mlsanityData to 14 Oct 2026

A deep learning energy method for hyperelasticity and viscoelasticity

arXiv 2201.08690 ↗First posted 15 Jan 2022cs.LG cs.NA math.NAcited by 20

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Abstract (arXiv)

The potential energy formulation and deep learning are merged to solve partial differential equations governing the deformation in hyperelastic and viscoelastic materials. The presented deep energy method (DEM) is self-contained and meshfree. It can accurately capture the three-dimensional (3D) mechanical response without requiring any time-consuming training data generation by classical numerical methods such as the finite element method. Once the model is appropriately trained, the response can be attained almost instantly at any point in the physical domain, given its spatial coordinates. Therefore, the deep energy method is potentially a promising standalone method for solving partial differential equations describing the mechanical deformation of materials or structural systems and other physical phenomena.

Key facts. Cited by 20 papers in the map (data through 14 Oct 2026). Builds on 13 of its 53 references that are in the map. First posted on arXiv 15 Jan 2022.

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13 of 53 references.

References · 13

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  3. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  4. 2020Physics informed deep learning for computational elastodynamics without labeled datacited by 80
  5. 2019Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transportcited by 61
  6. 2020Deep Generative Modeling for Mechanistic-based Learning and Design of Metamaterial Systemscited by 52
  7. 2020Unsupervised discovery of interpretable hyperelastic constitutive lawscited by 51
  8. 2020Deep learning of thermodynamics-aware reduced-order models from datacited by 38
  9. 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
  10. 2020Meshless physics-informed deep learning method for three-dimensional solid mechanicscited by 27
  11. 2020GINNs: Graph-Informed Neural Networks for Multiscale Physicscited by 12
  12. 2019Prediction and optimization of mechanical properties of composites using convolutional neural networkscited by 10
1 more
  1. 2020Discovery of Governing Equations with Recursive Deep Neural Networkscited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 2; 2024: 5; 2025: 8; 2026: 2.

20 papers.

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  1. 2024A review on data-driven constitutive laws for solidscited by 35
  2. 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
  3. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  4. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  5. 2025Solving engineering eigenvalue problems with neural networks using the Rayleigh quotientcited by 8
  6. 2022Learning the nonlinear dynamics of soft mechanical metamaterials with graph networkscited by 7
  7. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 7
  8. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  9. 2023DCEM: A deep complementary energy method for solid mechanicscited by 5
  10. 2024Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticitycited by 2
  11. 2025Variational volume reconstruction with the Deep Ritz Methodcited by 2
  12. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 2
8 more
  1. 2024Physics-informed DeepONet with stiffness-based loss functions for structural response predictioncited by 1
  2. 2025Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force methodcited by 1
  3. 2025Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structurescited by 1
  4. 2025Nonlinear discretizations and Newton's method: characterizing stationary points of regression objectivescited by 1
  5. 2025Finding geodesics with the Deep Ritz methodcited by 1
  6. 2026FEM-Informed Hypergraph Neural Networks for Efficient Elastoplasticitycited by 1
  7. 2025A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beamscited by 0
  8. 2026Nonlinear elliptic homogenization with the parametric Deep Ritz methodcited by 0