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

TL;DR

This work tackles the locking phenomenon in solving nearly incompressible linear elasticity with Physics-Informed Neural Networks (PINNs). It identifies an intrinsic imbalance between the volumetric and deviatoric stress terms as the root cause of non-robust PINN training in the large $\lambda/\mu$ regime and proposes a decomposition-based PINN that splits the problem into balanced subsystems handled by separate neural networks, enabling simultaneous forward and inverse learning. The authors establish an error framework showing quasi-optimality for the decomposed formulation and validate the approach through extensive 2D and 3D numerical experiments, including constant, variable, and parametric Lamé coefficients, where the method consistently outperforms standard PINNs in the nearly incompressible limit. The proposed framework promises robust, mesh-free elasticity solvers with inverse- and parametric-capable capabilities, and may generalize to other singular-perturbation problems such as small-diffusion or high-Reynolds-number regimes.

Abstract

Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lamé coefficient $λ\to\infty$, or equivalently as the Poisson ratio $ν\to1/2$. 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 Lamé coefficients, we illustrate the efficiency of the proposed methodology.

Figures (15)

• Figure 1: The locking phenomenon in solving linear elasticity equations using PINNs
• Figure 2: Comparison of schematic diagrams of PINN and parametric PINN.
• Figure 3: Cosine similarity between $\nabla_{{\boldsymbol{\theta}}}\mathcal{L}$ and $\nabla_{{\boldsymbol{\theta}}}\overline{\mathcal{L}}$ during training. This demonstrates that minimizing the loss $\mathcal{L}$ proceeds almost identically to minimizing the loss $\overline{\mathcal{L}}$, thereby explaining the occurrence of the locking phenomenon in PINN training.
• Figure 4: A schematic diagram of the proposed method using a decomposition. Three independent neural networks are utilized for the three intermediate functions $\widehat{{\bf u}}$, $\widetilde{{\bf u}}$, and $\widehat{{\bf f}}$, and the loss functions are defined based on the decomposition \ref{['eq:dec_1']}-\ref{['eq:dec_2']}.
• Figure 5: A comparison of solutions in two dimensions when $\lambda=10^5$: the exact solution (a), the solution computed by the standard PINN (b), and the solution obtained by our method (c). The experiment shows that standard PINNs fail to approximate the true solution in this example.

Theorems & Definitions (2)

• Theorem 5.1
• proof