# A deep learning energy method for hyperelasticity and viscoelasticity

arXiv: https://arxiv.org/abs/2201.08690 · HTML: https://mlsanity.com/paper/2201.08690/

## 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.

## Builds on (most cited first, 13 of 13)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Deep Generative Modeling for Mechanistic-based Learning and Design of Metamaterial Systems](https://mlsanity.com/paper/2006.15274/) (2020)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Meshless physics-informed deep learning method for three-dimensional solid mechanics](https://mlsanity.com/paper/2012.01547/) (2020)
- [GINNs: Graph-Informed Neural Networks for Multiscale Physics](https://mlsanity.com/paper/2006.14807/) (2020)
- [Prediction and optimization of mechanical properties of composites using convolutional neural networks](https://mlsanity.com/paper/1906.00094/) (2019)
- [Discovery of Governing Equations with Recursive Deep Neural Networks](https://mlsanity.com/paper/2009.11500/) (2020)

## Built on by (most cited first, 20 of 20)

- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient](https://mlsanity.com/paper/2506.04375/) (2025)
- [Learning the nonlinear dynamics of soft mechanical metamaterials with graph networks](https://mlsanity.com/paper/2202.13775/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [DCEM: A deep complementary energy method for solid mechanics](https://mlsanity.com/paper/2302.01538/) (2023)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Variational volume reconstruction with the Deep Ritz Method](https://mlsanity.com/paper/2508.08309/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Physics-informed DeepONet with stiffness-based loss functions for structural response prediction](https://mlsanity.com/paper/2409.00994/) (2024)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)
- [Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structures](https://mlsanity.com/paper/2505.07090/) (2025)
- [Nonlinear discretizations and Newton's method: characterizing stationary points of regression objectives](https://mlsanity.com/paper/2510.11987/) (2025)
- [Finding geodesics with the Deep Ritz method](https://mlsanity.com/paper/2510.15177/) (2025)
- [FEM-Informed Hypergraph Neural Networks for Efficient Elastoplasticity](https://mlsanity.com/paper/2602.07364/) (2026)
- [A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beams](https://mlsanity.com/paper/2501.06925/) (2025)
- [Nonlinear elliptic homogenization with the parametric Deep Ritz method](https://mlsanity.com/paper/2609.05778/) (2026)

## 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.