图神经网络算法与应用：实验数据分辨率提升与维度扩展的时间-空间图结构

Assessment of cardiovascular and neurovascular disease requires precise measurement of hemodynamic parameters for effective diagnosis and prognosis. Variations in velocity gradients are correlated with changes in the pressure gradient, a key hemodynamic biomarker, necessitating a super-resolved velocity field. We introduce PI-GNN (Physics-Informed Graph Neural Networks) that combines the strengths of Graph Neural Network (GNN) and Navier-Stokes equations as physical laws of fluid dynamics to enhance the resolution of 4D flow MRI. We employed the capacity of Graph Neural Networks (GNNs) to process unstructured Computational Fluid Dynamics (CFD) data for super-resolution tasks. GNNs, inherently size-agnostic, effectively capture both local and global features. This unique suitability is further enhanced by incorporating terms from the Navier-Stokes equations as edge features within their message-passing mechanisms. We trained, evaluated, and tested PI-GNN using high-fidelity simulated Computational Fluid Dynamics (CFD) data in a phantom model of vascular stenosis. Subsequent testing of our proposed network on in vitro 4D flow MRI data exhibited commendable precision in velocity field estimation for super-resolution. Our approach has the potential to impact broader fluid dynamics research to reduce computation time with applications in cardiovascular and biological systems.

The primary objective of flow field super-resolution technology is to enhance the resolution of flow field data while simultaneously reducing computational resource requirements through computational simulations and data processing methods. Building upon prior research, this study introduces a novel flow field super-resolution method composed of three essential components: an interpolation module, graph data preprocessing, and a graph neural network, referred to as RGHF. Within the interpolation module, the radial basis function interpolation method is employed. Furthermore, a pioneering graph convolution operation named GConv is introduced in the graph neural network and prediction modules. This approach effectively leverages low-resolution data to predict high-resolution data, resulting in reduced computational time while maintaining an acceptable level of precision. Experimental results on the RAE2822 airfoil dataset demonstrate an 81% reduction in computational time, along with robustness across varying flight conditions.

Modern engineering design requires high-fidelity simulations, which can impose an enormous computational burden and slow the speed of design iteration. Data-driven up-sampling methods like physics-informed neural networks (PINNs) help reduce the computational resources required. However, machine learning model capacity and hardware limitations still pose challenges when evaluating large engineering simulations with complex physics dynamics. Recently, methods have been proposed to enforce the principle of locality in physical systems to neural network layers, allowing for concurrent inference on smaller subdomains with improved efficiency and accuracy. Based on such an idea, we extend the theory of domain decomposition to complex three-dimensional geometries using graph neural networks (GNNs). We developed a graph decomposition method to improve the training and inference efficiency of machine learning models. Super-resolution GNNs are then trained on individual subdomains distributed among GPU nodes, and during the inference phase, their predictions are combined to achieve a close to linear time reduction as the number of parallel GPUs increases. This approach significantly reduces computational overhead while maintaining simulation accuracy. The parallel nature of our method allows for scalability across available hardware resources, making it suitable for industrial applications where time constraints are critical. We validate the method performance on the design and simulation of a low-pressure bleed duct from an Airbus A350 aircraft, achieving 0.9947 in R2 metric in velocity and 0.9996 in pressure compared with high-fidelity simulations. These results demonstrate that our approach can effectively bridge the gap between computational efficiency and simulation fidelity in complex engineering design tasks.

The applications of super-resolution (SR) technology in the field of image completion are successful. Nevertheless, industry applications demand not only image completion but also the topology and time-series completion. In this article, the SR technology on a topology graph is studied in the scenario of recovering measurements in power distribution systems for cost saving and security & stability improvement. The power flow and voltage magnitude measurements on feeders are reported at different frequencies. In this article, a new data completion method considering distribution system topology is proposed. Firstly, the graph convolutional neural network (GCN) is used for spatial-temporal convolution on a graph, and then the power system state estimation (SE) is used introducing the physical constraints. This method realizes the super-resolution of distribution system measurements, improves the state awareness of distribution systems. Hence, it helps to improve the efficiency of distribution network operation and to reduce equipment failures.

