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

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. 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 het-eroskedasticity, 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.1. Introduction Physics-Informed Neural Networks (PINNs) have emerged as a versatile framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training [1, 2]. Numerous variants have been developed to enhance accuracy, efficiency, and applicability [3, 4, 5, 6, 7, 8], enabling PINNs to address complex geometries [9, 10], high-dimensional and multiscale problems [11, 12, 13], and inverse formulations [14, 15] within a unified mesh-free paradigm. Applications span fluid mechanics [16, 17], heat transfer [18, 19], and materials science [20, 21]; see [16, 22, 23, 24, 25] for comprehensive reviews.