图神经网络在不同边界条件和不同流动构型下的泛化能力

Data-driven models often neglect the underlying physical principles, limiting generalization capabilities in water distribution systems (WDSs). This study presents a novel spatio-temporal graph physics-informed neural network (ST-GPINN) for water quality prediction in WDSs, integrating hydraulic simulations, physics-informed neural networks (PINNs), and graph neural networks (GNNs) to capture dynamics and graph-based network connectivity while approximating partial differential equations (PDEs). ST-GPINN discretizes WDSs using virtual nodes to enhance spatial granularity, employs an Encoder-Processor-Decoder architecture for predictions. Validated on Network A (a small-scale network with 9 junctions and 11 pipes) and Network B (a real large-scale WDS with 920 junctions and 1032 pipes), ST-GPINN outperforms others, achieving a MAE of 0.0073 mg/L, RMSE of 0.0121 mg/L, and R2 of 88.91% in Network A, and a MAE of 0.008 mg/L, RMSE of 0.0098 mg/L, and R² of 98.91% in Network B. Its scalability and accuracy highlight ST-GPINN’s potential for water quality predictions.

Graph Neural Networks (GNNs) show promising results for graph tasks. However, existing GNNs' generalization ability will degrade when there exist distribution shifts between testing and training graph data. The fundamental reason for the severe degeneration is that most GNNs are designed based on the I.I.D hypothesis. In such a setting, GNNs tend to exploit subtle statistical correlations existing in the training set for predictions, even though it is a spurious correlation. In this paper, we study the problem of the generalization ability of GNNs on Out-Of-Distribution (OOD) settings. To solve this problem, we propose the Learning to Reweight for Generalizable Graph Neural Network (L2R-GNN) to enhance the generalization ability for achieving satisfactory performance on unseen testing graphs that have different distributions with training graphs. We propose a novel nonlinear graph decorrelation method, which can substantially improve the out-of-distribution generalization ability and compares favorably to previous methods in restraining the over-reduced sample size. The variables of graph representation are clustered based on the stability of their correlations, and graph decorrelation method learns weights to remove correlations between the variables of different clusters rather than any two variables. Besides, we introduce an effective stochastic algorithm based on bi-level optimization for the L2R-GNN framework, which enables simultaneously learning the optimal weights and GNN parameters, and avoids the over-fitting issue. Experiments show that L2R-GNN greatly outperforms baselines on various graph prediction benchmarks under distribution shifts.

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Accurate, interpretable, and real-time modeling of multi-body dynamical systems is essential for predicting behaviors and inferring physical properties in natural and engineered environments. Traditional physics-based models face scalability challenges and are computationally demanding, while data-driven approaches like Graph neural networks (GNNs) often lack physical consistency, interpretability, and generalization. In this paper, we propose DYNAMI-CAL GRAPHNET, a Physics-Informed Graph Neural Network that integrates the learning capabilities of GNNs with physics-based inductive biases to address these limitations. DYNAMI-CAL GRAPHNET enforces pairwise conservation of linear and angular momentum for interacting nodes using edge-local reference frames that are equivariant to rotational symmetries, invariant to translations, and equivariant to node permutations. This design ensures physically consistent predictions of node dynamics while offering interpretable, edge-wise linear and angular impulses resulting from pairwise interactions. Evaluated on a 3D granular system with inelastic collisions, DYNAMI-CAL GRAPHNET demonstrates stable error accumulation over extended rollouts, effective extrapolation to unseen configurations, and robust handling of heterogeneous interactions and external forces. DYNAMI-CAL GRAPHNET offers significant advantages in fields requiring accurate, interpretable, and real-time modeling of complex multi-body dynamical systems, such as robotics, aerospace engineering, and materials science. By providing physically consistent and scalable predictions that adhere to fundamental conservation laws, it enables the inference of forces and moments while efficiently handling heterogeneous interactions and external forces. This makes it invaluable for designing control systems, optimizing mechanical processes, and analyzing dynamic behaviors in both natural and engineered systems. Learning complex dynamics from data often leads to unstable or unphysical predictions. Here, the authors introduce Dynami-CAL GraphNet, a physics-informed architecture that conserves linear and angular momentum and enables accurate rollouts across diverse dynamical systems.

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

Partial differential equations (PDEs) are a foundational tool for modeling physical systems. However, high-fidelity numerical simulations are often computationally intensive, which has prompted the development of surrogate models in the field of scientific machine learning. While recent approaches have achieved high predictive accuracy by employing complex architectures and extensive datasets, many scientific applications offer only a limited number of simulation trajectories. In this study, we examine the impact of architectural inductive biases on the learning of two-dimensional PDE in data-limited settings. Instead of proposing a highly expressive model, we deliberately adopt a simple multi-channel U-Net modified to enforce periodic boundary conditions. In this study, we compare this architecture with transformer-based and UNet variants such as vision transformer, adaptive Fourier neural operator, PDE-Transformer (PDE-T), and UNet with Kolmogorov–Arnold Layers under a common training setup across five PDE families. To keep comparisons uniform and to mimic partial observability in real measurements, we train all models on rollouts of a single representative scalar field per system. Despite its simplicity, the adapted U-Net consistently matches or outperforms these more complex models in forecasting accuracy, spectral similarity, and physics-based metrics while requiring substantially less training time. The findings of this study demonstrate that, in scenarios where data is limited, the judicious selection of inductive biases can often prove to be more advantageous than increasing the model complexity for the purpose of learning the dynamics of PDEs.

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Underwater sound field prediction conventionally relies on numerically solving the Helmholtz equation, which is computationally expensive and struggles to meet real-time requirements. This study presents the shallow water Fourier neural operator (SWFNO), an end-to-end model based on the Fourier neural operator, for the rapid and accurate prediction of the amplitude and phase of two-dimensional underwater sound fields in shallow water environments with variable sound speed profiles (SSPs). To enhance the model's generalization capability, we design a hybrid sample generation scheme combining Gaussian random fields with simulated shallow water SSPs, constructing a training dataset that balances randomness and physical plausibility. Simulation results demonstrate that SWFNO achieves high sound field prediction accuracy on both test sets and generalization datasets containing actual ocean SSPs, demonstrating its robustness to out-of-distribution samples. Furthermore, the model exhibits super-resolution prediction capability, enabling high-resolution sound field prediction without retraining, and achieves a speedup of over an order of magnitude compared to the traditional spectral method for large-scale samples. These results indicate that SWFNO is an efficient and accurate surrogate model for underwater acoustic propagation, offering a promising route toward real-time, high-fidelity, and intelligent sound field prediction in complex marine environments.

