3D-ESRGAN). In addition, a novel transfer-learning method based on tricubic interpolation is employed. Turbulent channel flow data at friction Reynolds numbers Reτ = 180 and Reτ = 500 were generated by direct numerical simulation (DNS) and used to estimate the performance of the deep-learning model as well as that of the tricubic interpolation-based transfer learning. The results, including instantaneous velocity fields and turbulence statistics, show that the reconstructed high-resolution data agree well with the reference DNS data. The findings also indicate that the proposed 3D-ESRGAN can reconstruct 3D high-resolution turbulent flows even with limited training data.

No abstract available

A super-resolution reconstruction model for the subgrid scale (SGS) turbulent flow field in large-eddy simulation (LES) is proposed, and it is called the meta-learning deep convolutional neural network (MLDCNN). Direct numerical simulation (DNS) data of isotropic turbulence are used as the dataset of the model. The MLDCNN is an unsupervised learning model, which only includes high-resolution DNS data without manually inputting preprocessed low-resolution data. In this model, the training process adopts the meta-learning method. First, in the a priori test, the SGS turbulent flow motions in the filtered DNS (FDNS) flow field are reconstructed, and the energy spectrum and probability density function of the velocity gradient of the DNS flow field are reconstructed with high accuracy. Then, in the a posteriori test, the super-resolution reconstruction of the LES flow field is carried out. The difficulty of LES flow field reconstruction is that it contains filtering loss and subgrid model errors relative to the DNS flow field. The super-resolution reconstruction of the LES flow field achieves good results through this unsupervised learning model. The proposed model makes a good prediction of small-scale motions in the LES flow field. This work improves the prediction accuracy of LES, which is crucial for the phenomena dominated by small-scale motions, such as relative motions of particles suspended in turbulent flows.

Two deep learning (DL) models addressing the super-resolution (SR) reconstruction of turbulent flows from low-resolution coarse flow field data are developed. One is the static convolutional neural network (SCNN), and the other is the novel multiple temporal paths convolutional neural network (MTPC). The SCNN model takes instantaneous snapshots as an input, while the MTPC model takes a time series of velocity fields as an input, and it includes spatial and temporal information simultaneously. Three temporal paths are designed in the MTPC to fully capture features in different time ranges. A weight path is added to generate pixel-level weight maps of each temporal path. These models were first applied to forced isotropic turbulence. The corresponding high-resolution flow fields were reconstructed with high accuracy. The MTPC seems to be able to reproduce many important features as well, such as kinetic energy spectra and the joint probability density function of the second and third invariants of the velocity gradient tensor. As a further evaluation, the SR reconstruction of anisotropic channel flow with the DL models was performed. The SCNN and MTPC remarkably improve the spatial resolution in various wall regions and potentially grasp all the anisotropic turbulent properties. It is also shown that the MTPC supplements more under-resolved details than the SCNN. The success is attributed to the fact that the MTPC can extract extra temporal information from consecutive fluid fields. The present work may contribute to the development of the subgrid-scale model in computational fluid dynamics and enrich the application of SR technology in fluid mechanics.Two deep learning (DL) models addressing the super-resolution (SR) reconstruction of turbulent flows from low-resolution coarse flow field data are developed. One is the static convolutional neural network (SCNN), and the other is the novel multiple temporal paths convolutional neural network (MTPC). The SCNN model takes instantaneous snapshots as an input, while the MTPC model takes a time series of velocity fields as an input, and it includes spatial and temporal information simultaneously. Three temporal paths are designed in the MTPC to fully capture features in different time ranges. A weight path is added to generate pixel-level weight maps of each temporal path. These models were first applied to forced isotropic turbulence. The corresponding high-resolution flow fields were reconstructed with high accuracy. The MTPC seems to be able to reproduce many important features as well, such as kinetic energy spectra and the joint probability density function of the second and third invariants of the veloc...

No abstract available

To overcome the limitations of traditional convolutional and recurrent neural networks in capturing spatio-temporal dynamics in flow fields on unstructured grids, this study proposes a novel Spatio-Temporal Graph Neural Network (ST-GNN) model that integrates a Graph Neural Network (GNN) with a Long Short-Term Memory (LSTM) network. The GNN component captures spatial dependencies among irregular grid nodes via message passing, while the LSTM component models temporal evolution through gated memory mechanisms. This hybrid framework enables the joint learning of spatial and temporal features in complex flow systems. Two variants of ST-GNN, namely, GCN-LSTM and GAT-LSTM, were developed and evaluated using the SUBOFF AFF-8 benchmark dataset. The results show that GAT-LSTM achieved higher accuracy than GCN-LSTM, with average relative errors of 2.51% for velocity and 1.43% for pressure at the 1000th time step. Both models achieved substantial speedups over traditional CFD solvers, with GCN-LSTM and GAT-LSTM accelerating predictions by approximately 350 and 150 times, respectively. These findings position ST-GNN as an efficient and accurate alternative for spatio-temporal flow modeling on unstructured grids, advancing data-driven fluid dynamics.

This study introduces a Spatio-Temporal Graph Neural Network (ST-GNN) for the multi-step prediction of highly dynamic cylinder water entry flow fields. The proposed model integrates a Graph Neural Network (GNN) to capture complex spatial dependencies on unstructured meshes with a Long Short-Term Memory (LSTM) network to model temporal evolution. We generate a dataset from numerical simulations across different cases, varying both Froude numbers and initial entry angles. Extensive ablation studies are conducted to systematically optimize the model's architecture and hyperparameters, revealing that a single-step prediction strategy with an input sequence of eight timesteps yields the most robust performance, significantly reducing error accumulation in long-term prediction. The optimized model achieves high accuracy (R2>0.97) and successfully predicts key physical phenomena during water entry. Furthermore, generalization tests on unseen cases demonstrate the model's excellent interpolation capability within the training data range and reveal the challenges of extrapolation, particularly for the complex phase interface. Crucially, while the multi-step prediction of the velocity field degrades due to error accumulation, the inferred prediction of the water volume fraction remains remarkably stable. This indicates that the model has learned not just the instantaneous spatial correlations between the fields, but more importantly, has captured the underlying temporal dynamics that govern the evolution of the phase interface.

No abstract available

This study introduces a novel physics-informed spatiotemporal graph neural network (PIST-GNN) for rapid, physically consistent, and accurate long-term prediction of complex water entry flow fields. The model integrates a MeshGraphNet-based GNN for spatial feature extraction on unstructured meshes and a long short-term memory (LSTM) network for temporal evolution, critically enhanced by embedding mass conservation and volume of fluid (VOF) transport equations as soft constraints during training. Comprehensive evaluations demonstrate that the combined physics constraints provide an optimal balance, achieving robust predictions with only a slight increase in data loss while significantly reducing physical inconsistencies like velocity divergence and VOF advection residuals. The PIST-GNN exhibits superior long-term stability and generalization capabilities, consistently maintaining the lowest prediction errors and the best physical consistency over 30 autoregressive steps on unseen Froude number and initial angle cases. Furthermore, through both quantitative and qualitative analysis of prediction results, we elucidate the complex influence of initial conditions on prediction accuracy. Ultimately, this study provides a practical and feasible approach to address the persistent challenge of error accumulation in long-term predictions of water entry flow fields by purely data-driven models. Future research will focus on overcoming the limitations of the current two-dimensional slice approximation method by optimizing physical constraints and model architecture, thereby more directly capturing the complete three-dimensional dynamic characteristics of the water entry flow field.

