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We introduce the concept of numerical Gaussian processes, which we define asGaussian processes with covariance functions resulting from temporaldiscretization of time-dependent partial differential equations. NumericalGaussian processes, by construction, are designed to deal with cases where: (1)all we observe are noisy data on black-box initial conditions, and (2) we areinterested in quantifying the uncertainty associated with such noisy data inour solutions to time-dependent partial differential equations. Our methodcircumvents the need for spatial discretization of the differential operatorsby proper placement of Gaussian process priors. This is an attempt to constructstructured and data-efficient learning machines, which are explicitly informedby the underlying physics that possibly generated the observed data. Theeffectiveness of the proposed approach is demonstrated through severalbenchmark problems involving linear and nonlinear time-dependent operators. Inall examples, we are able to recover accurate approximations of the latentsolutions, and consistently propagate uncertainty, even in cases involving verylong time integration.

Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations