用影响函数解析PINN，看懂数据如何影响物理方程求解结果。
PINNfluence: Interpreting PINNs through Influence Functions
- 基于影响函数，实现对PINN训练数据的细粒度归因
- 揭示不同训练状态下的结构特征差异，区分好坏模型
- 适合想理解或优化PINN可靠性的研究者使用
物理信息神经网络（PINNs）在解决物理科学中的偏微分方程（PDEs）方面展现出强大能力，但其行为仍不透明，通常依赖故障模式分析而非显式可解释性。为此，我们提出PINNfluence，一种基于影响函数的PINN训练数据归因框架。通过将影响函数扩展至复合的物理信息训练目标，实现了预测、损失组件与训练数据点之间的细粒度关联。在多种PDE上的基准实验表明，影响模式能提供精细诊断，区分良好与不良训练的PINN的结构特性。PINNfluence因此为通过数据视角理解并提升PINN可靠性开辟了新路径。
原文摘要 · Abstract (English)
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.
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