Modern engineering design requires high-fidelity simulations, which can impose an enormous computational burden and slow the speed of design iteration. Data-driven up-sampling methods like physics-informed neural networks (PINNs) help reduce the computational resources required. However, machine learning model capacity and hardware limitations still pose challenges when evaluating large engineering simulations with complex physics dynamics. Recently, methods have been proposed to enforce the principle of locality in physical systems to neural network layers, allowing for concurrent inference on smaller subdomains with improved efficiency and accuracy. Based on such an idea, we extend the theory of domain decomposition to complex three-dimensional geometries using graph neural networks (GNNs). We developed a graph decomposition method to improve the training and inference efficiency of machine learning models. Super-resolution GNNs are then trained on individual subdomains distributed among GPU nodes. This approach significantly reduces computational overhead while maintaining simulation accuracy. We validate the method performance on two engineering applications: a variable inlet-angle mixing elbow junction and a low-pressure bleed duct from an Airbus A350 aircraft. These results demonstrate that our approach can effectively bridge the gap between computational efficiency and simulation fidelity across different scales of complex engineering design tasks.

Accurate prediction of traffic flow plays an important role in maintaining traffic order and traffic safety, which is a key task in the application of intelligent transportation systems (ITS). However, the urban road network has complex dynamic spatial correlation and nonlinear temporal correlation, and achieving accurate traffic flow prediction is a highly challenging task. Traditional methods use sensors deployed on roads to construct the spatial structure of the road network and capture spatial information by graph convolution. However, they ignore that the spatial correlation between nodes is dynamically changing, and using a fixed adjacency matrix cannot reflect the real road spatial structure. To overcome these limitations, this paper proposes a new spatial-temporal deep learning model: gated fusion adaptive graph neural network (GFAGNN). GFAGNN first extracts long-term dependencies on raw data through stacking expansion causal convolution, Then the spatial features of the dynamics are learned by adaptive graph attention network and adaptive graph convolutional network respectively, Finally the fused information is passed through a lightweight channel attention to extract temporal features. The experimental results on two public data sets show that our model can effectively capture the spatiotemporal correlation in traffic flow prediction. Compared with GWNET-conv model on METR-LA dataset, the three indexes in the 60-minute task prediction improved by 2.27%,2.06% and 2.13%, respectively.

Space-time video super-resolution (STVSR) is a comprehensive task comprising two subtasks: video super resolution in space dimension and video frame interpolation in time dimension. Conventional decoupled two-stage approaches tend to overlook the intrinsic correlation between the two tasks. Overcoming this challenge requires the development of a unified model capable of simultaneously implementing space-time super-resolution across arbitrary scales. Most existing models are confined to training on fixed space upsampling scales or specific frame-rate videos, resulting in limited generalization capabilities for flexible space-time super-resolution scenarios. In response to this limitation, our approach draws inspiration from continuous implicit neural representation. We propose an enhanced Implicit Neural Alignment Network (INAN) based on the VideoINR framework, encompassing feature refinement, precise motion flow estimation, and multi-scale feature fusion to optimize the final implicit neural decoding. Our extensive experimental evaluations on multiple benchmarks underscore the efficacy of the INAN model, indicate its superior performance compared to prior STVSR methods.

This article introduces a dynamic weighted graph neural network (DWGNN) for precise reconstruction of impact force time histories under both fault and nonfault conditions in sensor networks. The DWGNN model integrates a one-dimensional convolutional neural network for extracting features from sensor signals and a graph neural network that dynamically adjusts its weight matrix based on node feature similarity, thereby reducing the impact of faulty sensor signals. Incorporating physical constraints, such as the distances between impact and sensor locations, improves the model’s performance. This innovative approach enables the accurate reconstruction of impact forces using only nonfaulty training data. Experimental validation on a stiffened metal plate, with mixed artificially generated and real measured signals, demonstrates the DWGNN model’s high accuracy and robustness. Data augmentation with fault signals significantly enhances the model’s accuracy under challenging fault conditions, proving the DWGNNs potential for real-time structural health monitoring and fault diagnosis in complex engineering structures.

