Bernstein-von Mises theorems for time evolution equations
Provides theoretical justification for Bayesian inference in nonlinear PDE models, enabling uncertainty quantification for practitioners in scientific computing and inverse problems.
The paper establishes Bernstein-von Mises theorems for infinite-dimensional dynamical systems governed by nonlinear parabolic PDEs, showing that posterior distributions over trajectories converge to a Gaussian process in Wasserstein distance. For periodic reaction-diffusion equations with smooth nonlinearities, the limiting Gaussian measure is characterized via a Schrödinger equation.
We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $θ$ 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 - Δu &= f(u) \\ u(0) &= θ\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 Schrödinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.