Pin-fins are widely used to enhance heat transfer in compact heat exchangers, turbine cooling passages, and electronic devices, but their complex geometries make accurate thermal–fluid prediction computationally expensive. This paper presents a geometry-aware multiscale (GAMS) graph neural network (GNN) for predicting steady turbulent flow and heat transfer in a two-dimensional channel containing arbitrarily shaped pin-fin geometries. An automated framework integrating geometry generation, meshing, and ANSYS Fluent simulations was developed to construct the training dataset. Pin-fin geometries were parameterized using piecewise cubic splines, generating 1000 unique configurations through Latin Hypercube Sampling. Each simulation was converted into a graph representation, where nodes contained spatial coordinates, normalized streamwise position, one-hot boundary indicators, and signed distance to the nearest wall. These graph-based features were used to train the GNN to predict the temperature, velocity magnitude, and pressure fields directly from geometry. The network achieved excellent predictive accuracy, successfully capturing boundary layers, recirculation zones, and upstream stagnation regions while reducing computational wall time by 2–3 orders of magnitude compared to conventional CFD simulations. Overall, the proposed GNN provides a fast, reliable surrogate modeling framework for complex thermal–fluid flow configurations.

This paper introduces a novel surrogate model for two-dimensional adaptive steady-state thermal convection fields based on deep learning technology. The proposed model aims to overcome limitations in traditional frameworks caused by network types, such as the requirement for extensive training data, accuracy loss due to pixelated preprocessing of original data, and inability to predict information near the boundaries with precision. We propose a new framework that consists primarily of a physical-informed neural network (PINN) and a graph convolutional neural network (GCN). The GCN serves as the prediction module and predicts thermal convection in the two-dimensional computational domain by considering the mutual influence between unstructured nodes and their neighbors. On the other hand, the PINN acts as the physical constraint module of the framework by embedding the control equation of thermal convection into the loss function of the neural network, ensuring that the inference and prediction results of the GCN comply with the constraints of the control equation. The advantages of this framework lie in two aspects. First, the computation mechanism of the GCN is more in line with the actual evolution of temperature fields. Second, the PINN enhances the cognitive ability of the surrogate model toward the convection field information. It accurately describes the changes of temperature gradient information at the boundary position and reduces the model's demand for training data. To validate the advantages of the proposed model, we gradually analyzed the model's geometric adaptability and predictive accuracy from the single cylinder case to the double cylinder case. We also investigated the impact of the number of sampling points on model training and compared the model's prediction results with those of a purely data-driven model. The results show that the proposed model exhibits good geometric adaptability and stability. With only 20 training data, the mean error of the proposed model in predicting the velocity and temperature field is less than 1% and 0.6% for the single cylinder, and less than 2% and 1% for the double cylinder case, while the mean error of the purely data-driven GCN model in predicting the velocity and temperature field is 9.4% and 6.4% for the double cylinder case. These findings demonstrate the effectiveness of the proposed physics-informed graph convolutional neural network, allowing for more accurate prediction of fluid flow and heat convection using surrogate model.

In the interdisciplinary field of data-driven models and computational fluid mechanics, the reduced-order model for flow field prediction is mainly constructed by a convolutional neural network (CNN) in recent years. However, the standard CNN is only applicable to data with Euclidean spatial structure, while data with non-Euclidean properties can only be convolved after pixelization, which usually leads to decreased data accuracy. In this work, a novel data-driven framework based on graph convolution network (GCN) is proposed to allow the convolution operator to predict fluid dynamics on non-uniform structured or unstructured mesh data. This is achieved by the fact that the graph data inherit the spatial characteristics of the mesh and by the message passing mechanism of GCN. The conversion method from the form of mesh data to graph data and the operation mechanism of GCN are clarified. Moreover, additional relevance features and weight loss function of the dataset are also investigated to improve the model performance. The model learns an end-to-end mapping between the mesh spatial features and the physical flow field. Through our studies of various cases of internal flow, it is shown that the proposed GCN-based model offers excellent adaptability to non-uniformly distributed mesh data, while also achieving a high accuracy and three-order speedup compared with numerical simulation. Our framework generalizes the graph convolution network to flow field prediction and opens the door to further extending GCN to most existing data-driven architectures of fluid dynamics in the future.

In this study, we propose a graph neural network (GNN)-based numerical model for efficiently predicting the flow behavior of non-Newtonian fluids with free surface dynamics. Conventional numerical approaches, such as the finite element method, often suffer from computational inefficiency and convergence issues when simulating non-Newtonian fluids whose viscosity varies dynamically with the local shear rate. To overcome these limitations, particle-based methods such as smoothed particle hydrodynamics (SPH) have been developed, offering greater robustness for complex free-surface flows. However, SPH remains constrained by hardware-dependent scalability, as even parallelization techniques using graphics processing units (GPUs) are limited by system-specific constraints and computational cost. To address these challenges, we extend the SPH framework by introducing a graph neural network (GNN)-based model, which achieves efficient, data-driven prediction of non-Newtonian fluid dynamics. In this work, we focus on power-law fluids, one of the simplest models of non-Newtonian behavior. Our GNN model is trained on SPH simulation data, learning the effects of particle accelerations in the presence of SPH particle interactions based on the fluid's power-law model parameters. We demonstrate that the GNN significantly accelerates computations while maintaining reliable accuracy in benchmark tests, including dam-break and droplet impact simulations. Quantitative evaluations demonstrate that the GNN reduces computation time by approximately 30% compared to conventional SPH simulations, while maintaining comparable accuracy across benchmark problems such as dam-break and droplet impact cases. The results highlight the potential of GNN-based frameworks for simulating non-Newtonian free-surface flows, paving the way for future data-driven non-Newtonian fluid modeling.

No abstract available

A novel Spatiotemporal Sequence Graph Convolutional Network (ST-SGCN) data-driven model is proposed to predict transient fluid dynamics around airfoils using complex and unstructured flow field data, with the aim of reducing dimensions and expediting predictions. Graph Neural Networks directly interact with the flow field grid, capturing spatiotemporal physical features of grid nodes and their interconnections, while eliminating the need for complex preprocessing steps. The ST-SGCN model integrates a Graph Convolutional Network and a Graph Attention Network with a Deep Recurrent Neural Network that uses a Gate Recurrent Unit as the kernel, adeptly extracting spatial and temporal physical features of the flow field to accurately predict transient flow states. Preliminary airfoil flow experiments demonstrated the model's ability to continuously predict transient flow fields, achieving an average accuracy of 97% for both velocity and pressure field predictions, with a maximum error of approximately 10% in the testing dataset. Further experiments, varying angles of attack, airfoils, and Reynolds numbers, demonstrated the model's generalizability, extensibility, and adaptability, with prediction errors below 5% and a speedup of over 20 times.

