Probabilistic simulation of partial differential equations

TL;DR

This paper develops a probabilistic spectral simulation method for PDEs using Gaussian processes, extending Bayesian filtering techniques from ODEs to PDEs with periodic boundary conditions, enabling uncertainty quantification.

Contribution

It introduces a novel Bayesian filtering approach for PDEs with periodic boundaries using continuous Gaussian processes, enhancing efficiency and uncertainty estimation.

Findings

Efficient probabilistic spectral simulation method for PDEs.

Joint estimation of PDE solution and prior power spectrum.

Extension of Bayesian filtering from ODEs to PDEs with periodic conditions.

Abstract

Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of a probabilistic simulation scheme can be regarded as Bayesian filtering by means of probabilistic numerics. Gaussian prior based filters, specifically Gauss-Markov priors, have successfully been applied to simulation of ordinary differential equations (ODEs) and give rise to filtering problems that can be solved efficiently. This work extends this approach to partial differential equations (PDEs) subject to periodic boundary conditions and utilizes continuous Gaussian processes in space and time to arrive at a Bayesian filtering problem structurally similar to the ODE setting. The usage of a process that is Markov in time and statistically homogeneous in space leads to a probabilistic spectral simulation method that allows for an efficient realization. Furthermore, the Bayesian perspective allows the incorporation of methods developed within the context of information field theory such as the estimation of the power spectrum associated with the prior distribution, to be jointly estimated along with the solution of the PDE.