mlsanityData to 14 Oct 2026

PINNfluence: Interpreting PINNs through Influence Functions

arXiv 2409.08958 ↗First posted 13 Sep 2024cs.LG cs.AI physics.comp-phcited by 1

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Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 43 of its 79 references that are in the map. First posted on arXiv 13 Sep 2024.

Builds on

43 of 79 references.

References · 43

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2017Understanding Black-box Predictions via Influence Functionscited by 1,741
  3. 2023Sparse Autoencoders Find Highly Interpretable Features in Language Modelscited by 1,420
  4. 2016Understanding intermediate layers using linear classifier probescited by 1,353
  5. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  6. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  7. 2019Data Shapley: Equitable Valuation of Data for Machine Learningcited by 629
  8. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  9. 2020Estimating Training Data Influence by Tracing Gradient Descentcited by 580
  10. 2020Captum: A unified and generic model interpretability library for PyTorchcited by 547
  11. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  12. 2023Studying Large Language Model Generalization with Influence Functionscited by 326
31 more
  1. 2017Right for the Right Reasons: Training Differentiable Models by Constraining their Explanationscited by 294
  2. 2014Identifying and attacking the saddle point problem in high-dimensional non-convex optimizationcited by 289
  3. 2020Influence Functions in Deep Learning Are Fragilecited by 252
  4. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  5. 2018Representer Point Selection for Explaining Deep Neural Networkscited by 187
  6. 2019On the Accuracy of Influence Functions for Measuring Group Effectscited by 147
  7. 2019Interpretations are useful: penalizing explanations to align neural networks with prior knowledgecited by 145
  8. 2022If Influence Functions are the Answer, Then What is the Question?cited by 144
  9. 2020Higher-Order Explanations of Graph Neural Networks via Relevant Walkscited by 138
  10. 2022Training Data Influence Analysis and Estimation: A Surveycited by 135
  11. 2021Scaling Up Influence Functionscited by 132
  12. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  13. 2019On Second-Order Group Influence Functions for Black-Box Predictionscited by 79
  14. 2021Input Similarity from the Neural Network Perspectivecited by 79
  15. 2022Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Samplingcited by 73
  16. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 46
  17. 2021An Empirical Comparison of Instance Attribution Methods for NLPcited by 35
  18. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  19. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 32
  20. 2023Theoretical and Practical Perspectives on what Influence Functions Docited by 29
  21. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  22. 2021Rethinking Influence Functions of Neural Networks in the Over-parameterized Regimecited by 26
  23. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  24. 2024The Challenges of the Nonlinear Regime for Physics-Informed Neural Networkscited by 21
  25. 2024xMIL: Insightful Explanations for Multiple Instance Learning in Histopathologycited by 16
  26. 2023Towards Fixing Clever-Hans Predictors with Counterfactual Knowledge Distillationcited by 14
  27. 2024Learning in PINNs: Phase transition, diffusion equilibrium, and generalizationcited by 13
  28. 2023Efficient Error Certification for Physics-Informed Neural Networkscited by 10
  29. 2022Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNscited by 8
  30. 2024Sparse, Efficient and Explainable Data Attribution with DualXDAcited by 6
  31. 2024From Neurons to Neutrons: A Case Study in Interpretabilitycited by 5

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Errorcited by 0