Solving partial differential equations of complex physical systems is a computationally expensive task, especially in Computational Fluid Dynamics(CFD). This drives the application of deep learning methods in solving physical systems. There exist a few deep learning models that are very successful in predicting flow fields of complex physical models, yet most of these still exhibit large errors compared to simulation. Here we introduce AMGNET, a multi-scale graph neural network model based on Encoder-Process-Decoder structure for flow field prediction. Our model employs message passing of graph neural networks at different mesh graph scales. Our method has significantly lower prediction errors than the GCN baseline on several complex fluid prediction tasks, such as airfoil flow and cylinder flow. Our results show that multi-scale representation learning at the graph level is more effective in improving the prediction accuracy of flow field.

In the fish passage facility design, understanding the coupled effects of hydrodynamics on fish behaviour is particularly important. The flow field caused by fish movement however are usually obtained via time-consuming transient numerical simulation. Hence, a hybrid deep neural network (HDNN) approach is designed to predict the unsteady flow field around fish. The basic architecture of HDNN includes the UNet convolution (UConv) module and the bidirectional convolutional long-short term memory (BiConvLSTM) module. Specifically, the UConv module extracts crucial features from the flow field graph, while the BiConvLSTM module learns the evolution of low-dimensional spatio-temporal features for prediction. The numerical results showcase that the HDNN achieves accurate multi-step rolling predictions of the effect of fish movement on flow fields under different tail-beat frequency conditions. Specifically, the average and standard deviation of PSNR and SSIM for the proposed HDNN model for 60 time-step rolling predictions on the entire sequences of four test sets being respectively larger than 34 dB and 0.9. The HDNN delivers a speedup of over 130 times compared to the numerical simulator. Moreover, the HDNN demonstrates commendable generalisation capabilities, enabling the prediction of spatial–temporal evolution within unsteady flow fields even at unknown tail-beat frequencies.

No abstract available

This study employs a finite-element inspired hypergraph neural network surrogate model to investigate the coupled dynamics of a freely oscillating cylinder in fluid flow, a well-established benchmark problem in vortex-induced and wake-induced vibration research. The presented methodology integrates high-fidelity numerical simulations based on variational finite-element methods with graph neural network surrogate modeling to forecast the spatio-temporal dynamics of flow-induced oscillations. The surrogate model utilizes graph-based representations to capture intricate fluid-structure interactions (FSIs), incorporating a multi-layer perceptron for mesh displacement updates and a hypergraph neural network for fluid-state predictions. Quantitative analyses demonstrate that the model could effectively learn from high-fidelity simulation data and provide stable and accurate predictions on unseen test data. These results underscore the potential of graph neural network-based methods for predicting fluid-structure interactions. This research supports the development of physics-based digital twins in marine and offshore engineering applications.

Computational fluid dynamics (CFD) discretizes the Navier-Stokes equations on computational meshes to obtain flow-field quantities like velocity, pressure, and density through numerical iteration, but traditional solvers, though reliable in accuracy, are computationally expensive and time-consuming for high-Reynolds-number turbulence and complex unsteady flows. With geometric deep learning, graph neural networks (GNNs) overcome the limitation of conventional neural network (CNNs) that only handle grid-structured Euclidean data; by message-passing, they naturally fit unstructured CFD meshes and capture spatial correlations. This paper systematically reviews GNN fundamentals and core fluid mechanics, including turbulence characteristics and mesh type classifications, and then focuses on typical GNN applications in flow-field prediction, turbulence modeling, and complex-flow simulation. Existing challenges such as high training costs, insufficient generalization, and limited computational efficiency are analyzed, and promising future research directions are discussed. The review indicates GNNs are efficient surrogates for traditional solvers, offering a pathway to intelligent, fast, high-fidelity simulation.

The urban flow wind field is a critical element for downstream research, such as mitigation of urban wind disasters, assessment of urban wind environment, and urban drone route planning. However, it is impractical to arrange a large number of sensors to monitor an urban wind flow field. Hence, acquiring the entire urban wind flow field via sparse sensors would be highly valuable. To date, no scheme including deep learning (DL) model has been specifically designed for this purpose. This study presents an innovative approach to reconstruct complex high‐resolution urban wind fields based on sparse sensors, using a physics‐informed graph neural network (GNN)‐assisted auto‐encoder. The proposed method leverages the relationship between sensors and their surrounding environment enabled by deep mining capabilities of GNNs. As a result, the utilization and emphasis on sparse sensors data are significantly enhanced. The continuity equation of fluid flow is incorporated into the loss function of the convolution neural network to improve the stability and performance of the model. The findings suggest that, in contrast to prevalent generative DL models, the proposed model yields an approximate 50% reduction in root mean square error for reconstructing high‐resolution urban wind fields for multiple wind attack angles.

Accurate prediction of the flow field is crucial to evaluating the aerodynamic performance of an aircraft. While traditional computational fluid dynamics (CFD) methods solve the governing equations to capture both global flow structures and localized gradients, they are computationally intensive. Deep learning-based surrogate models offer a promising alternative, yet often struggle to simultaneously model long-range dependencies and near-wall flow gradients with sufficient fidelity. To address this challenge, this paper introduces the Message-passing And Global-attention block (MAG-BLOCK), a graph neural network module that combines local message passing with global self-attention mechanisms to jointly learn fine-scale features and large-scale flow patterns. Building on MAG-BLOCK, we propose FLOW-GLIDE, a cross-architecture deep learning framework that learns a mapping from initial conditions to steady-state flow fields in a latent space. Evaluated on the AirfRANS dataset, FLOW-GLIDE outperforms existing models on key performance metrics. Specifically, it reduces the error in the volumetric flow field by 62% and surface pressure prediction by 82% compared to the state-of-the-art.

High-fidelity computational fluid dynamics (CFD) remains computationally expensive for steady aerodynamic prediction under multi-condition and variable-geometry configurations, which limits rapid design iteration. To address this issue, this study proposes a data-driven surrogate framework for aircraft surface flow-field prediction on irregular meshes. The framework combines a geometry-unification strategy for variable rudder-deflection configurations with KM-GAT, a hybrid neural architecture that integrates graph attention and KAN-based nonlinear feature transformation. Geometry unification maps the surface flow fields associated with different rudder-deflection states onto a common zero-deflection reference template, thereby establishing consistent mesh correspondence and fixed prediction locations across samples while retaining the rudder angle as an operating-condition variable. The KM-GAT model further combines topology-aware message passing with localized nonlinear refinement, while the Huber loss is adopted to improve training robustness for CFD-derived data. Experiments on the F-22 research model show that the proposed framework achieves lower prediction errors and more concentrated error distributions than baseline MLP and GNN-based models. Qualitative comparisons further indicate that KM-GAT better preserves localized high-gradient structures, including pressure transitions and vortex-dominated regions. These results suggest that the proposed framework provides an effective surrogate modeling strategy for variable-geometry aerodynamic flow field prediction.