Recent advances in machine learning (ML) have enabled efficient modelling of process-structure-property relationships in metallic additive manufacturing (AM), offering promising alternatives to conventional simulation-based methods. However, most ML models rely on input-output regression paradigms, which limit their ability to generalize to unseen scenarios. This paper proposes a graph neural operator that integrates deep operator network (DeepONet) with graph neural networks (GNNs) to simulate the thermo-mechanical constitutive behaviour in metallic AM. The proposed DeepONet-GNN framework decouples the thermal and structural fields, leveraging sparse temperature measurements to predict full-field z-direction distortion across unseen geometries. Through layer-wise evaluations on multiple structures, the model demonstrates strong generalization, data efficiency, and robustness to variations in sensor distribution, achieving a low RMSE of 0.0881 mm. Compared to a coupled GNN, DeepONet-GNN reaches convergence with similar accuracy using 50% fewer training epochs. The proposed DeepONet-GNN model demonstrates the ability to generalize to unseen geometries while leveraging only 5% of the temperature sensor data, highlighting the potential of graph neural operators as accurate and scalable surrogates for real-time prediction in AM processes.

Fourier Neural Operators (FNOs) exhibit mode saturation on high-contrast inhomogeneous media, and recent multi-scale extensions (MscaleFNO) further worsen out-of-distribution (OOD) generalization. We introduce the Helmholtz Neural Operator (HNO), a physics-informed, FFT-free branch–trunk operator in the DeepONet family, with a hybrid SIREN+learnable-Fourier trunk and a dual-path rank-32 hypernetwork branch, with bounded multiplicative gating on per-mode coefficients. At a matched parameter count (∼1.05 M, five seeds), HNO achieves a 2.6× lower OOD generalization gap than FNO (19.6% vs. 50.6%, p=1.7×10−3, Cohen’s d=5.1), 5.1× lower than vanilla DeepONet (19.6% vs. 99.9%, p=8.2×10−3), and 6.0× lower than MscaleFNO (19.6% vs. 117.4%, p=2.4×10−6); MscaleFNO’s deficit grows at 4.2× more parameters, ruling out capacity starvation. HNO is 4.6×/16.4× faster than FNO/MscaleFNO and 64×–245× faster than multi-threaded FD-PML (MKL PARDISO, 12 cores; 183×–698× vs. single-thread scipy.spsolve), making it suitable as a forward surrogate inside many-query workflows. Absolute accuracy on extreme-contrast (15:1) OOD samples is limited (relative L2≈1), so HNO is positioned as a many-query surrogate or warm start for refinement loops, not a stand-alone replacement for direct solvers. A scope limitation is that HNO underperforms FNO on elliptic Darcy Flow, confirming specialization for hyperbolic/wave equations rather than universal operator learning.

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To obtain a numerical model for wafer temperature distribution under multiple operating conditions in rapid thermal processing (RTP) systems, this paper proposes a modeling method based on Physics-Informed Neural Networks (PINNs). By embedding the partial differential equations (PDEs), initial conditions, and boundary conditions derived from the physical laws of RTP systems into the loss function of PINNs, and integrating them with the Deep Operator Network (DeepONet), a novel fusion network architecture is constructed to support numerical modeling of wafer temperature distribution across diverse operating conditions. Comparative experiments against the finite difference method (FDM) demonstrate the effectiveness of the proposed method. This fusion model strikes a balance between physical constraint compliance and multi-condition generalization capability, and can provide reliable modeling support for real-time temperature prediction and process parameter optimization of RTP systems.

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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 physics-informed neural network (PINN) method is extended to learn and predict compressible steady-state aerodynamic flows with a high Reynolds number. To better learn the thin boundary layer, the sampling distance function and hard boundary condition are explicitly introduced into the input and output layers of the deep neural network, respectively. A gradient weight factor is considered in the loss function to implement the PINN methods based on the Reynolds averaged Navier–Stokes (RANS) and Euler equations, respectively, denoted as PINN–RANS and PINN–Euler. Taking a transonic flow around a cylinder as an example, these PINN methods are first verified for the ability to learn complex flows and then are applied to predict the global flow based on a part of physical data. When predicting the global flow based on velocity data in local key regions, the PINN–RANS method can always accurately predict the global flow field including the boundary layer and wake, while the PINN–Euler method can accurately predict the inviscid region. When predicting the subsonic and transonic flows under different freestream Mach numbers (Ma∞= 0.3–0.7), the flow fields predicted by both methods avoid the inconsistency with the real physical phenomena of the pure data-driven method. The PINN–RANS method is insufficient in shock identification capabilities. Since the PINN–Euler method does not need the second derivative, the training time of PINN–Euler is only 1/3 times that of PINN–RANS at the same sampling point and deep neural network.

Many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. In this paper, we propose a new neural network model, called graph neural network (GNN) model, that extends existing neural network methods for processing the data represented in graph domains. This GNN model, which can directly process most of the practically useful types of graphs, e.g., acyclic, cyclic, directed, and undirected, implements a function tau(G,n) isin IRm that maps a graph G and one of its nodes n into an m-dimensional Euclidean space. A supervised learning algorithm is derived to estimate the parameters of the proposed GNN model. The computational cost of the proposed algorithm is also considered. Some experimental results are shown to validate the proposed learning algorithm, and to demonstrate its generalization capabilities.

Graph neural network (GNN) is a general framework for using deep neural networks on graph data. The defining feature of a GNN is that it uses a form of neural message passing where vector messages are exchanged between nodes and updated using neural networks. The message passing operation that underlies GNNs has recently been applied to develop neural approximate inference algorithms, but little work has been done on understanding under what conditions GNNs can be used as a core module for building general inference models. To study this question, we consider the task of out-of-distribution generalization where training and test data have different distributions, by systematically investigating how the graph size and structural properties affect the inferential performance of GNNs. We find that (1) the average unique node degree is one of the key features in predicting whether GNNs can generalize to unseen graphs; (2) the graph size is not a fundamental limiting factor of the generalization in GNNs when the average node degree remains invariant across training and test distributions; (3) despite the size-invariant generalization, training GNNs on graphs of high degree (and of large size consequently) is not trivial (4) neural inference by GNNs outperforms algorithmic inferences especially when the pairwise potentials are strong, which naturally makes the inference problem harder.

