Title: PINNfluence: Decoding PINNs via Influence Functions

Abstract: While physics-informed neural networks (PINNs) have established themselves as a potent deep learning methodology for resolving partial differential equations (PDEs) within the physical sciences, their internal mechanics remain largely obscure. Currently, insights into their behavior are predominantly derived from failure mode analyses rather than direct interpretability. To bridge this gap, we present PINNfluence, a novel framework for training data attribution designed to interpret PINNs using influence functions. By adapting influence functions to accommodate composite physics-informed loss objectives, our approach facilitates fine-grained attribution linking predictions and specific loss components to individual training data points. Our benchmark experiments, conducted across a range of PDEs, reveal that these influence patterns offer detailed diagnostics capable of distinguishing structural differences between well-trained and poorly-trained PINNs. Consequently, PINNfluence provides a new perspective for enhancing PINN reliability and understanding through the analysis of their underlying data.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC