Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients
This work addresses the challenge of integrating physical knowledge into machine learning for modeling physical systems, offering a generally applicable algorithmic approach with significant performance gains.
The authors tackled the problem of incorporating systems of linear PDEs with constant coefficients into machine learning models by proposing EPGP, a family of Gaussian process priors that ensure all realizations are exact solutions, and demonstrated improvements in computation time and precision by several orders of magnitude in experiments on heat, wave, and Maxwell's equations.
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.