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PINNfluence: Influence Functions for Physics-Informed Neural Networks

Jonas R. Naujoks, Aleksander Krasowski, Moritz Weckbecker, Thomas Wiegand, Sebastian Lapuschkin, Wojciech Samek, René P. Klausen

TL;DR

Addresses interpretability of PINNs by applying influence functions to assess the impact of individual training points. Proposes two physics-aware indicators to test whether PINNs capture flow physics in Navier-Stokes. Demonstrates on three PINN variants (good, broken, bad); good model aligns with indicators while others reveal misalignment. Discusses limitations and future directions including extensions to other PDEs and integration with training strategies.

Abstract

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.

PINNfluence: Influence Functions for Physics-Informed Neural Networks

TL;DR

Addresses interpretability of PINNs by applying influence functions to assess the impact of individual training points. Proposes two physics-aware indicators to test whether PINNs capture flow physics in Navier-Stokes. Demonstrates on three PINN variants (good, broken, bad); good model aligns with indicators while others reveal misalignment. Discusses limitations and future directions including extensions to other PDEs and integration with training strategies.

Abstract

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.
Paper Structure (19 sections, 3 theorems, 23 equations, 6 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 23 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Theoretical Background
  4. Physics-Informed Neural Networks
  5. Influence Functions
  6. Experiments
  7. Navier-Stokes Problem
  8. Experimental Setup
  9. Empirical Indicators
  10. Results
  11. Conclusion
  12. Appendix / Supplemental Material
  13. Influence Functions
  14. Adding & removing training points:
  15. Removing a complete loss term:
  16. ...and 4 more sections

Key Result

Lemma 1

For $\Theta\subseteq\mathbb R^n$ and $U\subseteq\mathbb R$ open, let $g : \Theta \times U \to \mathbb R$ be a twice continuously differentiable function for which there exists an $(\theta_0, \epsilon_0) \in \Theta \times U$ such that $\theta_0$ is a non-degenerate stationary point of the function $g If $g$ is analytic in a neighbourhood of $(\theta_0,\epsilon_0)$, then we can choose $U_0$ such tha

Figures (6)

  • Figure 1: Trained model ($\phi_\text{good}$) with predictions: $u_1$, $u_2$, $p$ (top to bottom).
  • Figure 2: Influence heatmap on the prediction of $u_1$ of a single training point ($\times$) close to the cylinder (bottom).
  • Figure 3: Average log values of $|\operatorname{Inf}_{\sum_i |\! | u(\boldsymbol{x}_i; \theta) |\! |}(\boldsymbol{x})|$ over the domain $\overline\Omega$. The area of $C_{1.5r}$ is outlined in black.
  • Figure 4: Trained models and target values with respective predictions for $u_1$, $u_2$, $p$, $|\! | \boldsymbol u |\! |$ (top to bottom).
  • Figure 5: Absolute errors between respective model predictions for each output dimension w.r.t. precomputed target values.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Theorem 1
  • Corollary 1
  • proof