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The rapid development of deep learning has significantly influenced computational studies in convective heat transfer. To facilitate broader applications of deep learning models in convective heat transfer, this paper proposes a physics-informed graph neural network based on the finite volume method (FVGP-Net) for unsupervised training and prediction of steady incompressible laminar convective heat transfer problems. In this model, mesh data generated by the finite volume method (FVM) are converted into graph data, preserving the mesh's topological properties. This conversion allows FVGP-Net to utilize a graph convolutional network for information aggregation, capturing both local and global flow features and enhancing the model's geometric adaptability and predictive performance. The model incorporates physical laws directly into its loss function, ensuring compliance to these laws without reliance on training data. Unlike traditional physics-informed neural networks (PINNs), FVGP-Net replaces automatic differentiation with FVM-based numerical differentiation, balancing training efficiency with prediction accuracy. Boundary conditions are handled in accordance with the FVM, ensuring that the model strictly satisfies these constraints. We validated FVGP-Net using representative test cases, also examining the effects of different initialization methods on model training. The results demonstrate that FVGP-Net achieves high accuracy in predicting incompressible laminar steady convective heat transfer. Compared to traditional PINNs, this model inherits the conservation properties of the FVM, enhancing velocity prediction accuracy in convective heat transfer problems by 70.03%. Furthermore, the application of transfer learning markedly accelerates training, achieving approximately 70% faster results compared to Xavier initialization.

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Physic-informed neural networks (PINNs) have been widely used in computational physics including electromagnetic simulations. Unlike conventional numerical solvers, PINNs leverage deep neural networks and optimization to find the solution of Maxwell’s equations given specific source and boundary conditions. When the material in the solving domain is heterogeneous, it’s not easy to enforce continuity of electromagnetic fields due to the meshless property of PINNs. On the contrary, graph neural networks (GNNs) explicitly construct a graph (mesh) and apply message passing between the nodes in the graph. In this work, GNNs and PINNs are combined to build physics-informed graph neural networks (PIGNNs) that are flexible in mesh discretization and straightforward to enforce continuity of electromagnetic fields. Moreover, PIGNNs get easier to be trained and converge by resorting to the variational formulation of Maxwell’s equations.

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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.

Fluid simulations in Computational Fluid Dynamics (CFD) often involve solving complex Partial Differential Equations (PDEs) with varying parameters, leading to high computational costs and slow inference times. Graph neural networks (GNNs) offer a promising alternative by efficiently representing mesh spaces and reducing these costs. In this work, we introduce a pioneering Physics-Informed Graph Neural Network (PIGNN) approach, utilizing a multi-GNN processor architecture that assigns each output parameter to a dedicated GNN processor with individual decoders. This innovative method reduces the training epochs for PIGNN by half while preserving accuracy. Additionally, we adopted mixed-precision training and achieved further improvements in reducing the training time to one-quarter of the original network while maintaining VRAM usage.

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This paper presents a generalizable data-driven computational electromagnetics (CEM) framework leveraging graph neural networks (GNNs). The proposed model supports training and inference for CEM scenarios with different simulation domain sizes and electromagnetic properties, while exploiting the locality of GNNs to achieve reduced complexity and enhanced accuracy. Our results indicate that GNNs can successfully infer the electromagnetic field spatiotemporal evolution for arbitrary simulation domain setups, paving the way for fully-fledged data-driven CEM models.

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This study introduces the Graph-Structured Physics-Informed DeepONet (GS-PI-DeepONet), a novel neural network framework designed to address the challenges of solving parametric Partial Differential Equations (PDEs) in structural analysis, particularly for problems with complex geometries and dynamic boundary conditions. By integrating Graph Neural Networks (GNNs), Deep Operator Networks (DeepONets), and Physics-Informed Neural Networks (PINNs), the proposed method employs graph-structured representations to model unstructured Finite Element (FE) meshes. In this framework, nodes encode physical quantities such as displacements and loads, while edges represent geometric or topological relationships. The framework embeds PDE constraints as soft penalties within the loss function, ensuring adherence to physical laws while reducing reliance on large datasets. Extensive experiments have demonstrated the GS-PI-DeepONet’s superiority over traditional Finite Element Methods (FEMs) and standard DeepONets. For benchmark problems, including cantilever beam bending and Hertz contact, the model achieves high accuracy. In practical applications, such as stiffness analysis of a recliner mechanism and strength analysis of a support bracket, the framework achieves a 7–8 speed-up compared to FEMs, while maintaining fidelity comparable to FEM, with R2 values reaching up to 0.9999 for displacement fields. Consequently, the GS-PI-DeepONet offers a resolution-independent, data-efficient, and physics-consistent approach for real-time simulations, making it ideal for rapid parameter sweeps and design optimizations in engineering applications.

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Fluid flows are present in various fields of science and engineering, so their mathematical description and modeling is of high practical importance. However, utilizing classical numerical methods to model fluid flows is often time consuming and a new simulation is needed for each modification of the domain, boundary conditions, or fluid properties. As a result, these methods have limited utility when it comes to conducting extensive parameter studies or optimizing fluid systems. By utilizing recently proposed physics-informed neural networks (PINNs), these limitations can be addressed. PINNs approximate the solution of a single or system of partial differential equations (PDEs) by artificial neural networks (ANNs). The residuals of the PDEs are used as the loss function of the ANN, while the boundary condition is imposed in a supervised manner. Hence, PDEs are solved by performing a nonconvex optimization during the training of the ANN instead of solving a system of equations. Although this relatively new method cannot yet compete with classical numerical methods in terms of accuracy for complex problems, this approach shows promising potential as it is mesh-free and suitable for parametric solution of PDE problems. This is achieved without relying on simulation data or measurement information. This study focuses on the impact of parametric boundary conditions, specifically a variable inlet velocity profile, on the flow calculations. For the first time, a physics-based penalty term to avoid the suboptimal solution along with an efficient way of imposing parametric boundary conditions within PINNs is presented.

