Solving Noisy Linear Operator Equations by Gaussian Processes: Application to Ordinary and Partial Differential Equations

We formulate the problem of solving stochastic linear operator equations in a Bayesian Ganssian process (GP) framework. The solution is obtained in the spirit of a collocation method based on noisy evaluations of the target function at randomly drawn or deliberately chosen points. Prior knowledge about the solution is encoded by the covariance kernel of the GP. As in GP regression, analytical expressions for the mean and variance of the estimated target function are obtained, from which the solution of the operator equation follows by a manipulation of the kernel. Linear initial and boundary value constraints can be enforced by embedding the non-parametric model in a form that automatically satisfies the boundary conditions. The method is illustrated on a noisy linear first-order ordinary differential equation with initial condition and on a noisy second-order partial differential equation with Dirichlet boundary conditions.