A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks
This provides a statistically rigorous uncertainty quantification method for PINN users in scientific computing, though it's an incremental advance applying existing conformal prediction techniques to PINNs.
The authors tackled the lack of rigorous statistical guarantees in uncertainty quantification for Physics-Informed Neural Networks by introducing a conformal prediction framework, achieving reliable calibration and locally adaptive uncertainty intervals that consistently outperformed heuristic approaches across multiple PDE benchmarks.
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.