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The inception of physics-constrained or physics-informed machine learning represents a paradigm shift, addressing the challenges associated with data scarcity and enhancing model interpretability. This innovative approach incorporates the fundamental laws of physics as constraints, guiding the training process of machine learning models. In this work, the physics-constrained convolutional recurrent neural network is further extended for solving spatial-temporal partial differential equations with arbitrary boundary conditions. Two notable advancements are introduced: the implementation of boundary conditions as soft constraints through finite difference-based differentiation, and the establishment of an adaptive weighting mechanism for the optimal allocation of weights to various losses. These enhancements significantly augment the network’s ability to manage intricate boundary conditions and expedite the training process. The efficacy of the proposed model is validated through its application to two-dimensional problems in heat transfer, phase transition, and fluid dynamics, which are pivotal in materials modeling. Compared to traditional physics-constrained neural networks, the physics-constrained convolutional recurrent neural network demonstrates a tenfold increase in prediction accuracy within a similar computational budget. Moreover, the model’s exceptional performance in extrapolating solutions for the Burgers’ equation underscores its utility. Consequently, this research establishes the physics-constrained recurrent neural network as a viable surrogate model for sophisticated spatiotemporal PDE systems, particularly beneficial in scenarios plagued by sparse and noisy datasets.

Graph distribution shifts between training and test graphs pose severe challenges to the generalization of graph neural networks (GNNs). In real-world deployment, application environments are continuously evolving, while retraining or redesigning GNNs is often costly and impractical. In light of this, test-time adaptation on graphs, which aims to dynamically adapt well-trained GNNs or adjust test graphs to improve inference performance, has attracted growing attention as a practical solution. In this survey, we provide a comprehensive review of test-time adaptation on graphs, an emerging yet underexplored research direction. We identify two fundamental challenges: (1) Data-level: complex graph distribution shifts; and (2) Model-level: limited test-time learning information. Upon this, we present a systematic taxonomy of existing methods into (a) model-centric, (b) data-centric, and (c) hybrid methods, followed by a summary of representative applications, benchmarks, and open opportunities. We aim to bridge the gap between laboratory GNN development and real-world deployment via test-time adaptation. The repository of papers, code, and datasets is at https://github.com/jiayichen1121/Test-Time-Adaptation-for-Graph-Learning.

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

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Optical simulation plays an important role in photonic hardware design flow. The finite-difference time-domain (FDTD) method is widely adopted to solve time-domain Maxwell equations. However, FDTD is known for its prohibitive runtime cost as it iteratively solves Maxwell equations and takes minutes to hours to simulate a single device. Recently, AI has been applied to realize orders-of-magnitude speedup in partial differential equation solving. However, AI-based FDTD solvers for photonic devices have not been clearly formulated. Directly applying off-the-shelf models to predict the optical field dynamics shows unsatisfying fidelity and efficiency since the model primitives are agnostic to the unique physical properties of Maxwell equations and lack algorithmic customization. In this work, we thoroughly investigate the synergy between neural operator designs and the physical property of Maxwell equations and introduce a physics-inspired AI-based FDTD prediction framework PIC2O-Sim. PIC2O-Sim features a causality-aware dynamic convolutional neural operator as its backbone model that honors the space–time causality constraints via careful receptive field configuration and explicitly captures the permittivity-dependent light propagation behavior via an efficient dynamic convolution operator. Meanwhile, we explore the trade-offs among prediction scalability, fidelity, and efficiency via a multi-stage partitioned time-bundling technique in autoregressive prediction. Multiple key techniques have been introduced to mitigate iterative error accumulation while maintaining efficiency advantages during autoregressive field prediction. Extensive evaluations on three challenging photonic device simulation tasks have shown the superiority of our PIC2O-Sim method, showing 51.2% lower roll-out prediction error, 23.5 times fewer parameters than state-of-the-art neural operators, providing 133–310× or 31–89× higher simulation speed than an open-source single-process or eight-process parallel FDTD numerical solver.

This paper introduces the Temporal Graph Neural Network-based Simulator (TGNNS), a novel physical-information-flow-constrained deep learning-based simulator for granular material modeling. The TGNNS leverages a series of frames, each representing material point positions, enabling particle dynamics to propagate through the sequence, resulting in a more physically grounded architecture for granular flow learning. The TGNNS has been thoroughly trained, validated, and tested using simulation data derived from a hierarchical multiscale modeling approach, DEMPM, which combines the Material Point Method (MPM) and the Discrete Element Method (DEM). Results demonstrate that the TGNNS performs robustly with previously unseen datasets of varying granular column sizes, even under manually incorporated barrier boundary conditions. Remarkably, the TGNNS operates at a speed 100 times faster than direct numerical simulation using the state-of-the-art GPU-based DEMPM. Employing a unique deep learning architecture that is constrained by the flow of physical information, the TGNNS offers a pioneering learning paradigm for multiscale emerging behaviors of granular materials and provides a potential solution to physics-based modeling in digital twins involving granular materials.

Abstract. Deep-learning-based surrogate models represent a powerful alternative to numerical models for speeding up flood mapping while preserving accuracy. In particular, solutions based on hydraulic-based graph neural networks (SWE-GNNs) enable transferability to domains not used for training and allow the inclusion of physical constraints. However, these models are limited due to four main aspects. First, they cannot model rapid differences in flow propagation speeds; secondly, they can face instabilities during training when using a large number of layers, needed for effective modelling; third, they cannot accommodate time-varying boundary conditions; and fourth, they require initial conditions from a numerical solver. To address these issues, we propose a multi-scale hydraulic-based graph neural network (mSWE-GNN) that models the flood at different resolutions and propagation speeds. We include time-varying boundary conditions via ghost cells, which enforce the solution at the domain’s boundary and drop the need for a numerical solver for the initial conditions. To improve generalization over unseen meshes and reduce the data demand, we use invariance principles and make the inputs independent from coordinates' rotations. Numerical results applied to dike-breach floods show that the model predicts the full spatio-temporal simulation of the flood over unseen irregular meshes, topographies, and time-varying boundary conditions, with mean absolute errors in time of 0.05 m for water depths and 0.003 m2 s−1 for unit discharges. We further corroborate the mSWE-GNN in a realistic case study in the Netherlands and show generalization capabilities with only one fine-tuning sample, with mean absolute errors of 0.12 m for water depth, a critical success index for a water depth threshold of 0.05 m of 87.68 %, and speed-ups of over 700 times. Overall, the approach opens up several avenues for probabilistic analyses of realistic configurations and flood scenarios.

