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：偏微分方程是建模物理系统的重要工具，将其融入机器学习模型是融入物理知识的重要途径。针对任意常系数线性偏微分方程系统，我们提出一类高斯过程先验族（称为EPGP），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GP核函数，该方法可镜像GP的标准谱方法。我们的方法能够从任意数据（如含噪测量值或逐点定义的初始与边界条件）推断线性PDE系统的可能解。EPGP先验的构造具有算法性、普适性，并附带稀疏版本S-EPGP，该版本可学习相关谱频率，在数据集较大时表现更优。我们在三类PDE系统族（热方程、波动方程和麦克斯韦方程组）上验证了该方法，在计算时间和精度方面均超越了现有技术水平，部分实验中的提升达数个数量级。