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 like 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 PDE, 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：部分差异方程式(PDEs)是模拟物理系统的重要工具,并将之纳入机器学习模型是纳入物理知识的重要方法。鉴于任何具有恒定系数的线性PDE系统,我们提出一个加萨进程(GP)前置(GP)系统,我们称之为EPGP(EPGP),这样所有实现都是这个系统的精确解决方案。我们应用Ehrenpriis-Palamodov基本原则,该原则像非线性Fourier变换一样,用来构建GP内核,反映GP的标准光谱方法。我们的方法可以从任何数据中推断出线性PDE系统的可能解决方案,例如噪音测量,或点性界定的初始和边界条件。构建EPGP-Priororas(GP)是算法,一般适用,并带有稀疏版本(S-EPGP),可以学习相关的光谱频率,对大数据集工作更好。我们展示了我们在PDE、热方程式、波方程式和Maxwell等的三种系统中的方法,我们从若干级实验中改进了计算时间和精确度状态。