In this study, we introduce a Graph network‐enhanced Finite Element approach to accelerate Finite Element simulations. We utilize the discretized geometry from a Finite Element pre‐processor to establish the graph and use the Graph Neural Network to solve the boundary value problem of the discretized domain. The advantage of graph neural networks is that they have a similar structure as compared to a discretized domain with nodes and elements. The underlying dynamics of the system are computed via a learned message‐passing. The goal here is to enhance and accelerate the FEM simulations using the proposed GNN network by incorporating the underlying mechanics knowledge into the network to enhance the generalizing ability of the network on various loading and boundary conditions. All the proposed studies in the literature where graph networks are applied to Finite Element Methods use images as input and output. The advantage of the proposed model is that it takes inputs such as the nodal information, their corresponding edges, nodal coordinates and the boundary conditions for each particular node from a Finite Element pre‐processor and computes the von‐Mises stress at each node along with their edge connections as output that can be read by a Finite Element post‐processor.

Physics-informed neural networks (PINNs) have emerged as a popular approach in scientific machine learning for solving both forward and inverse problems of partial differential equations (PDEs). However, complex physical systems are often characterized by parameters, such as viscosity and Reynolds number in fluid dynamics, which pose significant challenges for parameterized PDE solutions. The inherent limitations of PINNs include the need for repeated and time-consuming training under varying parameter conditions, and the minimization of PDE residuals with PDE-based soft constraints, which makes the “ill-conditioned” problem. To address these issues, this paper proposes an innovative framework: the pseudo-time stepping and parameterized physics-informed neural network (P2PINN). P2PINN leverages the explicit encoding of only two PDE parameters' latent representations to enable efficient interpolation and extrapolation across a wide range of parameters. By integrating the pseudo-time stepping method with deep learning, the framework significantly alleviates the ill-conditioned problem. We validated our method in the context of the Navier–Stokes equations, and experimental results demonstrate that P2PINN achieves solution speedups of up to 2–4 orders of magnitude compared to baseline PINNs and their variants, while also surpassing them in accuracy.

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.

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Recent advancements in connected vehicle technology have the potential to transform the efficiency and sustainability of transportation systems. Among these, truck platooning has emerged as a promising strategy to enhance freight delivery operations. However, the generalization of truck platoon modeling and the economic impacts of implementing such systems remain areas that require deeper exploration. This paper introduces a novel graph neural network-based surrogate model, namely Graph Network-based Platoon Simulator (GNPS), designed to predict aerodynamic drag forces in truck platoons across a wide range of traffic and environmental conditions. By representing the platoon and surrounding traffic as a heterogeneous graph, the proposed GNPS model leverages node-level features such as wind configurations and vehicle geometries, alongside edge-level features like inter-vehicle distances. In comparison to traditional CFD simulations, the proposed GNPS model provides a cost-effective and computationally efficient alternative, offering dynamic adaptability to real-world traffic scenarios.

High-fidelity computational fluid dynamics (CFD) simulations are widely used to analyse three-dimensional turbulent wake flows in large-scale engineering domains, but remain too expensive for iterative design optimisation and real-time decision making. This paper proposes an attention-based graph neural network (GNN) surrogate that learns a direct mapping from short inflow velocity histories to three-dimensional (3D) wake fields on an unstructured monitoring grid. The model operates on CFD sampling points connected by a radius-based graph, and combines a temporal inflow encoder with a graph message-passing decoder to reconstruct all three velocity components in a downstream region of interest. A proof-of-concept study is conducted on a high-fidelity CFD dataset emulating a canonical inflow–wake configuration representative of array-type bluff-body flows, where both inflow and interior data are extracted from the same solver. Numerical results show that the surrogate reproduces the main velocity deficit and its downstream recovery with sub-m/s errors, explaining a non-negligible portion of the variance in the dominant streamwise component while capturing large-scale lateral and vertical spreading patterns. Once trained, the GNN surrogate reduces the cost of full-field prediction from hours of CFD computation to milliseconds of neural-network inference, highlighting its potential as a bridge between mesoscale inflow information (e.g. NWP- or lidar-based reconstructions) and fast volumetric flow-field prediction in complex engineering environments.

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.

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

Accurate and efficient prediction of chemical nonequilibrium aerodynamic heating remains a fundamental challenge in spacecraft design. High-fidelity computational fluid dynamics (CFD) simulations are computationally prohibitive due to stiff chemical source terms, whereas purely data-driven approaches often lack physical interpretability and generalize poorly across varying catalytic wall conditions. To address these limitations, a physics-guided prior graph neural network is proposed. The central idea of the proposed framework is a physics-based multifidelity fusion strategy, in which a computationally inexpensive perfect-gas solution is employed as a low-fidelity physical model before capturing the topological structure of the flowfield. A dual-branch graph-based architecture is constructed to explicitly decouple the total wall heat flux into a conductive component, governed primarily by macroscopic thermal gradients, and a diffusive component driven by surface catalytic reactions. Numerical experiments conducted on double-ellipsoid and lifting-body configurations indicate that the proposed framework provides a sevenfold speedup relative to chemical-nonequilibrium CFD simulations while maintaining high predictive fidelity. The model exhibits robust performance even under complex regimes involving Mach number and wall temperature extrapolation alongside catalytic efficiency interpolation. This work effectively bridges the gap between physical mechanism decoupling and data-driven efficiency, offering a robust tool for rapid aerothermal design.

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.

A predictive model has been developed based on experimental data obtained from open channel flow past two horizontal cylinders, one above the other, using a graph neural network framework. Upon investigating the error plots, it is evident that the decreasing mean square error indicates the model's capability to predict near accurate data. Furthermore, second- and third-order structure functions have been illustrated to validate Kolmogorov's 2/3 and 4/5 laws for both original and predicted data, which were found to be a little inconsistent near the two cylinders in the near-wake zone. To investigate this inconsistency further, scaling exponents from extended self-similarity have been calculated and plotted with the conventional Kolmogorov 1941 theory in order to visualize the deviation. It is found that strong anisotropic flow occurs in the vicinity of the rough bed and two cylinders, which resulted in substantial intermittency. This article bears a significant contribution to the understanding of turbulence intermittency and deep learning implementation in turbulent open channel flow.

