# Energy-based physics-informed neural network for frictionless contact problems under large deformation

arXiv: https://arxiv.org/abs/2411.03671 · HTML: https://mlsanity.com/paper/2411.03671/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 68 references that are in the map.
- First posted on arXiv 6 Nov 2024.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [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)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches](https://mlsanity.com/paper/2010.06050/) (2020)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates](https://mlsanity.com/paper/2010.05698/) (2020)
- [A deep learning energy-based method for classical elastoplasticity](https://mlsanity.com/paper/2209.06467/) (2022)
- [BINN: A deep learning approach for computational mechanics problems based on boundary integral equations](https://mlsanity.com/paper/2301.04480/) (2023)
- [CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries](https://mlsanity.com/paper/2110.01359/) (2021)
- [Artificial intelligence for partial differential equations in computational mechanics: A review](https://mlsanity.com/paper/2410.19843/) (2024)
- [A Pretraining-Finetuning Computational Framework for Material Homogenization](https://mlsanity.com/paper/2404.07943/) (2024)
- [An introduction to programming Physics-Informed Neural Network-based computational solid mechanics](https://mlsanity.com/paper/2210.09060/) (2022)
- [DCEM: A deep complementary energy method for solid mechanics](https://mlsanity.com/paper/2302.01538/) (2023)
- [DEDEM: Discontinuity Embedded Deep Energy Method for solving fracture mechanics problems](https://mlsanity.com/paper/2407.11346/) (2024)
- [Internal contact modeling for finite strain topology optimization](https://mlsanity.com/paper/2010.14277/) (2020)
- [An interface-enriched generalized finite element formulation for locking-free coupling of non-conforming discretizations and contact](https://mlsanity.com/paper/2112.14234/) (2021)

## Built on by (most cited first, 4 of 4)

- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)
- [Distribution free uncertainty quantification in neuroscience-inspired deep operators](https://mlsanity.com/paper/2412.09369/) (2024)
- [Graph Neural Network Surrogates for Contacting Deformable Bodies with Necessary and Sufficient Contact Detection](https://mlsanity.com/paper/2507.13459/) (2025)
- [SplineNet: An Isogeometric Deep Learning Method for Complex Shells](https://mlsanity.com/paper/2607.06026/) (2026)

## Abstract (arXiv)

Numerical methods for contact mechanics are of great importance in engineering applications, enabling the prediction and analysis of complex surface interactions under various conditions. In this work, we propose an energy-based physics-informed neural network (PINNs) framework for solving frictionless contact problems under large deformation. Inspired by microscopic Lennard-Jones potential, a surface contact energy is used to describe the contact phenomena. To ensure the robustness of the proposed PINN framework, relaxation, gradual loading and output scaling techniques are introduced. In the numerical examples, the well-known Hertz contact benchmark problem is conducted, demonstrating the effectiveness and robustness of the proposed PINNs framework. Moreover, challenging contact problems with the consideration of geometrical and material nonlinearities are tested. It has been shown that the proposed PINNs framework provides a reliable and powerful tool for nonlinear contact mechanics. More importantly, the proposed PINNs framework exhibits competitive computational efficiency to the commercial FEM software when dealing with those complex contact problems. The codes used in this manuscript are available at https://github.com/JinshuaiBai/energy_PINN_Contact.(The code will be available after acceptance)