In experimental measurements of flow fields, the acquired data often fail to adequately resolve the fine structural characteristics of complex flows due to limitations in current technical conditions and measurement techniques. This constraint is particularly pronounced in analyzing turbulent random fluctuations and identifying small-scale vortices, hindering in-depth investigations into flow field dynamics. To address this challenge, this study proposes a novel super-resolution reconstruction method for flow fields based on pooling decomposition and a vortex attention mechanism. By introducing an innovative attention mechanism for high-order flow field quantities, the model effectively focuses on vortices within the flow field and extracts high-order information to guide velocity field reconstruction. Simultaneously, a pooling decomposition strategy is employed to specifically learn the null-space components of the flow field while preserving range-space components to enhance reconstruction performance. Notably, the introduction of a physics-constrained loss function ensures kinetic energy spectrum consistency in the reconstructed results. Experimental results demonstrate that the proposed model significantly improves the accuracy of flow field data reconstruction, exhibiting outstanding performance in turbulent kinetic energy spectrum recovery and small-scale vortex reconstruction, along with excellent cross-condition generalization capability. This research provides a new technical approach to overcome the resolution limitations of experimental measurements and achieve high-precision analysis of complex flow structures.

Traditional super-resolution reconstruction methods for flow fields use end-to-end mapping to determine the relationship between high- and low-resolution flow field data. The reconstruction quality of these methods depends on the accuracy of the low-resolution data. Ensuring the accuracy of low-resolution data has, thus, become a precondition for super-resolution tasks, and it imposes strict limitations on the applicability of super-resolution reconstruction methods in practical engineering applications. This paper proposes a flow field super-resolution reconstruction method coupled with feature recognition (FRNet) to reduce the dependence on the accuracy of low-resolution data. FRNet uses a feature extractor with identification capabilities to determine the effectiveness of low-resolution flow field characteristics. It recognizes the effective characteristics using a feature distance distribution. Meanwhile, a representation of the obstacle shape and freestream information is introduced to compensate for invalid features and to suppress the influence of low-precision flow field characteristics on the reconstruction results. Different downsampling factors, different density grids, and noise are used to simulate a variety of engineering application scenarios to verify the effectiveness and applicability of the proposed method. The results demonstrate that FRNet has significant advantages over traditional super-resolution reconstruction methods. Our method does not rely on the accuracy of low-resolution data and can effectively mitigate the impact of low-resolution flow field data that do not conform to physical phenomena. This characteristic allows FRNet to exhibit outstanding performance when handling flow field data affected by noise from wind tunnel wall and rack interferences. Consequently, FRNet should prove highly beneficial for the optimization of complex flow fields using super-resolution reconstruction methods.

This study proposes a perception-augmented hierarchical learning framework (PAHLF) for reconstructing unsteady combustion processes in scramjets from sparse pressure signals, addressing limitations in capturing transient dynamics under extreme flight conditions. Six Mach 7.0 wind-tunnel tests with kerosene combustion generated synchronized pressure–schlieren datasets across varied equivalence ratios and multiport injections. PAHLF integrates spectral–pixel domain co-augmentation for feature enhancement, a multiscale feature-mapping network for combustion dynamics extraction, and a focused training strategy to resolve shock–flame interactions. Validated against the architectures of Chen et al. (“Intelligent Reconstruction of the Flow Field in a Supersonic Combustor Based on Deep Learning,” Physics of Fluids, Vol. 34, No. 3, 2022, Paper 035128) and Kong et al. (“Data-Driven Super-Resolution Reconstruction of Supersonic Flow Field by Convolutional Neural Networks,” AIP Advances, Vol. 11, No. 6, 2021, Paper 065021) via dual-axis evaluation metrics, PAHLF achieves absolute gains of [Formula: see text] in peak signal-to-noise ratio and [Formula: see text] in structural similarity while reducing parameters by [Formula: see text] (80.9 MB vs [Formula: see text]). The framework demonstrates superior capability in maintaining flowfield structural integrity, precisely mapping combustion intensity, and stabilizing predictions during transient evolution with intense shock oscillations. By enabling high-fidelity visualization of unsteady combustion from minimal sensors, PAHLF establishes new potential for real-time monitoring of hypersonic combustors. This approach advances intelligent modeling of combustion characteristics and provides foundational support for active combustion control through early flame-stability prediction in scramjet propulsion systems.

No abstract available

Deep learning-based super-resolution ultrasound (DL-SRU) framework has been successful in improving spatial resolution and measuring the velocity field information of a blood flows by localizing and tracking speckle signals of red blood cells (RBCs) without using any contrast agents. However, DL-SRU can localize only a small part of the speckle signals of blood flow owing to ambiguity problems encountered in the classification of blood flow signals from ultrasound B-mode images and the building up of suitable datasets required for training artificial neural networks, as well as the structural limitations of the neural network itself. An artificial intelligence-based speckle featurization and localization (AI-SFL) framework is proposed in this study. It includes a machine learning-based algorithm for classifying blood flow signals from ultrasound B-mode images, dimensionality reduction for featurizing speckle patterns of the classified blood flow signals by approximating them with quantitative values. A novel and robust neural network (ResSU-net) is trained using the online data generation (ODG) method and the extracted speckle features. The super-resolution performance of the proposed AI-SFL and ODG method is evaluated and compared with the results of previous U-net and conventional data augmentation methods under in silico conditions. The predicted locations of RBCs by the AI-SFL and DL-SRU for speckle patterns of blood flow are applied to a PTV algorithm to measure quantitative velocity fields of the flow. Finally, the feasibility of the proposed AI-SFL framework for measuring real blood flows is verified under in vivo conditions.

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.

Positron emission tomography (PET) technology, with its advantages of strong γ-photon penetration and results unaffected by temperature or electromagnetic fields, has emerged as a novel non-contact monitoring technique for industrial flow fields under harsh conditions. However, dynamic sampling leads to a severe lack of photon data within individual time frames, resulting in an ill-posed nature of positron image reconstruction, which introduces uncertainty in noise statistical characteristics and degradation in imaging quality. This paper proposes a novel noise-suppressing super-resolution enhancement module for positron flow field imaging. The module, based on convolution and SwinTransformer structures, achieves noise reduction and enhancement of positron images under conditions of severe photon scarcity. Furthermore, a multi-loss fusion performance evaluation system is constructed to extract texture and hierarchical feature information from the images. Experimental results demonstrate that the proposed module effectively reduces image noise while preserving critical information, achieving significant improvements in the quality of generated positron flow field images.

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.