Cylindrical members of offshore platforms and sea-crossing bridges are continuously exposed to traveling-wave-induced hydrodynamic loads, where rapid prediction and mechanistic interpretation of free-surface gas–liquid flow are both desirable. Conventional finite-volume simulations for complex two-phase configurations remain expensive and are difficult to reuse across boundary or geometric changes. To address these issues, we develop a physics-informed neural network framework that combines a residual-connected convolutional neural network geometric encoder with a Transformer attention mechanism in a transfer-learning architecture, termed the physics-informed residual-connected convolutional transformer network (PRCT-Net). High-fidelity reference solutions are generated using the Reynolds-averaged Navier–Stokes equations coupled with the volume of fluid method and the shear stress transport turbulence model and are used to validate traveling-wave-driven two-phase flow past two cylinders under multiple operating conditions. PRCT-Net achieves high-fidelity reconstruction of the free surface and the associated velocity and pressure fields while substantially accelerating inference; under the baseline condition, the maximum relative error of wave height is about 3.2%, velocity errors remain below 4% at all sampled instants, and the overall error is within 3.5%. A hierarchical freeze–unfreeze fine-tuning strategy further enables efficient cross-condition adaptation, reaching the target accuracy within 119 epochs while preserving a stable geometric–temporal latent space. From a mechanistic perspective, an overlapping V-shaped free-surface wake forms downstream at the steady stage and gradually decouples as cylinder spacing increases. The Ω field reveals V-shaped downstream vortex bands whose trajectory angles closely match those extracted from the free-surface wake. Leveraging PRCT-Net for rapid sampling over operating conditions, an empirical correlation for the wake–vortex trajectory angles is fitted, providing a quantitative and efficient tool for characterizing wake interference and vortex-band dynamics in free-surface flow past two cylinders.

Stable density stratification introduces buoyancy effects, vortex–wave coupling, and internal-wave radiation into body wakes, making full-field prediction more challenging than in homogeneous flows. This study develops a residual-gated mesh graph neural network (RGGNN) for coupled density–velocity prediction of two-dimensional stratified circular-cylinder wakes using large-eddy-simulation data. The model combines a mesh-based encoder–processor–decoder architecture with residual and gated incremental updates to reduce autoregressive error accumulation and preserve local flow structures. It is evaluated through single-condition autoregressive rollout, cross-Froude-number prediction under linear stratification, and cross-condition prediction under hyperbolic-tangent stratification, together with a direct-transfer test on an unseen tandem-cylinder wake. At the 10th rollout step, the RGGNN reduces the root mean square errors of density, horizontal velocity, and vertical velocity by 96.5%, 84.4%, and 86.1%, respectively, relative to the baseline GNN. The extended 49-step rollout further shows substantially slower error growth. For the unseen-Froude-number case, the density root mean square error decreases from 0.099 37 to 0.005 12, while the coefficient of determination increases from 0.989 29 to 0.999 97. Under hyperbolic-tangent stratification, the density root mean square error is reduced by 76.3%–83.1%. In the unseen tandem-cylinder test, the RGGNN maintains coefficients of determination above 0.994 for all predicted variables without retraining. These results demonstrate improved rollout stability, local-feature preservation, and preliminary transferability across unseen geometry and mesh topology, indicating the potential of residual-gated mesh graph learning for coupled density–velocity prediction of stratified wakes within controlled numerical settings.

As a mesh-free solving paradigm, Physics-Informed Neural Networks (PINNs) demonstrate potential in both forward and inverse problems by embedding physical equations into the loss function. However, they still face challenges in capturing the spatiotemporal evolution of complex physical processes. When applied to time-dependent complex flows, such as high-Reynolds-number cylinder flow, they often rely on supervised data, which is frequently difficult to obtain accurately in practice. To address these issues, this paper proposes a novel unsupervised solving framework—the Adaptive Hard-Constraint Physics-Informed Neural Network (AHC-PINN). This method integrates an adaptive sampling mechanism based on partial differential equation residuals with a hard-constraint strategy. By dynamically evaluating the contribution of collocation points to the loss and incorporating analytically embedded boundary constraints, it directs the network training entirely toward solving the governing equations. Using two-dimensional unsteady cylinder flow as a validation case, experimental results show that AHC-PINN significantly improves the prediction accuracy of wake evolution under unsupervised conditions. Its performance surpasses that of traditional soft-constraint PINNs by an order of magnitude and is even superior to methods using sparse supervised data. Furthermore, through analysis of the PDE loss and gradient distribution, the study explicitly identifies the impact of large-gradient regions on PINN training stability and prediction accuracy, providing a basis for subsequent optimization.

No abstract available

No abstract available

Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a novel family of neural operators based on our Graph Fourier Neural Kernels, designed to learn solution generators for nonlinear PDEs in which the highest-order term is diffusive, across multiple domains and parameters. G-FuNK combines components that are parameter- and domain-adapted with others that are not. The domain-adapted components are constructed using a weighted graph on the discretized domain, where the graph Laplacian approximates the highest-order diffusive term, ensuring boundary condition compliance and capturing the parameter and domain-specific behavior. Meanwhile, the learned components transfer across domains and parameters using our variant Fourier Neural Operators. This approach naturally embeds geometric and directional information, improving generalization to new test domains without need for retraining the network. To handle temporal dynamics, our method incorporates an integrated ODE solver to predict the evolution of the system. Experiments show G-FuNK's capability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations across various geometries and anisotropic diffusivity fields. G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions compared to traditional finite-element solvers.

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

This work develops a neural operator preconditioned subspace method for sequences of linear systems arising from the discretization of the two-dimensional Helmholtz equation in transcranial ultrasound applications. The problem involves strongly heterogeneous, patient-dependent velocity fields that induce severe wave distortion and pose significant challenges for standard iterative solvers. Building on neural network preconditioning framework of Giraud et al. (HAL RR-9593, 2025) and the idealized skull dataset used for the learned optimizer of Stanziola et al. (JCP 441, 2021), neural operator preconditioners are trained on six mixed velocity-source datasets combining randomized source configurations and idealized skull-based velocity fields with random noise. The proposed mixed-dataset strategy aims to improve both computational efficiency and generalization across varying configurations. The neural operator is trained on a coarse grid using a physics-informed loss based on the relative residual of the discrete Helmholtz equation and is incorporated as a nonlinear preconditioner within flexible GMRES (FGMRES). Numerical experiments demonstrate that the resulting hybrid method efficiently solves practical transcranial ultrasound problems on grids 64 times larger than those used during training, whereas both classical GMRES and the learned optimizer fail to converge within comparable computational budgets. Moreover, the proposed method achieves arbitrary solution accuracies and exhibits strong out-of-distribution generalization across diverse source and velocity configurations. This work highlights the importance of dataset design in scientific machine learning and provides a practical framework for integrating matrix-free neural operator preconditioning with Krylov subspace methods for solving practical large-scale Helmholtz problems.

