Physics-informed neural networks (PINNs) have emerged as a powerful deep learning approach for solving partial differential equations (PDEs) in the physical sciences, yet their behavior remains largely opaque and is typically understood through failure mode analyses rather than explicit interpretability. To address this issue, we introduce PINNfluence, a training data attribution framework for interpreting PINNs based on influence functions. By extending influence functions to composite physics-informed training objectives, we enable fine-grained attribution between predictions, loss components, and training data points. Through benchmark experiments across various PDEs, we demonstrate that influence patterns provide granular diagnostics that distinguish structural characteristics across well-trained and poorly-trained PINNs. PINNfluence thus opens a new avenue for understanding and improving the reliability of PINNs through the lens of their data.

翻译：物理信息神经网络（PINNs）已成为物理科学中求解偏微分方程（PDEs）的强大深度学习方法，但其行为在很大程度上仍不透明，通常通过失败模式分析而非显式可解释性来理解。针对这一问题，我们提出了PINNfluence，一种基于影响函数对物理信息神经网络进行训练数据归因的解释框架。通过将影响函数扩展到复合型物理信息训练目标，我们实现了预测、损失分量与训练数据点之间的细粒度归因。通过对多种PDEs的基准实验，我们证明影响模式能够提供区分训练良好与训练不佳PINNs结构特征的精细化诊断。因此，PINNfluence通过数据视角为理解和提升物理信息神经网络的可靠性开辟了新途径。