The coalescence of liquid droplets and lenses is of great practical and theoretical importance in fluid dynamics and the statistical mechanics of multiphase flows. During such coalescence, there is an interesting and intricate interplay between the shapes of the interfaces, separating two phases, and the background flow field. In experiments, it is easier to visualize concentration fields than to obtain the flow field. We demonstrate that two-dimensional (2D) encoder-decoder CNNs, 2D U-Nets, and three-dimensional (3D) U-Nets can be used to obtain flow fields from concentration fields here. To train these networks, we use concentration and flow fields from our numerical simulations of the Cahn-Hilliard-Navier-Stokes equations. We show that, given test images of concentration fields, our trained models accurately predict the flow fields. Finally, we use data from recent experiments on droplet coalescence to show how our method can be used to obtain the flow field from measurements of the concentration field. Liquid-droplet coalescence and liquid-lens mergers are relevant in pharmaceuticals, oil recovery, food industries, and climate-system science. In order to obtain flow fields from experimentally measured concentration fields in both viscous and inertial regimes, this paper offers a framework of convolutional neural network machine learning models trained using data from direct numerical simulations.

Cerebral aneurysms (CAs) can have adverse consequences if ruptured, including severe disability or death. Minimally invasive endovascular procedures to treat cerebral aneurysms involve the intricate placement of coiling or flow diverters guided by 2D angiography; however, retreatment is often necessary. However, cerebral vasculature and the corresponding hemodynamics are highly three-dimensional, with complex 3D curvature and tortuosity. Current imaging techniques used during procedures are 2D projections that ‘flatten’ the complex flow characteristics. In this study, we developed a novel method for using three orthogonal angiographic image projections to reconstruct the 3D volumetric angiographic contrast transport in cerebral angiography from sparse angiographic data using a convolutional neural network. We utilize a large dataset of aneurysm geometries of patient-specific cerebral aneurysms and surrounding vasculature with complex morphologies and sizes. For each aneurysm, angiography was simulated using a transient computational fluid dynamic model quantifying blood and contrast flow characteristics. Virtual angiographic images were obtained using 2D projections of the gridded data in 3 orthogonal directions for 2,639 unique time frames. The 2D projected virtual angiograms and an anatomical mask were inputs to a custom-built model architecture. We developed the 3D Angiographic Reconstruction Neural Network as a 3D convolutional neural network that reconstructs contrast in 3D in a frame-by-frame approach. For the tested patient-specific models, the heterogeneous 3D nature of the contrast was identified, and the contrast transport through the vasculature was successfully reconstructed across successive time frames. The average mean square error across all the unseen test patients was 5.27E-04. Providing clinicians with additional information regarding 3D CA hemodynamics can improve the assessment and treatment of CAs.

ABSTRACT

This paper investigates deep learning methods in the framework of convolutional neural networks for reconstructing compressible turbulent flow fields. The aim is to develop methods capable of up-scaling coarse turbulent data into fine-resolution images. The method is based on a parallel computational framework that accepts five image sets of various resolutions, trained to correspond to the respective fine resolution. The network architecture mainly consists of convolutional layers, constructing an encoder/decoder network. Based on the U-Net scheme, three different implementations are presented, with residual and skip connections. The methods are implemented in a supersonic shock-boundary-layer interaction problem. The results suggest that simple networks perform better when trained on limited data, and this can be a practical and fast solution when dealing with turbulent flow data, where the computational burden is most of the time difficult to decrease. In such a way, a coarse simulation grid can be upscaled to a fine grid.

Deep learning technologies have been widely used in fluid data processing to reconstruct various flow fields. However, due to the complex particle dynamics, relying exclusively on data-driven methods lacks reflection of physical mechanisms. In this article, an electron density reconstruction model of sensor data based on a physics-informed convolutional transposed neural network (PICTNN) is proposed. Employing the continuity equation of plasmas, a physics-informed loss function is constructed to enhance model stability during training through logarithmic maximum normalization. As a validation of the method, based on the density dataset of wakes obtained using the computational fluid dynamics method, the 2-D reconstruction of plasma wakes under different Mach numbers and angles of attack (AOAs) is tested. The results demonstrate excellent preservation of physical features, with Pearson correlation coefficients between the reconstructed data and the computational fluid dynamics simulations reaching up to 0.95. Additionally, this model has been successfully applied to reconstruct 2-D wake distributions from 1-D measurement data. The wake electron density reconstruction model may enhance the effective use of experimental data and extend the measurement capabilities of hypersonic wake devices, offering significant engineering implications.

We propose a spatiotemporal coupling deep neural network approach for time-resolved reconstruction of the velocity field around a circular cylinder. The neural network leverages two distinct data types: (1) non-time-resolved velocity field around the cylinder, consisting of fixed frequency sampling and variable frequency sampling velocity field, and (2) the time-resolved surface pressure sequence around the cylinder. The deep neural network comprises two sub-networks: a convolutional autoencoder (CAE) for nonlinear mode extraction and a Transformer for sequence-to-sequence learning. We refer to this architecture as CTNet (CAE-Transformer Network). The encoder in the CAE maps non-time-resolved velocity field to a latent vector, enabling the extraction of nonlinear modal coefficients. An appropriate time window length for the surface pressure sequence is then selected to establish a Transformer sequence learning model, using the chosen sequence as input to predict the corresponding nonlinear modal coefficients. Once the Transformer is well trained, the time-resolved nonlinear modal coefficients of velocity field can be achieved. Along with the well-trained decoder in the CAE, the time-resolved velocity field can be reconstructed from the output of the Transformer. We verify the performance of CTNet by a simulated dataset at a representative Reynolds number of 3900. The results show a relative reconstruction error of just 6.3% for the time-resolved velocity field, demonstrating high reliability in the reconstruction. We further compare the reconstructed velocity field obtained with and without the utilization of variable frequency sampling velocity field. Notably, the inclusion of variable frequency sampling velocity field significantly improves the reconstruction quality.

Artificial intelligence based on neural network technology has provided innovative methods for predicting unsteady flow fields. However, both purely data-driven and single physics-driven methods can only perform short-term predictions for unsteady flow fields and are unable to achieve medium- to long-term predictions. A composite neural network CNN-GRU-PINN (CGPINN) is proposed by combining convolutional neural network (CNN), gated recurrent unit (GRU), and physics-informed neural network (PINN). CNN and GRU are used to learn the spatial and temporal characteristics of unsteady flows, respectively. PINN is adopted to constrain flow field prediction data according to physical laws. The flow around a circular cylinder is employed to verify the medium- and long-term prediction performances of the CGPINN. The test results show that compared to PINN, the reconstruction accuracy of the CGPINN is improved by about 86.10% on average, and the prediction accuracy is improved by about 96.18%. Compared to pure data-driven approaches, the prediction accuracy of the CGPINN is improved by an average of 65.71%. Additionally, CGPINN exhibits better robustness, demonstrating insensitivity to variations in sample size and noise levels, thereby ensuring stable and reliable performances across diverse data conditions. This study has provided a more accurate and robust method for the reconstruction and prediction of unsteady flow fields.

