AI 中文总结
针对神经算子求解PDE缺乏严格不确定性估计的问题，提出物理信息保形预测框架，将PDE残差嵌入非一致性得分，生成具有无分布覆盖保证且空间自适应的预测区间，并揭示FNO平移等变性的近似障碍，在六个物理场景中验证了89-91%的稳定覆盖率。
AI 中文摘要
神经算子（如傅里叶神经算子FNO）在逼近偏微分方程（PDE）解方面取得了显著精度。然而，提供严格的不确定性估计仍是一个开放挑战。我们提出物理信息保形预测（PI-CP），该框架将PDE残差嵌入分裂保形预测的非一致性得分中，生成预测区间，这些区间（i）无分布且具有可证明的覆盖保证，（ii）当PDE残差与预测误差相关时具有空间自适应性——在物理满足良好处更紧，在违反处更宽。此外，我们证明FNO的平移等变性对具有Dirichlet边界条件的PDE造成根本性近似障碍，并表明坐标通道可将误差降低高达63倍。我们在六个物理场景中验证PI-CP——热传导（2D/3D）、结构力学（2D/3D）、达西流和纳维-斯托克斯方程——证明所有四种保形方法均实现一致的89-91%覆盖，而MC Dropout和深度集成不稳定（82-100%）。FNO比CNN和DeepONet性能提升10-12倍。
英文摘要
Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs). However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are (i) distribution-free with provable coverage guarantees, and (ii) spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable (82-100%). FNO outperforms CNN and DeepONet by 10-12x.
Comments14 pages, 11 tables, 5 figures
