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

arXiv: https://arxiv.org/abs/2509.13717 · HTML: https://mlsanity.com/paper/2509.13717/

## 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 (most cited first, 30 of 35)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Conformal Prediction With Conditional Guarantees](https://mlsanity.com/paper/2305.12616/) (2023)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networks](https://mlsanity.com/paper/2105.00862/) (2021)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)

## Built on by (most cited first, 5 of 5)

- [Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss](https://mlsanity.com/paper/2510.15664/) (2025)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [Conformal risk control for model-form uncertainty in parametric non-intrusive reduced-order models](https://mlsanity.com/paper/2608.03360/) (2026)
- [Structure-preserving uncertainty quantification for GENERIC dynamics](https://mlsanity.com/paper/2608.12624/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)

## 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.