Gaussian Process Regression for Data Fulfilling Linear Differential Equations with Localized Sources

Specialized Gaussian process regression is presented for data that are known to fulfill a given linear differential equation with vanishing or localized sources. The method allows estimation of system parameters as well as strength and location of point sources. It is applicable to a wide range of data from measurement and simulation. The underlying principle is the well-known invariance of the Gaussian probability distribution under linear operators, in particular differentiation. In contrast to approaches with a generic covariance function/kernel, we restrict the Gaussian process to generate only solutions of the homogeneous part of the differential equation. This requires specialized kernels with a direct correspondence of certain kernel hyperparameters to parameters in the underlying equation and leads to more reliable regression results with less training data. Inhomogeneous contributions from linear superposition of point sources are treated via a linear model over fundamental solutions. Maximum likelihood estimates for hyperparameters and source positions are obtained by nonlinear optimization. For differential equations representing laws of physics the present approach generates only physically possible solutions, and estimated hyperparameters represent physical properties. After a general derivation, modeling of source-free data and parameter estimation is demonstrated for Laplace's equation and the heat/diffusion equation. Finally, the Helmholtz equation with point sources is treated, representing scalar wave data such as acoustic pressure in the frequency domain.