解决物理神经网络在近不可压缩弹性问题中的精度下降难题
A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity
- 将弹性方程分解为平衡子系统，避免病态条件导致的锁死
- 同时求解正反问题，恢复分解变量与外部条件
- 在多种参数条件下验证方法高效稳定
由于发散不稳定性，低阶连续有限元方法在拉梅系数λ→∞（即泊松比ν→1/2）时，近不可压缩弹性方程的精度会显著下降，这种现象称为锁死或非鲁棒性，至今仍不完全清楚。本文首次揭示，将流行的物理信息神经网络（PINNs）应用于近不可压缩弹性问题时，同样会出现类似不稳定性，导致精度大幅降低和收敛困难。为此，我们提出一种基于分解的鲁棒PINN框架，将弹性方程重构为平衡子系统，从而消除引发锁死的病态性。该方法可同时求解正问题与逆问题，以恢复分解后的场变量及相应的外部条件。我们还进行了收敛性分析，进一步提升方法可靠性。通过包含常数、变系数和参数化拉梅系数在内的多种数值实验，验证了该方法的高效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。