Bernstein-von Mises theorems for time evolution equations

TL;DR

This paper establishes conditions under which posterior distributions for solutions to certain non-linear parabolic PDEs can be approximated by Gaussian measures, with applications to reaction diffusion equations.

Contribution

It provides a general framework for Gaussian approximation of non-Gaussian posteriors in infinite-dimensional dynamical systems driven by PDEs.

Findings

Posterior measures are approximated by Gaussian laws in Wasserstein distance.

Applicable to reaction diffusion equations with smooth, compactly supported reaction functions.

Gaussian limits characterized via solutions to a time-dependent Schrödinger equation.

Abstract

We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $\theta$ modelled by a Gaussian process `prior' probability measure. Given discrete samples of the state of the system evolving in space-time, one obtains updated `posterior' measures on a function space containing all possible trajectories. We give a general set of conditions under which these non-Gaussian posterior distributions are approximated, in Wasserstein distance for the supremum-norm metric, by the law of a Gaussian random function. We demonstrate the applicability of our results to periodic non-linear reaction diffusion equations \begin{align*} \frac{\partial}{\partial t} u - \Delta u &= f(u) \\ u(0) &= \theta \end{align*} where $f$ is any smooth and compactly supported reaction function. In this case the limiting Gaussian measure can be characterised as the solution of a time-dependent Schr\"odinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.