Dynamic graph representation learning aims to generate low-dimensional latent vector representations of graphs or nodes at various time points from evolving graph datas, which are then used for downstream tasks like link prediction and node classification. Existing graph neural network-based approaches primarily focus on modeling node and edge interactions, but fail to capture the spatial structure of these interactions, limiting their ability to represent nodes at different time points within the dynamic graph. To overcome the limitations, we propose a novel dynamic graph neural network representation learning method based on motif reconstruction. The method begins with a dynamic motif sampling strategy that considers both temporal and spatial dimensions to capture the evolving interaction patterns of node neighborhoods. Next, a dynamic graph neural network model based on motif reconstruction is developed, using dynamic motif reconstruction to create a new computational graph for aggregating information from node neighborhoods. Additionally, we propose a motif information aggregation method based on neural ordinary differential equations, which enables the representation of nodes at any given time. We conduct comparative experiments on seven publicly available dynamic graph datasets, benchmarking our method against six state-of-the-art dynamic graph representation learning methods. The results demonstrate that our approach outperforms others in both dynamic link prediction and node classification tasks.

The graph neural network (GNN) has shown outstanding performance in processing unstructured data. However, the downstream task performance of GNN strongly depends on the accuracy of data graph structural features and, as a type of deep learning (DL) model, the size of the training dataset is equally crucial to its performance. This paper is based on graph neural networks to predict and complete the target radio environment map (REM) through multiple complete REMs and sparse spectrum monitoring data in the target domain. Due to the complexity of radio wave propagation in space, it is difficult to accurately and explicitly construct the spatial graph structure of the spectral data. In response to the two above issues, we propose a multi-source domain adaptive of GNN for regression (GNN-MDAR) model, which includes two key modules: (1) graph structure alignment modules are used to capture and learn graph structure information shared by cross-domain radio propagation and extract reliable graph structure information for downstream reference signal receiving power (RSRP) prediction task; and (2) a spatial distribution matching module is used to reduce the feature distribution mismatch across spatial grids and improve the model’s ability to remain domain invariant. Based on the measured REMs dataset, the comparative results of simulation experiments show that the GNN-MDAR outperforms the other four benchmark methods in accuracy when there is less RSRP label data in the target domain.

Effective condition monitoring of motors is crucial in diverse electric powertrain systems, including applications in electric transportation and nuclear power plants. Vibration analysis, a key component of motor condition testing, aids in identifying equipment failures, assessing operational status, and guiding preventive maintenance. However, achieving high evaluation accuracy while minimizing the number of vibration sensors to reduce operational and maintenance costs poses a significant challenge. This article introduces a novel approach for vibration testing using a spatial–spectral-based inductive graph neural network. The proposed algorithm focuses on mining vibration sensor-based clusters to reveal spatial connectivity and spectral correlation patterns. It efficiently aggregates and extracts features from sensor graph signals near the target location, subsequently reconstructing vibration signals using convolutional networks to create a virtual sensor. To validate the effectiveness of the proposed method, experimental verification was conducted on a 21 kW interior permanent magnet synchronous motor testing rig equipped with Brüel and Kjær's vibration sensing equipment. The results demonstrate the algorithm's ability to enhance evaluation accuracy and reliability. This innovative approach not only contributes to the field of motor condition monitoring but also addresses the challenge of minimizing the number of vibration sensors, thereby reducing manual operation and maintenance costs associated with sensor networks.

A new graph neural network (GNN) subgrid-scale (SGS) stress modeling framework based on multi-scale self-similarity input features is investigated as an improvement in generalizability over a previous GNN model developed by the authors' group. Training and a priori testing are conducted with an incompressible direct numerical simulation (DNS) database comprising homogeneous isotropic turbulence (HIT), turbulent mixing layer, and turbulent channel flow (CH) cases, each across five Reynolds numbers. DNS fields are filtered at two different scales using both Gaussian and box kernels to create a diverse large eddy simulation training dataset. The multi-scale Leonard stress tensor and spatial velocity fluctuations are used as inputs to capture scale interactions from a broader portion of the turbulent energy spectrum. Galilean invariance and scale equivariance are ensured by input definition and nondimensionalization, respectively. Randomized data augmentation is used to improve model robustness to changes in orientation. Spline-based graph-convolution layers enable non-local feature extraction while maintaining model compatibility with unstructured grids. A priori analysis of four GNN SGS models shows that graph convolution and multi-scale input features provide significant improvement in accuracy for incompressible flows at Reynolds numbers below, within and above the training range, with particular impact in the near-wall region. Model generalizability to compressible reactive flows is high except in a wall-bounded case. In a posteriori testing with incompressible turbulent CH, one multi-scale GNN was found to better match reference DNS profiles than conventional eddy-viscosity models, with computational overhead comparable to that of a dynamic k-equation model.