Reconstruction of field quantities from sparse measurements is a problem arising in a broad spectrum of applications. This task is particularly challenging when the mapping between sparse measurements and field quantities is performed in an unsupervised manner. Further complexity is added for moving sensors and/or random on-off status. Under such conditions, the most straightforward solution is to interpolate the scattered data onto a regular grid. However, the spatial resolution achieved with this approach is ultimately limited by the mean spacing between the sparse measurements. In this work, we propose a super-resolution generative adversarial network (GAN) framework to estimate field quantities from random sparse sensors without needing any full-field high-resolution training. The algorithm exploits random sampling to provide incomplete views of the {high-resolution} underlying distributions. It is hereby referred to as RAndomly-SEEDed super-resolution GAN (RaSeedGAN). The proposed technique is tested on synthetic databases of fluid flow simulations, ocean surface temperature distributions measurements, and particle image velocimetry data of a zero-pressure-gradient turbulent boundary layer. The results show excellent performance even in cases with high sparsity or with levels of noise. To our knowledge, this is the first GAN algorithm for full-field high-resolution estimation from randomly-seeded fields with no need of full-field high-resolution representations.

Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where $128\times128$ vorticity fields are reconstructed from extremely coarse $8\times8$ observations representing a $16\times$ downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative $\ell_2$ error of $26.04\%$ with the lowest error variance among all methods, reducing reconstruction error by $31.7\%$ over FNO, $26.0\%$ over EDSR, and $9.3\%$ over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.

The accurate prediction of small scales in underresolved flows is still one of the main challenges in predictive simulations of complex configurations. Over the last few years, data-driven modeling has become popular in many fields as large, often extensively labeled datasets are now available and training of large neural networks has become possible on graphics processing units (GPUs) that speed up the learning process tremendously. In fact, the successful application of deep neural networks in fluid dynamics, such as for underresolved reactive flows, is still challenging. This work advances the recently introduced PIESRGAN to reactive finite-rate-chemistry flows. However, since combustion chemistry typically acts on the smallest scales, the original approach needs to be extended. Therefore, the modeling approach of PIESRGAN is modified to accurately account for the challenges in the context of laminar finite-rate-chemistry flows. The modified PIESRGAN-based model gives good agreement in a priori and a posteriori tests in a laminar lean premixed combustion setup. Furthermore, a reduced PIESRGAN-based model is presented that solves only the major species on a reconstructed field and employs PIERSGAN lookup for the remaining species, utilizing staggering in time. The advantages of the discriminator-supported training are shown, and the usability of the new model demonstrated in the context of a model gas turbine combustor.

Changes in cardiovascular hemodynamics are closely related to the development of aortic regurgitation, a type of valvular heart disease. Metrics derived from blood flows are used to indicate aortic regurgitation onset and evaluate its severity. These metrics can be non-invasively obtained using four-dimensional (4D) flow magnetic resonance imaging (MRI), where accuracy is primarily dependent on spatial resolution. However, insufficient resolution often results from limitations in 4D flow MRI and complex aortic regurgitation hemodynamics. To address this, computational fluid dynamics simulations were transformed into synthetic 4D flow MRI data and used to train a variety of neural networks. These networks generated super resolution, full-field phase images with an upsample factor of 4. Results showed decreased velocity error, high structural similarity scores, and improved learning capabilities from previous work. Further validation was performed on two sets of in-vivo 4D flow MRI data and demonstrated success in de-noising flow images. This approach presents an opportunity to comprehen-sively analyse aortic regurgitation hemodynamics in a non-invasive manner.

In this paper, a novel deep learning framework is proposed for temporal super-resolution simulation of blood vessel flows, in which a high-temporal-resolution time-varying blood vessel flow simulation is generated from a low-temporal-resolution flow simulation result. In our framework, point-cloud is used to represent the complex blood vessel model, resistance-time aided PointNet model is proposed for extracting the time-space features of the time-varying flow field, and finally we can reconstruct the high-accuracy and high-resolution flow field through the Decoder module. In particular, the amplitude loss and the orientation loss of the velocity are proposed from the vector characteristics of the velocity. And the combination of these two metrics constitutes the final loss function for network training. Several examples are given to illustrate the effective and efficiency of the proposed framework for temporal super-resolution simulation of blood vessel flows.

Turbulence is still one of the main challenges for accurately predicting reactive flows. Therefore, the development of new turbulence closures which can be applied to combustion problems is essential. Data-driven modeling has become very popular in many fields over the last years as large, often extensively labeled, datasets became available and training of large neural networks became possible on GPUs speeding up the learning process tremendously. However, the successful application of deep neural networks in fluid dynamics, for example for subgrid modeling in the context of large-eddy simulations (LESs), is still challenging. Reasons for this are the large amount of degrees of freedom in realistic flows, the high requirements with respect to accuracy and error robustness, as well as open questions, such as the generalization capability of trained neural networks in such high-dimensional, physics-constrained scenarios. This work presents a novel subgrid modeling approach based on a generative adversarial network (GAN), which is trained with unsupervised deep learning (DL) using adversarial and physics-informed losses. A two-step training method is used to improve the generalization capability, especially extrapolation, of the network. The novel approach gives good results in a priori as well as a posteriori tests with decaying turbulence including turbulent mixing. The applicability of the network in complex combustion scenarios is furthermore discussed by employing it to a reactive LES of the Spray A case defined by the Engine Combustion Network (ECN).

We introduce FLEX (FLow EXpert), a backbone architecture for generative modeling of spatio-temporal physical systems using diffusion models. FLEX operates in the residual space rather than on raw data, a modeling choice that we motivate theoretically, showing that it reduces the variance of the velocity field in the diffusion model, which helps stabilize training. FLEX integrates a latent Transformer into a U-Net with standard convolutional ResNet layers and incorporates a redesigned skip connection scheme. This hybrid design enables the model to capture both local spatial detail and long-range dependencies in latent space. To improve spatio-temporal conditioning, FLEX uses a task-specific encoder that processes auxiliary inputs such as coarse or past snapshots. Weak conditioning is applied to the shared encoder via skip connections to promote generalization, while strong conditioning is applied to the decoder through both skip and bottleneck features to ensure reconstruction fidelity. FLEX achieves accurate predictions for super-resolution and forecasting tasks using as few as two reverse diffusion steps. It also produces calibrated uncertainty estimates through sampling. Evaluations on high-resolution 2D turbulence data show that FLEX outperforms strong baselines and generalizes to out-of-distribution settings, including unseen Reynolds numbers, physical observables (e.g., fluid flow velocity fields), and boundary conditions.

Simulating turbulence is critical for many societally important applications in aerospace engineering, environmental science, the energy industry, and biomedicine. Large eddy simulation (LES) has been widely used as an alternative to direct numerical simulation (DNS) for simulating turbulent flows due to its reduced computational cost. However, LES is unable to capture all of the scales of turbulent transport accurately. Reconstructing DNS from low-resolution LES is critical for many scientific and engineering disciplines, but it poses many challenges to existing super-resolution methods due to the spatio-temporal complexity of turbulent flows. In this work, we propose a new physics-guided neural network for reconstructing the sequential DNS from low-resolution LES data. The proposed method leverages the partial differential equation that underlies the flow dynamics in the design of spatio-temporal model architecture. A degradation-based refinement method is also developed to enforce physical constraints and further reduce the accumulated reconstruction errors over long periods. The results on two different types of turbulent flow data confirm the superiority of the proposed method in reconstructing the high-resolution DNS data and preserving the physical characteristics of flow transport.

