为线性偏微分方程边界问题设计可精确满足方程与边界的高斯过程先验。
Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations
- 构建满足线性PDE与线性边界条件的高斯过程先验，直接嵌入物理规律。
- 在多个典型PDE系统上验证，精度显著优于现有方法，计算效率更高。
- 适合需高保真物理约束的科学计算场景，如流体力学、电磁场模拟。
求解偏微分方程（PDE）是计算科学中的基础任务。适定系统通常由数值求解器或神经算子处理，而基于数据的系统则常采用物理信息神经网络（PINNs）或高斯过程（GP）。本文提出边界埃伦普雷伊-帕拉莫多夫高斯过程（B-EPGP），一种新颖的概率框架，可构造同时满足具有常系数的线性PDE系统及线性边界条件的高斯过程先验，并能对有限数据集进行条件化。我们显式构建了若干具有实际边界的代表性PDE系统的GP先验。提供形式化正确性证明，并通过实证结果表明，在精度和计算资源消耗方面均显著优于当前最优方法。
原文摘要 · Abstract (English)
Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy and computational resource improvements over state-of-the-art approaches.
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