mlsanityData to 14 Oct 2026

A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations

arXiv 2404.00074 ↗First posted 28 Mar 2024cs.LG cs.CE cs.NAcited by 10

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Abstract (arXiv)

To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This approach generalizes and enhances each method, learning the parametric solution for mechanical problems without relying on data from other resources (e.g. other numerical solvers). We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics-based training yields higher accuracy compared to purely data-driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high-resolution solutions, avoiding the need to use basic interpolation techniques. First is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid point. Next, Fourier-based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. Comparisons with other well-known operator learning algorithms, further emphasize the advantages of the newly proposed method.

Key facts. Cited by 10 papers in the map (data through 14 Oct 2026). Builds on 41 of its 78 references that are in the map. First posted on arXiv 28 Mar 2024.

Builds on

41 of 78 references.

References · 41

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29 more
  1. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 129
  2. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
  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. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
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  11. 2022Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarioscited by 46
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  13. 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
  14. 2023Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loadscited by 33
  15. 2021Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Datacited by 31
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  23. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  24. 2021Predicting Mechanically Driven Full-Field Quantities of Interest with Deep Learning-Based Metamodelscited by 10
  25. 2022On the use of graph neural networks and shape-function-based gradient computation in the deep energy methodcited by 10
  26. 2023Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINNcited by 6
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  29. 2024Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equationscited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 2; 2026: 5.

10 papers.

  1. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  2. 2025Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learningcited by 7
  3. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  4. 2024Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric spacecited by 3
  5. 2026A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materialscited by 1
  6. 2024Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materialscited by 0
  7. 2026Tackling multiphysics problems via finite element-guided physics-informed operator learningcited by 0
  8. 2026Differentiable Chemistry in PINNs for Solving Parameterized and Stiff Reaction Systemscited by 0
  9. 2026Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanicscited by 0
  10. 2026A variational physics-informed graph neural network for heterogeneous solid mechanicscited by 0