Recently, physics-informed neural networks (PINNs) have emerged as a flexible and promising application of deep learning to partial differential equations in the physical sciences. While offering strong performance and competitive inference speeds on forward and inverse problems, their black-box nature limits interpretability, particularly regarding alignment with expected physical behavior. In the present work, we explore the application of influence functions (IFs) to validate and debug PINNs post-hoc. Specifically, we apply variations of IF-based indicators to gauge the influence of different types of collocation points on the prediction of PINNs applied to a 2D Navier-Stokes fluid flow problem. Our results demonstrate how IFs can be adapted to PINNs to reveal the potential for further studies. The code is publicly available at https://github.com/aleks-krasowski/PINNfluence.

翻译：近年来，物理信息神经网络（PINNs）已成为深度学习在物理科学偏微分方程中一种灵活且富有前景的应用。尽管在正问题和反问题上展现出优异的性能和具有竞争力的推理速度，其黑箱特性限制了可解释性，特别是在与预期物理行为的一致性方面。在本工作中，我们探索了应用影响函数（IFs）对PINNs进行事后验证与调试的方法。具体而言，我们应用基于IF的指标变体来评估不同类型配置点对应用于二维Navier-Stokes流体流动问题的PINNs预测的影响。我们的结果表明，IFs如何能够适配于PINNs，从而揭示进一步研究的潜力。代码已公开于https://github.com/aleks-krasowski/PINNfluence。