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

TL;DR

This paper identifies instability issues in physics-informed neural networks (PINNs) for nearly incompressible elasticity and proposes a decomposition-based framework to improve robustness and accuracy.

Contribution

It introduces a novel decomposition-based PINN method that reformulates elasticity equations to overcome locking and divergence instability.

Findings

The proposed method effectively mitigates locking in PINNs for nearly incompressible elasticity.

Numerical experiments demonstrate improved accuracy and convergence across various Lamé coefficients.

The approach is applicable to both forward and inverse elasticity problems.

Abstract

Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lam\'e coefficient $\lambda\to\infty$, or equivalently as the Poisson ratio $\nu\to1/2$. 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.