物理信息神经网络不适条件的分析与解决
摘要
物理信息神经网络 (PINNs) 最近作为求解偏微分方程 (PDE) 的前向和逆问题的新型且流行的方法而出现。然而，在许多情况下实现稳定训练并获取正确结果仍是一个挑战，常归因于 PINNs 的不适条件。尽管如此，进一步的分析仍不足，严重限制了 PINNs 在复杂工程问题中的进展和应用。drawing inspiration from the ill-conditioning analysis in traditional numerical methods, we establish a connection between the ill-conditioning of PINNs and the ill-conditioning of the Jacobian matrix of the PDE system. Specifically, for any given PDE system, we construct its controlled system. This controlled system allows for adjustment of the condition number of the Jacobian matrix while retaining the same solution as the original system. Our numerical findings suggest that the ill-conditioning observed in PINNs predominantly stems from the Jacobian matrix. As the condition number of the Jacobian matrix decreases, PINNs exhibit faster convergence rates and higher accuracy. Building upon this understanding and the natural extension of controlled systems, we present a general approach to mitigate the ill-conditioning of PINNs, leading to successful simulations of the three-dimensional flow around the M6 wing at a Reynolds number of 5,000. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such complex systems, offering a promising new technique for addressing industrial complexity problems. Our findings also offer valuable insights guiding the future development of PINNs.
引用
@article{arxiv.2405.01957,
title = {An analysis and solution of ill-conditioning in physics-informed neural networks},
author = {Wenbo Cao and Weiwei Zhang},
journal= {arXiv preprint arXiv:2405.01957},
year = {2024}
}