mlsanityData to 14 Oct 2026

A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks

arXiv 2509.13717 ↗First posted 17 Sep 2025cs.LG cs.NA math.NAcited by 5

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Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving PDEs, yet existing uncertainty quantification (UQ) approaches for PINNs generally lack rigorous statistical guarantees. In this work, we bridge this gap by introducing a distribution-free conformal prediction (CP) framework for UQ in PINNs. This framework calibrates prediction intervals by constructing nonconformity scores on a calibration set, thereby yielding distribution-free uncertainty estimates with rigorous finite-sample coverage guarantees for PINNs. To handle spatial heteroskedasticity, we further introduce local conformal quantile estimation, enabling spatially adaptive uncertainty bands while preserving theoretical guarantee. Through systematic evaluations on typical PDEs (damped harmonic oscillator, Poisson, Allen-Cahn, and Helmholtz equations) and comprehensive testing across multiple uncertainty metrics, our results demonstrate that the proposed framework achieves reliable calibration and locally adaptive uncertainty intervals, consistently outperforming heuristic UQ approaches. By bridging PINNs with distribution-free UQ, this work introduces a general framework that not only enhances calibration and reliability, but also opens new avenues for uncertainty-aware modeling of complex PDE systems.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 35 of its 67 references that are in the map. First posted on arXiv 17 Sep 2025.

Builds on

35 of 67 references.

References · 35

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  4. 2021A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantificationcited by 792
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  11. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  12. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
23 more
  1. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  2. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  3. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  4. 2019Predictive inference with the jackknife+cited by 211
  5. 2017A Conceptual Introduction to Hamiltonian Monte Carlocited by 209
  6. 2023Conformal Prediction With Conditional Guaranteescited by 152
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  8. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  9. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  10. 2020Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantificationcited by 78
  11. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  12. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  13. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  14. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  15. 2024Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networkscited by 28
  16. 2021Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networkscited by 27
  17. 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
  18. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  19. 2024A Survey on Universal Approximation Theoremscited by 15
  20. 2024Uncertainty Quantification of Surrogate Models using Conformal Predictioncited by 11
  21. 2022Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditionscited by 7
  22. 2024Randomized Physics-Informed Neural Networks for Bayesian Data Assimilationcited by 7
  23. 2025Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learningcited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 4.

5 papers.

  1. 2025Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Losscited by 1
  2. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  3. 2026Conformal risk control for model-form uncertainty in parametric non-intrusive reduced-order modelscited by 0
  4. 2026Structure-preserving uncertainty quantification for GENERIC dynamicscited by 0
  5. 2026Conformal Uncertainty Quantification Guarantees for Neural Operatorscited by 0