No abstract available

Land surface temperature (LST) is a key parameter in the land surface energy cycle. However, due to cloud contamination, LST products derived from thermal infrared (TIR) remote sensing exhibit significant missing areas. Although many studies have attempted to reconstruct missing areas using reanalysis data, such as the land component of the fifth generation of European ReAnalysis (ERA5 Land), existing methods neglect the temporal differences between MODIS and reanalysis data, suffering from several limitations in applicability. Therefore, a two-stage surface energy balance-guided graph neural network (TSSEB-GNN) was developed, with the innovation of coupling the network with physical mechanisms. First, the physical factors influencing LST were determined based on surface energy balance (SEB) theory and the Bowen ratio. Second, the stage one model based on a graph neural network (GNN) module was designed to generate an initial gap-free LST. Third, the second-stage model fuses the initial LST by adding sparse and dense self-attention (SDSA) mechanisms and incorporating 12 physical factors to capture spatial and temporal differences. The results show that the TSSEB-GNN model achieves a root-mean-square error (RMSE) of 2.56–3.45 K under clear-sky conditions and 3.39–3.63 K under cloudy-sky conditions. Compared with existing methods, TSSEB-GNN not only provides higher accuracy but is also less affected by long-term gaps in MODIS LST. Ablation experiments demonstrate that the TSSEB-GNN model achieves the best performance by accounting for both temporal and spatial differences. The results highlight the effectiveness of the TSSEB-GNN, its interpretability, improved applicability, and greater potential for large-scale applications.

In the realm of oil and gas exploration, accurately predicting subsurface fluid types is crucial. Traditional techniques such as core sampling, x-ray diffraction, and x-ray fluorescence, despite providing essential data, are hampered by high costs, time consumption, or limited applications. This paper introduces an interpretable spatiotemporal deep learning network, ISTNet, utilizing well log data to predict fluid types. The framework enhances prediction accuracy and model robustness through a dual-branch design integrating spatial and temporal branches. The spatial branch employs graph neural networks to capture spatial features of well log data, while the temporal branch analyzes time series features using bidirectional long short-term memory networks (BiLSTM). Additionally, ISTNet incorporates the SHapley Additive exPlanations (SHAP) model to augment the interpretability of predictions. Empirical studies in the Tarim Basin demonstrated that ISTNet outperforms seven other advanced models, achieving an average accuracy exceeding 97% on datasets from two distinct wells. ISTNet not only improves the accuracy and robustness of fluid predictions in oil and gas exploration but also enhances transparency and interpretability through the SHAP model, providing geologists and engineers with tools to deeply understand subsurface geological processes and refine exploration and development strategies.

Hemodynamic and cardiovascular systems modelling hemodynamics and cardiovascular systems modelling is a challenging problem because blood flow is complex in geometry (spatial) and in pulsation (temporal). The conventional computational fluid dynamics (CFD) models are accurate but in most of the cases they are computationally heavy and cannot be executed in real-time in clinical practice. This paper suggests the use of Graph Recurrent Neural Networks (GRNNs) to estimate the dynamics of pressure and flow in the cardiovascular system. With the vascular network presented as a graph, i.e., the vessels are the edges and the bifurcations are the nodes, we use Graph Convolutional Networks (GCNs) to learn the geometric constraints and local connectivity of the circulatory system. The spatial characteristics are combined with Recurrent Neural Networks (RNNs) including Long Short-Term Memory (LSTM) units that can model the time-varying waveforms of the cardiac cycle. Our findings prove that the GRNN architecture is highly effective in comparison with the traditional sequence models because it considers the non-Euclidean topology of blood vessels. The model is effective in the propagation of waves, reflection and attenuation of waves in different arterial structures. The method is an alternative to patient-specific hemodynamic monitoring that offers high fidelity and computationally-efficient methods that hold potential to conduct real-time surgical planning and the timely diagnosis of cardiovascular diseases including stenosis or aneurysms.

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.

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.

