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An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications

arXiv 1908.10407 ↗First posted 27 Aug 2019stat.ML cs.LG math.APcited by 118

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Abstract (arXiv)

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

Key facts. Cited by 118 papers in the map (data through 13 Oct 2026). First posted on arXiv 27 Aug 2019.

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Citing papers per year: 2021: 24; 2022: 24; 2023: 20; 2024: 25; 2025: 11; 2026: 14.

118 papers.

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  1. 2022Physics-Informed Deep Neural Operator Networkscited by 112
  2. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  3. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  4. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 88
  5. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  6. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 66
  7. 2021Physics informed neural networks for continuum micromechanicscited by 46
  8. 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
  9. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  10. 2023Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networkscited by 32
  11. 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
  12. 2023Physics informed WNOcited by 27
38 more
  1. 2022MAgNET: A Graph U-Net Architecture for Mesh-Based Simulationscited by 26
  2. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 25
  3. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  4. 2022Topology Optimization via Machine Learning and Deep Learning: A Reviewcited by 22
  5. 2022A deep learning energy method for hyperelasticity and viscoelasticitycited by 20
  6. 2021Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligencecited by 17
  7. 2023Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networkscited by 17
  8. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  9. 2025Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptationcited by 15
  10. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  11. 2021A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machinescited by 13
  12. 2021Probabilistic Deep Learning for Real-Time Large Deformation Simulationscited by 11
  13. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  14. 2023Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlookcited by 11
  15. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  16. 2021CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometriescited by 10
  17. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  18. 2023BINN: A deep learning approach for computational mechanics problems based on boundary integral equationscited by 10
  19. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  20. 2021Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learningscited by 9
  21. 2021Bayesian neural networks for weak solution of PDEs with uncertainty quantificationcited by 8
  22. 2022Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networkscited by 8
  23. 2022A Neural Network-enhanced Reproducing Kernel Particle Method for Modeling Strain Localizationcited by 8
  24. 2021Online structural health monitoring by model order reduction and deep learning algorithmscited by 7
  25. 2022Koopman operator for time-dependent reliability analysiscited by 7
  26. 2023Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equationscited by 7
  27. 2021Physics-informed attention-based neural network for solving non-linear partial differential equationscited by 6
  28. 2021A deep learning based surrogate model for stochastic simulatorscited by 6
  29. 2022Neural PDE Solvers for Irregular Domainscited by 6
  30. 2023Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINNcited by 6
  31. 2023DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networkscited by 6
  32. 2024Separable Physics-Informed Neural Networks for the solution of elasticity problemscited by 6
  33. 2024Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured gridcited by 6
  34. 2021Distributed Multigrid Neural Solvers on Megavoxel Domainscited by 5
  35. 2022Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equationscited by 5
  36. 2022Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanicscited by 5
  37. 2023DCEM: A deep complementary energy method for solid mechanicscited by 5
  38. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4

50 most cited of 118 All 118 (JSON)