为物理信息神经网络提供有统计保证的不确定性量化方法
A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks
- 用非对称性评分构建校准集，实现无需分布假设的置信区间
- 在典型PDE问题上实现可靠校准与自适应不确定性带，优于传统启发式方法
- 适合需要可信预测区间的科学计算与工程建模场景
物理信息神经网络（PINNs）已成为求解偏微分方程的强大框架，但现有的不确定性量化（UQ）方法普遍缺乏严格的统计保障。本文引入一种无需分布假设的共形预测（CP）框架，用于PINNs的不确定性量化。该框架通过在校准集上构造非对称性评分，生成具有有限样本覆盖率保证的置信区间。为处理空间异方差性，进一步提出局部共形分位数估计，实现空间自适应的不确定性带，同时保持理论保障。在典型PDE问题（阻尼谐振子、泊松方程、Allen-Cahn方程和赫姆霍兹方程）上的系统评估显示，所提框架在多种不确定性度量下均表现出可靠的校准性和局部自适应性，显著优于启发式方法。本工作将PINNs与无分布假设的不确定性量化结合，不仅提升了模型校准与可靠性，也为复杂PDE系统的不确定性感知建模开辟了新路径。
原文摘要 · Abstract (English)
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.
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