In this paper we demonstrate the application of state-of-the-art deep learning using hybrid neural networks (HNN) that generalize and scale to multi-million, structurally diverse reservoir model grids and generate long-term spatio-temporal predictions of fluid and pressure propagation. The HNN simulator (HNNS) is a surrogate framework that consists of a subsurface graph neural network (SGNN) to model the evolution of fluids, and a 3D-U-Net to model the evolution of pressure. We benchmark the HNNS with two conceptually different reservoir models: a) modified SPE-10 model, with approx. 1 million grid size and variable number and positioning of vertical producers and injectors, b) synthetic fractured model, 15+ million grid size and 100+ injector and producer wells with variable geometry. We construct the network graph, where graph objects (nodes), representing reservoir grid cells are encoded with tens of static, dynamic, computed (relative permeability, gradients) and control (well rates) features. The graph edges represent interactions between the nodes with encoded features like transmissibility, direction and fluxes. We implement sector-based training with multi-step rollout to avail for the use of large-scale models. To properly perform the sector-based training the masking of sector boundary effects, sector stride and mixing sectors were used. We present the comparative results between the HNNS and the full-physics simulation for up to 30-year prediction of the 3D flow dynamics.

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.

Abstract Classical approaches for flood prediction apply numerical methods for the solution of partial differential equations that capture the physics of inundation processes (e.g., the 2D Shallow Water equations). However, traditional inundation models are still unable to satisfy the requirements of many relevant applications, including early-warning systems, high-resolution (or large spatial domain) simulations, and robust inference over distributions of inputs (e.g., rainfall events). Machine learning (ML) approaches are a promising alternative to physics-based models due to their ability to efficiently capture correlations between relevant inputs and outputs in a data-driven fashion. In particular, once trained, ML models can be tested/deployed much more efficiently than classical approaches. Yet, few ML-based solutions for spatio-temporal flood prediction have been developed, and their reliability/accuracy is poorly understood. In this paper, we propose FloodGNN-GRU, a spatio-temporal flood prediction model that combines a graph neural network (GNN) and a gated recurrent unit (GRU) architecture. Compared to existing approaches, FloodGNN-GRU (i) employs a graph-based model (GNN); (ii) operates on both spatial and temporal dimensions; and (iii) processes the water flow velocities as vector features, instead of scalar features. We evaluate FloodGNN-GRU using a LISFLOOD-FP simulation of Hurricane Harvey (2017) in Houston, Texas. Our results, based on several metrics, show that FloodGNN-GRU outperforms several data-driven alternatives in terms of accuracy. Moreover, our approach can be trained 100x faster and tested 1000x faster than the time required to run a comparable simulation. These findings illustrate the potential of ML-based methods to efficiently emulate physics-based inundation models, especially for short-term predictions.

No abstract available

In the field of big data, the phenomenon of missing data is widespread, particularly pronounced in time-series datasets, where its impact on subsequent processing is significant. However, existing data imputation methods have not comprehensively and adequately addressed the bidirectional spatio-temporal dependencies within the data, especially the dependencies among different dimensions at various timestamps in multivariate time-series data. To address this issue, this paper proposes a hybrid model that integrates Graph Neural Networks (GNNs) and Bidirectional Recurrent Neural Networks (RNNs) for imputing missing values in multivariate time-series data. The model introduces an enhanced bidirectional Gated Recurrent Unit (GRU) to capture the bidirectional temporal dependencies in the data. Additionally, it employs graph structure learning and attention mechanisms to construct a correlation graph, aggregating inter-dimensional correlation information through graph neural networks. Experimental results indicate that our model has significantly improved the imputation accuracy on three public datasets, particularly under conditions of low missing rates.

No abstract available

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.

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) 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.

This study designs a deep learning framework to obtain high-precision velocity fields of a pump-jet propulsor (PJP) from low-resolution (LR) velocity fields using super-resolution (SR) methods. In actual engineering or experiments, the velocity fields obtained via particle image velocimetry have low spatial resolution, which is limited by equipment and technology. This study investigates the performance of convolutional neural network (CNN) and hybrid downsampled skip-connection/multi-scale (DSC/MS) models in reconstructing the LR velocity fields of PJP. Moreover, the variational Bayesian (VB) idea is considered in two SR methods to design VB-DSC/MS and VB-CNN neural networks, which exhibit superior performance on small datasets and can analyze the uncertainty distribution. The optimal weight and bias distributions for four different SR methods are determined, which efficiently and accurately afford the SR velocity fields from the LR velocity fields of the wake and stator domain fields. Furthermore, the reconstruction ability of the SR method under different scaling factors is analyzed. The results show that the VB-DSC/MS method has higher accuracy and generalization ability than the other three SR methods in terms of reconstructing the velocity field gradient and velocity profile of PJP. It can enhance the LR velocity field by 256 times, which is difficult for CNN-related SR methods to improve the LR velocity field by a higher factor. Among the methods considered, the VB-DSC/MS method has the smallest uncertainty distribution under different scaling factors and different rotational speeds.

