用高斯过程高效求解线性偏微分方程的逆问题
Gaussian Process Regression for Inverse Problems in Linear PDEs
- 基于交换代数构建先验，用Macaulay2实现算法
- 从噪声数据中准确反演波动方程的波速
- 适合需要高精度逆问题求解的物理建模者
本文提出一种在系统理论中求解由线性偏微分方程（PDEs）支配的逆问题的计算高效算法。通过结合先进的交换代数与代数分析，将线性PDE的解建模为高斯过程，并定义相应的先验分布。这些先验的实现是算法化的，利用Macaulay2计算机代数软件完成。一个具体应用是从含噪数据中识别经典波动方程中的波速，该类方程广泛应用于物理学。该方法在保持高精度的同时显著提升了计算效率。
原文摘要 · Abstract (English)
This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency.
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