mlsanityData to 14 Oct 2026

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

arXiv 2308.12716 ↗First posted 24 Aug 2023math.NA cs.LG cs.NAcited by 18

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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.

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

21 of 65 references.

References · 21

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  4. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  5. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
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9 more
  1. 2022Physics-informed deep-learning applications to experimental fluid mechanicscited by 21
  2. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  3. 2022Constitutive model characterization and discovery using physics-informed deep learningcited by 17
  4. 2022Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanicscited by 13
  5. 2022Physics-informed neural networks for solving parametric magnetostatic problemscited by 10
  6. 2021HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networkscited by 5
  7. 2022Mesh-free Eulerian Physics-Informed Neural Networkscited by 4
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 11; 2026: 4.

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  1. 2025Solving engineering eigenvalue problems with neural networks using the Rayleigh quotientcited by 8
  2. 2024Energy-based physics-informed neural network for frictionless contact problems under large deformationcited by 4
  3. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 2
  4. 2025Data-Driven Surrogate Modeling Techniques to Predict the Effective Contact Area of Rough Surface Contact Problemscited by 1
  5. 2025Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force methodcited by 1
  6. 2025Robust Physics-Informed Neural Network Approach for Estimating Heterogeneous Elastic Properties from Noisy Displacement Datacited by 1
  7. 2025Performance of Machine Learning Methods for Gravity Inversion: Successes and Challengescited by 1
  8. 2025Finding geodesics with the Deep Ritz methodcited by 1
  9. 2025Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AIcited by 1
  10. 2024Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materialscited by 0
  11. 2024Physics-Informed Neural Network Based Digital Image Correlation Methodcited by 0
  12. 2025Snap-and-tune: combining deep learning and test-time optimization for high-fidelity cardiovascular volumetric meshingcited by 0
6 more
  1. 2025Graph Neural Network Surrogates for Contacting Deformable Bodies with Necessary and Sufficient Contact Detectioncited by 0
  2. 2025Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equationscited by 0
  3. 2026A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimizationcited by 0
  4. 2026QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwearcited by 0
  5. 2026Probabilistic Physics-Informed Neural Networks for Estimating Heterogeneous Elastic Properties from Low-Resolution and Noisy Displacement Datacited by 0
  6. 2026Deep learning-based prediction of time-resolved adhesive forces in viscoelastic Hertzian contactscited by 0