# A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity

arXiv: https://arxiv.org/abs/2505.21994 · HTML: https://mlsanity.com/paper/2505.21994/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 11 of 11)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior](https://mlsanity.com/paper/2406.06287/) (2024)
- [Separable Physics-Informed Neural Networks for the solution of elasticity problems](https://mlsanity.com/paper/2401.13486/) (2024)
- [The Deep Ritz Method for Parametric p-Dirichlet Problems](https://mlsanity.com/paper/2207.01894/) (2022)

## Abstract (arXiv)

Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lam'e coefficient λ→∞, or equivalently as the Poisson ratio nu→1/2. This phenomenon, known as locking or non-robustness, remains not fully understood despite extensive investigation. In this work, we illustrate first that an analogous instability arises when applying the popular Physics-Informed Neural Networks (PINNs) to nearly incompressible elasticity problems, leading to significant loss of accuracy and convergence difficulties. Then, to overcome this challenge, we propose a robust decomposition-based PINN framework that reformulates the elasticity equations into balanced subsystems, thereby eliminating the ill-conditioning that causes locking. Our approach simultaneously solves the forward and inverse problems to recover both the decomposed field variables and the associated external conditions. We will also perform a convergence analysis to further enhance the reliability of the proposed approach. Moreover, through various numerical experiments, including constant, variable and parametric Lam'e coefficients, we illustrate the efficiency of the proposed methodology.