Hyperelastic deformations are highly sensitive to domain geometry and boundary conditions, making generalization across both a critical capability for neural operators applied to these problems. However, existing benchmarks for neural operators on hyperelasticity rely on simple or few geometries, which makes it difficult to assess this capability rigorously. To address this gap, we introduce HyperShape, an extensible framework designed to generate synthetic shapes and their corresponding hyperelastic simulation data, producing a suite of 2D and 3D datasets with adjustable complexity and controllable shape variations. This design enables systematic assessment of generalization across in-distribution, out-of-distribution, and synthetic-to-real transfer settings. Using this framework, we evaluated the performance of several state-of-the-art neural operators over diverse shape distributions. Our findings reveal that neural operators perform well on simple shapes but struggle as shape complexity, geometric diversity, and boundary condition variability increase, requiring large amounts of training data in such regimes. Performance degrades consistently and predictably with geometric complexity highlighting the need for further model development. As an open and extensible benchmark, HyperShape is designed to grow alongside the field: new geometries, material models, loading conditions, and evaluation settings can be easily incorporated to validate hyperelastic surrogate models.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of PDEs. However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre-discretized temporal grids. Drawing inspiration from MeanFlow, a continuous-time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi-step numerical integration into an efficient prediction with a freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the spatio-temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time-dependent and stationary PDEs. Comprehensive experiments on benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out-of-distribution initial conditions and varying spatial resolutions.

Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi-steady-state values in a single differentiable solve. Embedded in a physics-informed DeepONet, the layer recovers all fast and algebraic states exactly from slow-state predictions, removes the per-window stiffness-amplification pathway, and yields a stiffness-scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi-component systems with provable convergence. On a grid-forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross-domain validation on the Robertson stiff DAE (stiffness ratio up to $10^5$) confirms generalization. Conformal prediction provides 90% coverage with automatic out-of-distribution detection.

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods.

Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications. In practical applications, variations in physical parameters induce distribution shifts between training and prediction regimes, making extrapolation a central challenge. As a result, the way parameters are incorporated into neural operator models plays a key role in their ability to generalize, particularly when state and parameter representations are entangled. In this work, we introduce the Late Fusion Neural Operator, an architecture that disentangles learning state dynamics from parameter effects, improving predictive performance both within and beyond the training distribution. Our approach combines neural operators for learning latent state representations with sparse regression to incorporate parameter information in a structured manner. Across four benchmark PDEs including advection, Burgers, and both 1D and 2D reaction-diffusion equations, the proposed method consistently outperforms Fourier Neural Operator and CAPE-FNO. Late Fusion Neural Operators achieve consistently the best performance in all experiments, with an average RMSE reduction of 72.9% in-domain and 71.8% out-domain compared to the second-best method. These results demonstrate strong generalization across both in-domain and out-domain parameter regimes.

AI for science (AI4Science) models often suffer from discretization: learned representations remain tied to the training grid, limiting transfer across resolutions, solvers and applications. We introduce Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator that learns nonlinear, orthogonal basis functions directly in function space and can be integrated in both projection-based reduced order models and operator-learning frameworks such as DeepONet. Neural-POD replaces SVD-derived, resolution-dependent linear modes with continuous, resolution-invariant bases learned via sequential residual minimization, analogous to Gram-Schmidt orthogonalization. The framework supports training under task-specific norms (e.g., $L^2$, $L^1$), improves out-of-distribution generalization to unseen parameter regimes, and captures nonlinear structure in complex systems. Because the learned bases are interpretable and reusable, Neural-POD serves as a general representation module for AI4Science workflows. We demonstrate Neural-POD on Burgers'and Navier-Stokes equations.

We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries. To this end, we introduce General-Geometry Neural Whitney Forms (Geo-NeW): a data-driven finite element method. We jointly learn a differential operator and compatible reduced finite element spaces defined on the underlying geometry. The resulting model is solved to generate predictions, while exactly preserving physical conservation laws through Finite Element Exterior Calculus. Geometry enters the model as a discretized mesh both through a transformer-based encoding and as the basis for the learned finite element spaces. This explicitly connects the underlying geometry and imposed boundary conditions to the solution, providing a powerful inductive bias for learning neural PDEs, which we demonstrate improves generalization to unseen domains. We provide a novel parameterization of the constitutive model ensuring the existence and uniqueness of the solution. Our approach demonstrates state-of-the-art performance on several steady-state PDE benchmarks, and provides a significant improvement over conventional baselines on out-of-distribution geometries.

Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics. Prior works address this limitation with large-scale multiple physics pretraining followed by fine-tuning, but this still require examples from the new dynamics, falling short of true zero-shot generalization. In this work, we propose a method to enhance generalization at test time, i.e., without modifying pretrained weights. Building on DISCO, which provides a dictionary of neural operators trained across different dynamics, we introduce a neural operator splitting strategy that, at test time, searches over compositions of training operators to approximate unseen dynamics.On challenging out-of-distribution tasks including parameter extrapolation and novel combinations of physics phenomena, our approach achieves state-of-the-art zero-shot generalization results, while being able to recover the underlying PDE parameters. These results underscore test-time computation as a key avenue for building flexible, compositional, and generalizable neural operators.

Accurate modeling of spatiotemporal dynamics is crucial to understanding complex phenomena across science and engineering. However, this task faces a fundamental challenge when the governing equations are unknown and observational data are sparse. System stiffness, the coupling of multiple time-scales, further exacerbates this problem and hinders long-term prediction. Existing methods fall short: purely data-driven methods demand massive datasets, whereas physics-aware approaches are constrained by their reliance on known equations and fine-grained time steps. To overcome these limitations, we introduce an equation-free learning framework, namely, the Stable Spectral Neural Operator (SSNO), for modeling stiff partial differential equation (PDE) systems based on limited data. Instead of encoding specific equation terms, SSNO embeds spectrally inspired structures in its architecture, yielding strong inductive biases for learning the underlying physics. It automatically learns local and global spatial interactions in the frequency domain, while handling system stiffness with a robust integrating factor time-stepping scheme. Demonstrated across multiple 2D and 3D benchmarks in Cartesian and spherical geometries, SSNO achieves prediction errors one to two orders of magnitude lower than leading models. Crucially, it shows remarkable data efficiency, requiring only very few (2--5) training trajectories for robust generalization to out-of-distribution conditions. This work offers a robust and generalizable approach to learning stiff spatiotemporal dynamics from limited data without explicit \textit{a priori} knowledge of PDE terms.