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.

Abstract Velocity measurement techniques, such as particle image velocimetry (PIV), face a trade-off between field of view, spatial resolution and sampling rate, so that small-scale vortices, shear layers and high-frequency turbulent motions are often under-resolved. Most physics-informed reconstructions use a velocity–pressure formulation, even though pressure is not measured in typical PIV experiments, so the Navier–Stokes constraints are only weakly enforced. We address this issue by formulating a vorticity–velocity physics-informed network (VVPINN), in which pressure is eliminated and incompressibility is enforced together with a vorticity transport equation, thereby directly constraining the velocity field and its derivatives. We then compare this formulation with a conventional velocity–pressure PINN (VPPINN) for spatio-temporal super-resolution of planar PIV data in three cases: a laminar multi-cylinder wake, a two-dimensional Taylor–Green vortex and an experimental two-cylinder wake. In the Taylor–Green vortex case, with identical architectures and training strategies, the VVPINN yields smaller velocity errors, reduces the $L_2$ errors in vorticity and shear by approximately $10\,\%$ , and the pressure gradient errors by up to approximately $30\,\%$ at moderate super-resolution factors, and produces instantaneous fields with more physically plausible vorticity, shear and fine-scale pressure gradient patterns. Spectral analysis shows that the temporal energy spectrum is recovered accurately, whereas the wavenumber spectra, particularly beyond the Nyquist wavenumber, remain more difficult to match because the training data strongly constrain the time histories at sampled locations, but only indirectly inform the smallest spatial scales. Overall, the results indicate that vorticity-based constraints provide a more effective route to physics-consistent super-resolution of sub-sampled PIV data than the conventional velocity–pressure formulation.

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.

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.

No abstract available

Particle image velocimetry (PIV) data of high Reynolds number unsteady turbulent flows are often undersampled in time; this leads to aliasing of important spectral content. The present work proposes a novel data-driven estimation technique that uses oversampled sparsely placed surface-mounted pressure sensors and long short-term memory neural networks to resolve the aliased transient velocity dynamics from undersampled PIV data. The method leverages the time-resolved pressure dynamics to estimate the temporal evolution of a proper orthogonal decomposition-based low-dimensional subspace of the velocity field. The proposed approach is demonstrated on a PIV dataset of a high Reynolds number turbulent separated flow over a Gaussian speed-bump benchmark geometry ( $$\text{Re}_{H}=2.26\times 10^{5}$$ Re H = 2.26 × 10 5 , where H is the Bump height). The 15 Hz PIV data is super-resolved to 2 kHz, and spectral analysis of the flowfields is conducted to educe the originally aliased unsteady dynamics of the turbulent separation bubble. The estimator is shown to accurately reconstruct the Reynolds shear stress from unseen sensor data, demonstrating its generalizability to resolve the coherent motions. The estimated velocity spectra show distributions consistent with those of other separated flows.

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High-resolution (HR) fluid-flow velocity information is important to reliably analyze fluid measurements in particle image velocimetry (PIV), such as the boundary layer and turbulent flow. Efforts in PIV to enhance the resolution of flow fields are mainly based on single-frame information, which follows the velocity field estimation and may influence the final reconstruction accuracy. In this study, we propose a novel super-resolution (SR) reconstruction technology from another perspective, which consists of two parts: a multi-frame imaging system and a Bayesian-based multi-frame SR reconstruction algorithm. First, a splitbased imaging system is developed to obtain particle image pairs with fixed displacements. Subsequently, we present a Bayesian-based multi-frame SR (BMFSR) reconstruction algorithm to obtain an SR particle image. Multi-frame particle images collected by the developed system are used as the input low-resolution images for the following novel SR reconstruction algorithm. Synthetic and experimental particle images have been tested to verify the performance of the proposed technology, and the results are compared with the traditional and advanced reconstruction methods in PIV. The results and comparisons show that the proposed technology successfully achieves good performance in obtaining finer particle images and a more accurate velocity field.

