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Partial differential equations (PDEs) are important tools to model physicalsystems and including them into machine learning models is an important way ofincorporating physical knowledge. Given any system of linear PDEs with constantcoefficients, we propose a family of Gaussian process (GP) priors, which wecall EPGP, such that all realizations are exact solutions of this system. Weapply the Ehrenpreis-Palamodov fundamental principle, which works as anon-linear Fourier transform, to construct GP kernels mirroring standardspectral methods for GPs. Our approach can infer probable solutions of linearPDE systems from any data such as noisy measurements, or pointwise definedinitial and boundary conditions. Constructing EPGP-priors is algorithmic,generally applicable, and comes with a sparse version (S-EPGP) that learns therelevant spectral frequencies and works better for big data sets. Wedemonstrate our approach on three families of systems of PDEs, the heatequation, wave equation, and Maxwell's equations, where we improve upon thestate of the art in computation time and precision, in some experiments byseveral orders of magnitude.

Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients