Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

翻译：偏微分方程（PDE）是模拟物理系统的重要工具，将其纳入机器学习模型是融入物理知识的关键途径。针对任意常系数线性偏微分方程系统，我们提出一类称为EPGP的高斯过程（GP）先验族，使得所有实现均为该系统的精确解。我们应用作为非线性傅里叶变换的Ehrenpreis-Palamodov基本原理，构建镜像标准GP谱方法的高斯过程核函数。该方法可从任意数据（如含噪声的测量值、逐点定义的初始条件及边界条件）推断线性PDE系统的可能解。EPGP先验的构建具有算法化特性、普遍适用性，并包含稀疏版本（S-EPGP），可通过学习相关频谱频率适用于大规模数据集。我们在三类PDE系统（热传导方程、波动方程和麦克斯韦方程组）上验证该方法，在计算时间和精度上均超越现有技术，部分实验中的提升幅度达数个数量级。