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With the increasing use of deep neural networks as surrogate models for accelerating computational simulations in mechanics, the application of artificial intelligence in computational fluid dynamics has seen renewed interest in recent years. However, the application of deep neural networks for flow simulations has mainly concentrated on relatively simple cases of incompressible flows. The strongly discontinuous structures that appear in compressible flows dominated by convection, such as shock waves, introduce significant challenges when approximating the nonlinear solutions or governing equations. In this work, we propose a novel physics-constrained, flow-field-message-informed (FFMI) graph neural network for spatiotemporal flow simulations of compressible flows involving strong discontinuities. To enhance the nonlinear approximation capability of strong discontinuities, a shock detector method is leveraged to extract the local flow-field messages. These messages are embedded into the graph representation to resolve the discontinuous solutions accurately. A new FFMI sample-and-aggregate-based message-passing layer, which aggregates the edge-weighted attributes with node features on different hop layers, is then developed to diffuse and process the flow-field messages. Furthermore, an end-to-end paradigm is established within the encoder–decoder framework to transform the extracted information from the flow field into latent knowledge about the underlying fluid mechanics. Finally, a variety of one- and two-dimensional cases involving strong shock waves are considered to demonstrate the effectiveness and generalizability of the proposed FFMI graph neural network.

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

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

The Physics-informed Neural Network (PINN) has been a popular method for solving partial differential equations (PDEs) due to its flexibility. However, PINN still faces challenges in characterizing spatio-temporal correlations when solving parametric PDEs due to network limitations. To address this issue, we propose a Physics-Informed Neural Implicit Flow (PINIF) framework, which enables a meshless low-rank representation of the parametric spatio-temporal field based on the expressiveness of the Neural Implicit Flow (NIF), enabling a meshless low-rank representation. In particular, the PINIF framework utilizes the Polynomial Chaos Expansion (PCE) method to quantify the uncertainty in the presence of noise, allowing for a more robust representation of the solution. In addition, PINIF introduces a novel transfer learning framework to speed up the inference of parametric PDEs significantly. The performance of PINIF and PINN is compared on various PDEs especially with variable coefficients and Kolmogorov flow. The comparative results indicate that PINIF outperforms PINN in terms of accuracy and efficiency.

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Driven by the abundant data generated from computational fluid dynamics (CFD) simulations, machine learning (ML) methods surpass the deterministic criteria on flow phenomena identification in the way that they are independent of a case-by-case threshold by combining the flow field properties and the topological distribution of the phenomena. The current most popular and successful ML models based on convolutional neural networks are limited to structured meshes and unable to directly digest the data generated from unstructured meshes, which are more widely used in real industrial CFD simulations. We proposed a framework based on graph neural networks with the proposed fast Gaussian mixture model as the convolution kernel and U-Net architecture to detect flow phenomena based on a graph hierarchy generated by the algebraic multigrid method embedded in the open-source CFD solver, code_saturne. We demonstrate the superiority of the proposed kernel and U-Net architecture, along with the generality of the framework to unstructured mesh and unseen case, on detecting the vortices once trained on a single backward-facing step case. Our proposed framework can be trivially extended to detect more flow phenomena in three dimensional cases, which is ongoing work.

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.

To address the challenges of limited labeled data and insufficient global feature extraction in flow field prediction, this paper proposes a modeling approach that combines self-supervised learning and Graph Transformer. The self-supervised learning module leverages feature reconstruction tasks and contrastive learning tasks to fully exploit the latent information in unlabeled data, thereby enhancing the joint modeling capability for local and global features. The Graph Transformer incorporates a self-attention mechanism, enabling effective modeling of long-range dependencies and multiscale features in complex flow fields. Experimental results demonstrate that, under 100% labeled data conditions, the proposed method reduces the root mean squared error achieved by graph convolutional network and a multiscale graph neural network model on the cylinder flow and airfoil flow datasets from 0.970 and 0.561 to 0.616 and 0.305, achieving significant accuracy improvements of 36.5% and 45.6%, respectively. Under 50% labeled data conditions, the method still exhibits outstanding robustness, with RMSEs of 0.792 and 0.390, respectively. The ablation studies reveal that the feature reconstruction and contrastive learning tasks exhibit strong complementarity, achieving optimal performance when jointly employed. Furthermore, the self-attention mechanism significantly enhances the modeling of global features, demonstrating its effectiveness in capturing complex dependencies. The proposed method demonstrates superior prediction accuracy and robustness under limited labeled data and complex flow field conditions, providing an efficient solution for flow field modeling with broad application potential.