Spatio-temporal processes often exhibit highly heterogeneous and non-intuitive responses to localized disruptions, limiting the effectiveness of conventional message passing approaches in modeling local heterogeneity. We reformulate spatio-temporal forecasting as the problem of learning information flow over locally structured spaces, rather than propagating globally aligned node representations. To this end, we introduce a spatio-temporal sheaf diffusion graph neural network (ST-Sheaf GNN) that embeds graph topology into sheaf-based vector spaces connected by learned linear restriction maps. Unlike prior approaches relying on static or globally shared transformations, our model learns dynamic restriction maps that evolve over time and adapt to local spatio-temporal patterns, enabling more expressive interactions. The proposed framework both theoretically guarantees and empirically demonstrates evidence that the proposed diffusion mechanism mitigates oversmoothing, preserving discriminative node representations even with increasing diffusion layer depth. Experiments on diverse real-world spatio-temporal forecasting benchmarks across multiple domains demonstrate state-of-the-art performance, highlighting the effectiveness of sheaf topological representations as a principled foundation for spatio-temporal graph learning. The code is available at: https://anonymous.4open.science/r/ST-SheafGNN-6523/.

Effective traffic prediction is a cornerstone of intelligent transportation systems, enabling precise forecasts of traffic flow, speed, and congestion. While traditional spatio-temporal graph neural networks (ST-GNNs) have achieved notable success in short-term traffic forecasting, their performance in long-term predictions remains limited. This challenge arises from over-squashing problem, where bottlenecks and limited receptive fields restrict information flow and hinder the modeling of global dependencies. To address these challenges, this study introduces a novel framework that incorporates virtual nodes, which are additional nodes added to the graph and connected to existing nodes, in order to aggregate information across the entire graph within a single GNN layer. Our proposed model incorporates virtual nodes by constructing a semi-adaptive adjacency matrix. This matrix integrates distance-based and adaptive adjacency matrices, allowing the model to leverage geographical information while also learning task-specific features from data. Experimental results demonstrate that the inclusion of virtual nodes significantly enhances long-term prediction accuracy while also improving layer-wise sensitivity to mitigate the over-squashing problem. Virtual nodes also offer enhanced explainability by focusing on key intersections and high-traffic areas, as shown by the visualization of their adjacency matrix weights on road network heat maps. Our advanced approach enhances the understanding and management of urban traffic systems, making it particularly well-suited for real-world applications.

This study aims to overcome the limitations of conventional deep-learning approaches based on convolutional neural networks in complex geometries and unstructured meshes by exploring the potential of Graph U-Nets for unsteady flow-field prediction. We present a comprehensive investigation of Graph U-Nets, originally developed for classification tasks, now tailored for mesh-agnostic spatio-temporal forecasting of fluid dynamics. Our focus is on enhancing their performance through systematic hyperparameter tuning and architectural modifications. We propose novel approaches to improve mesh-agnostic spatio-temporal prediction of transient flow fields using Graph U-Nets, enabling accurate prediction on diverse mesh configurations. Key enhancements to the Graph U-Net architecture, including the Gaussian-mixture-model convolutional operator and noise injection approaches, provide increased flexibility in modeling node dynamics: the former reduces prediction error by 95\% compared to conventional convolutional operators, while the latter improves long-term prediction robustness, resulting in an error reduction of 86\%. We demonstrate the effectiveness of these enhancements in both transductive and inductive learning settings, showcasing the adaptability of Graph U-Nets to various flow conditions and mesh structures. This work contributes to the field of reduced-order modeling for computational fluid dynamics by establishing Graph U-Nets as a viable and flexible alternative to convolutional neural networks, capable of accurately and efficiently predicting complex fluid flow phenomena across diverse scenarios.

Industrial design in fields such as vehicle and aerospace engineering often relies on large-scale numerical simulations to evaluate fluid dynamics performance, which can incur substantial computational costs. Deep neural networks have shown promise in improving simulation efficiency, especially graph neural networks (GNNs), which demonstrate great potential due to their flexibility with unstructured data. However, GNNs face challenges when dealing with tasks involving complex geometries and large-scale meshes. In this paper, we propose the Multi-scale Feature Enhanced Graph Neural Network (ME-GNN) to tackle these challenges. ME-GNN employs a graph neural network with a two-step message-passing mechanism to capture detailed local features effectively. Additionally, it integrates an Attention U-Net with uniform grid discretization, enabling the extraction of both fine and coarse features. The model also utilizes K-hop sampling to construct subgraphs, facilitating efficient training on large datasets while preserving detailed local features. We evaluated ME-GNN on three benchmark datasets and achieved state-of-the-art results: a relative L2 error of 0.0196 for the velocity field and 0.0556 for the surface pressure on ShapeNet-Car, a normalized mean squared error of 0.0033 for the flow field on AirfRANS, and a relative L2 error of 0.1416 for the surface pressure on DrivAerNet.

Intracranial aneurysms remain a major cause of neurological morbidity and mortality worldwide, where rupture risk is tightly coupled to local hemodynamics particularly wall shear stress and oscillatory shear index. Conventional computational fluid dynamics simulations provide accurate insights but are prohibitively slow and require specialized expertise. Clinical imaging alternatives such as 4D Flow MRI offer direct in-vivo measurements, yet their spatial resolution remains insufficient to capture the fine-scale shear patterns that drive endothelial remodeling and rupture risk while being extremely impractical and expensive. We present a graph neural network surrogate model that bridges this gap by reproducing full-field hemodynamics directly from vascular geometries in less than one minute per cardiac cycle. Trained on a comprehensive dataset of high-fidelity simulations of patient-specific aneurysms, our architecture combines graph transformers with autoregressive predictions to accurately simulate blood flow, wall shear stress, and oscillatory shear index. The model generalizes across unseen patient geometries and inflow conditions without mesh-specific calibration. Beyond accelerating simulation, our framework establishes the foundation for clinically interpretable hemodynamic prediction. By enabling near real-time inference integrated with existing imaging pipelines, it allows direct comparison with hospital phase-diagram assessments and extends them with physically grounded, high-resolution flow fields. This work transforms high-fidelity simulations from an expert-only research tool into a deployable, data-driven decision support system. Our full pipeline delivers high-resolution hemodynamic predictions within minutes of patient imaging, without requiring computational specialists, marking a step-change toward real-time, bedside aneurysm analysis.

Urban wind flow modeling and simulation play an important role in air quality assessment and sustainable city planning. A key challenge for modeling and simulation is handling the complex geometries of the urban landscape. Low order models are limited in capturing the effects of geometry, while high-fidelity Computational Fluid Dynamics (CFD) simulations are prohibitively expensive, especially across multiple geometries or wind conditions. Here, we propose a generative diffusion framework for synthesizing steady-state urban wind fields over unstructured meshes that requires only geometry information. The framework combines a hierarchical graph neural network with score-based diffusion modeling to generate accurate and diverse velocity fields without requiring temporal rollouts or dense measurements. Trained across multiple mesh slices and wind angles, the model generalizes to unseen geometries, recovers key flow structures such as wakes and recirculation zones, and offers uncertainty-aware predictions. Ablation studies confirm robustness to mesh variation and performance under different inference regimes. This work develops is the first step towards foundation models for the built environment that can help urban planners rapidly evaluate design decisions under densification and climate uncertainty.