Solving partial differential equations (PDEs) can be prohibitively expensive using traditional numerical methods. Deep learning-based surrogate models typically specialize in a single PDE with fixed parameters. We present a framework for equation-aware emulation that generalizes to unseen PDEs, conditioning a neural model on a vector encoding representing the terms in a PDE and their coefficients. We present a baseline of four distinct modeling technqiues, trained on a family of 1D PDEs from the APEBench suite. Our approach achieves strong performance on parameter sets held out from the training distribution, with strong stability for rollout beyond the training window, and generalization to an entirely unseen PDE. This work was developed as part of a broader effort exploring AI systems that automate the creation of expert-level empirical software for scorable scientific tasks. The data and codebase are available at https://github.com/google-research/generalized-pde-emulator.

PDEs arise ubiquitously in science and engineering, where solutions depend on parameters (physical properties, boundary conditions, geometry). Traditional numerical methods require re-solving the PDE for each parameter, making parameter space exploration prohibitively expensive. Recent machine learning advances, particularly physics-informed neural networks (PINNs) and neural operators, have revolutionized parametric PDE solving by learning solution operators that generalize across parameter spaces. We critically analyze two main paradigms: (1) PINNs, which embed physical laws as soft constraints and excel at inverse problems with sparse data, and (2) neural operators (e.g., DeepONet, Fourier Neural Operator), which learn mappings between infinite-dimensional function spaces and achieve unprecedented generalization. Through comparisons across fluid dynamics, solid mechanics, heat transfer, and electromagnetics, we show neural operators can achieve computational speedups of $10^3$ to $10^5$ times faster than traditional solvers for multi-query scenarios, while maintaining comparable accuracy. We provide practical guidance for method selection, discuss theoretical foundations (universal approximation, convergence), and identify critical open challenges: high-dimensional parameters, complex geometries, and out-of-distribution generalization. This work establishes a unified framework for understanding parametric PDE solvers via operator learning, offering a comprehensive, incrementally updated resource for this rapidly evolving field

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

Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the \textbf{Graph Spectral Neural Operator (GSNO)}, a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.

Accurate yet efficient surrogate models are essential for large-scale simulations of partial differential equations (PDEs), particularly for uncertainty quantification (UQ) tasks that demand hundreds or thousands of evaluations. We develop a physics-inspired graph neural network (GNN) surrogate that operates directly on unstructured meshes and leverages the flexibility of graph attention. To improve both training efficiency and generalization properties of the model, we introduce a domain decomposition (DD) strategy that partitions the mesh into subdomains, trains local GNN surrogates in parallel, and aggregates their predictions. We then employ transfer learning to fine-tune models across subdomains, accelerating training and improving accuracy in data-limited settings. Applied to ice sheet simulations, our approach accurately predicts full-field velocities on high-resolution meshes, substantially reduces training time relative to training a single global surrogate model, and provides a ripe foundation for UQ objectives. Our results demonstrate that graph-based DD, combined with transfer learning, provides a scalable and reliable pathway for training GNN surrogates on massive PDE-governed systems, with broad potential for application beyond ice sheet dynamics.

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled input-output pairs, and sampling is performed once, uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).

Operator learning seeks to approximate mappings from input functions to output solutions, particularly in the context of partial differential equations (PDEs). While recent advances such as DeepONet and Fourier Neural Operator (FNO) have demonstrated strong performance, they often rely on regular grid discretizations, limiting their applicability to complex or irregular domains. In this work, we propose a Graph-based Operator Learning with Attention (GOLA) framework that addresses this limitation by constructing graphs from irregularly sampled spatial points and leveraging attention-enhanced Graph Neural Netwoks (GNNs) to model spatial dependencies with global information. To improve the expressive capacity, we introduce a Fourier-based encoder that projects input functions into a frequency space using learnable complex coefficients, allowing for flexible embeddings even with sparse or nonuniform samples. We evaluated our approach across a range of 2D PDEs, including Darcy Flow, Advection, Eikonal, and Nonlinear Diffusion, under varying sampling densities. Our method consistently outperforms baselines, particularly in data-scarce regimes, demonstrating strong generalization and efficiency on irregular domains.

Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on arbitrary discretization schemes for computational domain. In this work, we propose a novel operator learning algorithm, referred to as Dynamic Gaussian Graph Operator (DGGO) that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The Dynamic Gaussian Graph (DGG) kernel learns to map the observation vectors defined in general Euclidean space to metric vectors defined in high-dimensional uniform metric space. The DGG integral kernel is parameterized by Gaussian kernel weighted Riemann sum approximating and using dynamic message passing graph to depict the interrelation within the integral term. Fourier Neural Operator is selected to localize the metric vectors on spatial and frequency domains. Metric vectors are regarded as located on latent uniform domain, wherein spatial and spectral transformation offer highly regular constraints on solution space. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems in comparison with mainstream neural operators. Ablation experiments are implemented to demonstrate the effectiveness of spatial transformation in the DGG kernel. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge - the Edge Augmented GNN and the Multi-GNN. We show that both these networks perform significantly better (by a factor of 1.5 to 2) than baseline methods when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings.

Graph-based spatio-temporal neural networks are effective to model the spatial dependency among discrete points sampled irregularly from unstructured grids, thanks to the great expressiveness of graph neural networks. However, these models are usually spatially-transductive – only fitting the signals for discrete spatial nodes fed in models but unable to generalize to ‘unseen’ spatial points with zero-shot. In comparison, for forecasting tasks on continuous space such as temperature prediction on the earth’s surface, the spatially-inductive property allows the model to generalize to any point in the spatial domain, demonstrating models’ ability to learn the underlying mechanisms or physics laws of the systems, rather than simply fit the signals. Besides, in temporal domains, irregularly-sampled time series, e.g. data with missing values, urge models to be temporally-continuous. Motivated by the two issues, we pro-pose a spatio-temporal framework based on neural operators for PDEs, which learn the underlying mechanisms governing the dynamics of spatially-continuous physical quantities. Experiments show our model’s improved performance on forecasting spatially-continuous physic quantities, and its superior generalization to unseen spatial points and ability to handle temporally-irregular data.