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When calculating the transient flow around a bridge structure, traditional computational fluid dynamics methods are extremely time-consuming, especially for multiparameter optimization analyses. Inspired by the development of deep graph neural networks with a mesh structure, this paper describes a spatiotemporal prediction framework for the rapid reconstruction and prediction of transient flows on large-scale unstructured grids. To ensure stability and reliability during self-supervised training, a causal self-attention mechanism is employed in the temporal model. The framework is trained and tested on a dataset containing 40 000 snapshots of bridge flow fields with Reynolds numbers ranging from 104 to 105. The relative mean square error of the model in predicting the velocity and pressure fields is found to be in the order of 10−3 and the relative error does not exceed 10%. This demonstrates that the model is capable of reconstructing high-dimensional flow field information from low-dimensional data. Furthermore, the proposed model achieves a computational speedup by two orders of magnitude compared with traditional computational fluid dynamics methods with respect to the temporal inference. To validate its ability to infer bridge aerodynamic characteristics, the model is used to predict the bridge surface pressure, aerodynamic coefficients, streamlines, and vorticity. The results demonstrate that the proposed model has reliable accuracy, representation, and stability in predicting bridge flow fields and identifying multiscale characteristics.

Abstract Accurate prediction of turbulent flows remains a critical challenge in computational fluid dynamics, particularly for complex, non-equilibrium environments where traditional Reynolds-averaged Navier–Stokes (RANS) models fail to capture deep anisotropy and nonlocal interactions. While data-driven turbulence modeling, such as tensor basis neural networks (TBNNs) offers a promising alternative, these approaches frequently struggle with physical realisability and the inherent nonlocality of turbulent transport. To address these limitations, this work introduces a realisability-preserving E(3)-equivariant graph neural network (E(3)-GNN) for Reynolds-stress anisotropy closure. By leveraging equivariant message passing and a spherical-tensor formulation, the architecture explicitly captures nonlocal neighborhood interactions while guaranteeing Galilean invariance. Furthermore, a stabilized positive semi-definite projection ensures that all predicted states permanently reside within the physically admissible Lumley triangle, completely eliminating the unphysical predictions that traditionally destabilize coupled solvers. Because accurate a priori stress predictions do not guarantee a posteriori stability during active RANS simulation, we additionally propose a solver-in-the-loop fine-tuning methodology using Group Relative Policy Optimization (GRPO). This step directly optimizes a scalar production-correction policy within the active RANS environment to capture dynamic energy propagation. Evaluated on the parameterised periodic hill benchmark family, with training and testing performed on distinct hill geometries, our E(3)-GNN achieves strict physical realisability by construction. Through GRPO fine-tuning, the fully coupled model successfully recovers the intense turbulent kinetic energy (k) peaks and mean velocity profiles in the separated shear layer that purely supervised models fail to capture, bridging the gap between offline regression accuracy and stable, coupled solver performance.

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 deep learning (DL)-based method for predicting remaining useful life (RUL) has gained lots of attention in the industrial equipment sector. Due to the complexity of modern industrial equipment and the necessity of monitoring multivariate time-series data to obtain comprehensive health information, DL models with spatio-temporal feature extraction have been developed to achieve accurate RUL prediction results. However, in the three typical approaches for spatio-temporal feature extraction (parallel, sequential, and nested), each module is independent and separate, making them ineffective in fusing spatio-temporal information at the same time. Most existing approaches use separated modules to extract spatial and temporal features, where convolution, graph, and recurrent neural networks (RNNs) are often applied. To overcome these limitations, this article introduces a unified paradigm for spatio-temporal feature fusion. By extending the message-passing graph neural network (GNN), the spatio-temporal propagation (STP) model is constructed. Using GNN as a single structure, the model can simultaneously match information propagation at both spatial and temporal scales. STP allows a flexible and intuitive way to construct RUL prediction models. An implementation of STP-GNN with an attention mechanism is given and discussed, and a case study on RUL prediction of turbofan engines is reported. Experimental results verify the effectiveness of STP-GNN, and highlight its interpretability in the process of RUL prediction.