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Tackling multiphysics problems via finite element-guided physics-informed operator learning

arXiv 2603.01420 ↗First posted 2 Mar 2026cs.LGcited by 0

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Abstract (arXiv)

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

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References · 36

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  4. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  5. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  6. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  7. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  8. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  9. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 80
  10. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  11. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  12. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 71
24 more
  1. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  2. 2024Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEscited by 58
  3. 2020A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applicationscited by 53
  4. 2020Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacturecited by 48
  5. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  6. 2023A graph convolutional autoencoder approach to model order reduction for parametrized PDEscited by 44
  7. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  8. 2023Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loadscited by 33
  9. 2023GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencodercited by 29
  10. 2023Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionalscited by 26
  11. 2023Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networkscited by 17
  12. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  13. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  14. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  15. 2024Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbonecited by 10
  16. 2024A Spectral-based Physics-informed Finite Operator Learning for Prediction of Mechanical Behavior of Microstructurescited by 9
  17. 2024NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elementscited by 6
  18. 2025Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learningcited by 6
  19. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  20. 2025Sharp-PINNs: staggered hard-constrained physics-informed neural networks for phase field modelling of corrosioncited by 4
  21. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  22. 2025High-fidelity Multiphysics Modelling for Rapid Predictions Using Physics-informed Parallel Neural Operatorcited by 3
  23. 2025Multiphysics Bench: Benchmarking and Investigating Scientific Machine Learning for Multiphysics PDEscited by 3
  24. 2025Universal Fourier Neural Operators for periodic homogenization problems in linear elasticitycited by 3

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