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

2025-09-17

AI Summary

Physics-informed neural networks (PINNs) lack uncertainty quantification (UQ) methods with rigorous statistical guarantees. Method: This paper proposes a distribution-free conformal prediction framework for PINNs, constructing nonconformity scores from a calibration set and incorporating local conformal quantile estimation to adaptively model spatial heteroscedasticity—yielding prediction intervals with finite-sample coverage guarantees. Contribution/Results: It is the first work to deeply integrate conformal prediction with PINNs while preserving theoretical validity and enabling spatially adaptive uncertainty band estimation. Experiments across multiple canonical PDE benchmarks demonstrate that our method significantly outperforms existing heuristic UQ approaches: calibration error is reduced by 30–50%, and empirical coverage consistently approaches the nominal level, achieving both statistical reliability and practical utility.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Develops conformal prediction for uncertainty quantification in PINNs
• Addresses lack of statistical guarantees in existing UQ methods
• Handles spatial heteroskedasticity with adaptive uncertainty bands

Innovation

• Conformal prediction for uncertainty quantification
• Local quantile estimation for spatial adaptation
• Rigorous finite-sample coverage guarantees