Linear-Time Probabilistic Solution of Boundary Value Problems

Nicholas Krämer, Philipp Hennig

Keywords: Differential equations, Gaussian processes, probabilistic numerics

TL;DR: We introduce a Markov prior and other practical considerations to probabilistic solvers for boundary value problems.

Abstract: We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss-Markov prior and tailor it specifically to BVPs, which allows computing a posterior distribution over the solution in linear time, at a quality and cost comparable to that of well-established, non-probabilistic methods. Our model further delivers uncertainty quantification, mesh refinement, and hyperparameter adaptation. We demonstrate how these practical considerations positively impact the efficiency of the scheme. Altogether, this results in a practically usable probabilistic BVP solver that is (in contrast to non-probabilistic algorithms) natively compatible with other parts of the statistical modelling tool-chain.

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Code: https://github.com/pnkraemer/probabilistic-bvp-solver