mlsanityData to 14 Oct 2026

Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations

arXiv 2401.02363 ↗First posted 4 Jan 2024cs.LG cs.CEcited by 4

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Abstract (arXiv)

We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase contrast. Similar equations manifest in diverse applications like chemical diffusion, electrostatics, and Darcy flow. The neural network aims to establish the link between the complex thermal conductivity profiles and temperature distributions, as well as heat flux components within the microstructure, under fixed boundary conditions. A distinctive aspect is our independence from classical solvers like finite element methods for data. A noteworthy contribution lies in our novel approach to defining the loss function, based on the discretized weak form of the governing equation. This not only reduces the required order of derivatives but also eliminates the need for automatic differentiation in the construction of loss terms, accepting potential numerical errors from the chosen discretization method. As a result, the loss function in this work is an algebraic equation that significantly enhances training efficiency. We benchmark our methodology against the standard finite element method, demonstrating accurate yet faster predictions using the trained neural network for temperature and flux profiles. We also show higher accuracy by using the proposed method compared to purely data-driven approaches for unforeseen scenarios.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 34 of its 56 references that are in the map. First posted on arXiv 4 Jan 2024.

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34 of 56 references.

References · 34

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  12. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
22 more
  1. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
  2. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  3. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  4. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  5. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  6. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  7. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  8. 2023Learning Physical Models that Can Respect Conservation Lawscited by 59
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  10. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  11. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  12. 2021Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Datacited by 31
  13. 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
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  16. 2023Learning in latent spaces improves the predictive accuracy of deep neural operatorscited by 19
  17. 2021Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligencecited by 17
  18. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
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  21. 2023Solving the Discretised Neutron Diffusion Equations using Neural Networkscited by 5
  22. 2023Comparison of Neural FEM and Neural Operator Methods for applications in Solid Mechanicscited by 3

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Cited by year

Citing papers per year: 2024: 4.

4 papers.

  1. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  2. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  3. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  4. 2024Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric spacecited by 3