# Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2308.12716 · HTML: https://mlsanity.com/paper/2308.12716/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 65 references that are in the map.
- First posted on arXiv 24 Aug 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [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)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios](https://mlsanity.com/paper/2205.07731/) (2022)
- [Physics-informed deep-learning applications to experimental fluid mechanics](https://mlsanity.com/paper/2203.15402/) (2022)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics](https://mlsanity.com/paper/2207.09908/) (2022)
- [Physics-informed neural networks for solving parametric magnetostatic problems](https://mlsanity.com/paper/2202.04041/) (2022)
- [HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networks](https://mlsanity.com/paper/2101.03271/) (2021)
- [Mesh-free Eulerian Physics-Informed Neural Networks](https://mlsanity.com/paper/2206.01545/) (2022)
- [Deep NURBS -- Admissible Physics-informed Neural Networks](https://mlsanity.com/paper/2210.13900/) (2022)
- [Optimization with Equality and Inequality Constraints Using Parameter Continuation](https://mlsanity.com/paper/1909.01422/) (2019)

## Built on by (most cited first, 18 of 18)

- [Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient](https://mlsanity.com/paper/2506.04375/) (2025)
- [Energy-based physics-informed neural network for frictionless contact problems under large deformation](https://mlsanity.com/paper/2411.03671/) (2024)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Data-Driven Surrogate Modeling Techniques to Predict the Effective Contact Area of Rough Surface Contact Problems](https://mlsanity.com/paper/2504.17354/) (2025)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)
- [Robust Physics-Informed Neural Network Approach for Estimating Heterogeneous Elastic Properties from Noisy Displacement Data](https://mlsanity.com/paper/2506.14036/) (2025)
- [Performance of Machine Learning Methods for Gravity Inversion: Successes and Challenges](https://mlsanity.com/paper/2510.09632/) (2025)
- [Finding geodesics with the Deep Ritz method](https://mlsanity.com/paper/2510.15177/) (2025)
- [Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI](https://mlsanity.com/paper/2511.04564/) (2025)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [Physics-Informed Neural Network Based Digital Image Correlation Method](https://mlsanity.com/paper/2409.00956/) (2024)
- [Snap-and-tune: combining deep learning and test-time optimization for high-fidelity cardiovascular volumetric meshing](https://mlsanity.com/paper/2506.08280/) (2025)
- [Graph Neural Network Surrogates for Contacting Deformable Bodies with Necessary and Sufficient Contact Detection](https://mlsanity.com/paper/2507.13459/) (2025)
- [Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations](https://mlsanity.com/paper/2512.07755/) (2025)
- [A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimization](https://mlsanity.com/paper/2602.13811/) (2026)
- [QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwear](https://mlsanity.com/paper/2604.23702/) (2026)
- [Probabilistic Physics-Informed Neural Networks for Estimating Heterogeneous Elastic Properties from Low-Resolution and Noisy Displacement Data](https://mlsanity.com/paper/2607.14563/) (2026)
- [Deep learning-based prediction of time-resolved adhesive forces in viscoelastic Hertzian contacts](https://mlsanity.com/paper/2607.19060/) (2026)

## Abstract (arXiv)

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush-Kuhn-Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer-Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.