Cardiovascular disease is the leading global cause of mortality, with coronary artery disease (CAD) as its most prevalent form, necessitating early risk prediction. While 3D coronary artery digital twins reconstructed from imaging offer detailed anatomy for personalized assessment, their analysis relies on computationally intensive computational fluid dynamics (CFD), limiting scalability. Data-driven approaches are hindered by scarce labeled data and lack of physiological priors. To address this, we present PINS-CAD, a physics-informed self-supervised learning framework. It pre-trains graph neural networks on 200,000 synthetic coronary digital twins to predict pressure and flow, guided by 1D Navier-Stokes equations and pressure-drop laws, eliminating the need for CFD or labeled data. When fine-tuned on clinical data from 635 patients in the multicenter FAME2 study, PINS-CAD predicts future cardiovascular events with an AUC of 0.73, outperforming clinical risk scores and data-driven baselines. This demonstrates that physics-informed pretraining boosts sample efficiency and yields physiologically meaningful representations. Furthermore, PINS-CAD generates spatially resolved pressure and fractional flow reserve curves, providing interpretable biomarkers. By embedding physical priors into geometric deep learning, PINS-CAD transforms routine angiography into a simulation-free, physiology-aware framework for scalable, preventive cardiology.

The accurate prediction of flow fields around airfoils is crucial for aerodynamic design and optimisation. Computational Fluid Dynamics (CFD) models are effective but computationally expensive, thus inspiring the development of surrogate models to enable quicker predictions. These surrogate models can be based on deep learning architectures, such as Convolutional Neural Networks (CNNs), Graph Neural Networks (GNNs), and Diffusion Models (DMs). Diffusion models have shown significant promise in predicting complex flow fields. In this work, we propose FoilDiff, a diffusion-based surrogate model with a hybrid-backbone denoising network. This hybrid design combines the power of convolutional feature extraction and transformer-based global attention to generate more adaptable and accurate representations of flow structures. FoilDiff takes advantage of Denoising Diffusion Implicit Model (DDIM) sampling to optimise the efficiency of the sampling process at no additional cost to model generalisation. We used encoded representations of Reynolds number, angle of attack, and airfoil geometry to define the input space for generalisation across a wide range of aerodynamic conditions. When evaluated against state-of-the-art models, FoilDiff shows significant performance improvements, with mean prediction errors reducing by up to 85\% on the same datasets. The results have demonstrated that FoilDiff can provide both more accurate predictions and better-calibrated predictive uncertainty than existing diffusion-based models.

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. 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.

Graph neural networks (GNNs) have shown promise in learning unstructured mesh-based simulations of physical systems, including fluid dynamics. In tandem, geometric deep learning principles have informed the development of equivariant architectures respecting underlying physical symmetries. However, the effect of rotational equivariance in modeling fluids remains unclear. We build a multi-scale equivariant GNN to forecast fluid flow and study the effect of modeling invariant and non-invariant representations of the flow state. We evaluate the model performance of several equivariant and non-equivariant architectures on predicting the evolution of two fluid flows, flow around a cylinder and buoyancy-driven shear flow, to understand the effect of equivariance and invariance on data-driven modeling approaches. Our results show that modeling invariant quantities produces more accurate long-term predictions and that these invariant quantities may be learned from the velocity field using a data-driven encoder.

Neural operators are promising surrogates for dynamical systems but when trained with standard L2 losses they tend to oversmooth fine-scale turbulent structures. Here, we show that combining operator learning with generative modeling overcomes this limitation. We consider three practical turbulent-flow challenges where conventional neural operators fail: spatio-temporal super-resolution, forecasting, and sparse flow reconstruction. For Schlieren jet super-resolution, an adversarially trained neural operator (adv-NO) reduces the energy-spectrum error by 15x while preserving sharp gradients at neural operator-like inference cost. For 3D homogeneous isotropic turbulence, adv-NO trained on only 160 timesteps from a single trajectory forecasts accurately for five eddy-turnover times and offers 114x wall-clock speed-up at inference than the baseline diffusion-based forecasters, enabling near-real-time rollouts. For reconstructing cylinder wake flows from highly sparse Particle Tracking Velocimetry-like inputs, a conditional generative model infers full 3D velocity and pressure fields with correct phase alignment and statistics. These advances enable accurate reconstruction and forecasting at low compute cost, bringing near-real-time analysis and control within reach in experimental and computational fluid mechanics. See our project page: https://vivekoommen.github.io/Gen4Turb/

The paper presents a Graph Attention Convolutional Network (GACN) for flow reconstruction from very sparse data in time-varying geometries. The model incorporates a feature propagation algorithm as a preprocessing step to handle extremely sparse inputs, leveraging information from neighboring nodes to initialize missing features. In addition, a binary indicator is introduced as a validity mask to distinguish between the original and propagated data points, enabling more effective learning from sparse inputs. Trained on a unique data set of Direct Numerical Simulations (DNS) of a motored engine at a technically relevant operating condition, the GACN shows robust performance across different resolutions and domain sizes and can effectively handle unstructured data and variable input sizes. The model is tested on previously unseen DNS data as well as on an experimental data set from Particle Image Velocimetry (PIV) measurements that were not considered during training. A comparative analysis shows that the GACN consistently outperforms both a conventional Convolutional Neural Network (CNN) and cubic interpolation methods on the DNS and PIV test sets by achieving lower reconstruction errors and better capturing fine-scale turbulent structures. In particular, the GACN effectively reconstructs flow fields from domains up to 14 times larger than those observed during training, with the performance advantage increasing for larger domains.

High-fidelity simulation of complex physical systems is exorbitantly expensive and inaccessible across spatiotemporal scales. Recently, there has been an increasing interest in leveraging deep learning to augment scientific data based on the coarse-grained simulations, which is of cheap computational expense and retains satisfactory solution accuracy. However, the major existing work focuses on data-driven approaches which rely on rich training datasets and lack sufficient physical constraints. To this end, we propose a novel and efficient spatiotemporal super-resolution framework via physics-informed learning, inspired by the independence between temporal and spatial derivatives in partial differential equations (PDEs). The general principle is to leverage the temporal interpolation for flow estimation, and then introduce convolutional-recurrent neural networks for learning temporal refinement. Furthermore, we employ the stacked residual blocks with wide activation and sub-pixel layers with pixelshuffle for spatial reconstruction, where feature extraction is conducted in a low-resolution latent space. Moreover, we consider hard imposition of boundary conditions in the network to improve reconstruction accuracy. Results demonstrate the superior effectiveness and efficiency of the proposed method compared with baseline algorithms through extensive numerical experiments.