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Bernstein-von Mises theorems for time evolution equations

Richard Nickl

TL;DR

The paper addresses statistical inference for initial conditions of infinite-dimensional time-evolution PDEs under discrete noisy observations. It develops a functional Bernstein–von Mises theory showing that, under a set of regularity and identifiability conditions, the non-Gaussian posterior over trajectories is well-approximated by a Gaussian measure in a negative Sobolev space, with convergence quantified in Wasserstein distance at rate $1/\sqrt{N}$. The Gaussian limit is constructed from a time-dependent Schrödinger equation with a rough Gaussian initial condition and is characterized via the inverse Fisher information operator, providing both posterior uncertainty quantification and practical computational implications. The analysis is instantiated for periodic non-linear reaction-diffusion equations, with detailed PDE results on existence, regularity, forward maps, linearisation, spectral properties, and the information operator, collectively enabling rigorous frequentist Bayesian guarantees for high-dimensional Bayesian data assimilation. These results offer a rigorous pathway to reliable uncertainty quantification and efficient computation in complex dynamical systems governed by parabolic PDEs.

Abstract

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.

Bernstein-von Mises theorems for time evolution equations

TL;DR

The paper addresses statistical inference for initial conditions of infinite-dimensional time-evolution PDEs under discrete noisy observations. It develops a functional Bernstein–von Mises theory showing that, under a set of regularity and identifiability conditions, the non-Gaussian posterior over trajectories is well-approximated by a Gaussian measure in a negative Sobolev space, with convergence quantified in Wasserstein distance at rate . The Gaussian limit is constructed from a time-dependent Schrödinger equation with a rough Gaussian initial condition and is characterized via the inverse Fisher information operator, providing both posterior uncertainty quantification and practical computational implications. The analysis is instantiated for periodic non-linear reaction-diffusion equations, with detailed PDE results on existence, regularity, forward maps, linearisation, spectral properties, and the information operator, collectively enabling rigorous frequentist Bayesian guarantees for high-dimensional Bayesian data assimilation. These results offer a rigorous pathway to reliable uncertainty quantification and efficient computation in complex dynamical systems governed by parabolic PDEs.

Abstract

We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition 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 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.
Paper Structure (26 sections, 22 theorems, 287 equations)

This paper contains 26 sections, 22 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Bayesian inference in dynamical systems
  3. Asymptotics for posterior measures in reaction-diffusion equations
  4. Notation and preliminaries
  5. Functional Bernstein--von Mises theorems
  6. A general result for non-linear parabolic PDEs
  7. Proof of Theorem \ref{['mainbvm']}
  8. Posterior contraction, regularity, and localisation
  9. Inverting the Fisher information operator, and the limiting process
  10. Perturbation expansion of the posterior Laplace transform
  11. Convergence in function space
  12. Application to the reaction diffusion model and proof of Theorem \ref{['ganzwien']}
  13. PDE results
  14. Periodic non-linear reaction diffusion equations
  15. Existence and uniqueness of strong solutions
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\mu_{N}= \mu(\cdot|Z^{(N)})$ be the conditional law in $\mathscr C$ of the stochastic process where $\theta \sim \Pi(\cdot|Z^{(N)})$ arises from posterior (post) with data (data) in the reaction-diffusion system (evol), prior $\Pi=\Pi_N$ in (prior) for $\rho$ as in (ronacher), integer $\gamma>2+3d$, $d \le 3$, and where $\tilde{\theta}_N = E^\Pi[\theta|Z^{(N)}]$ is the posterior mean in $L_0

Theorems & Definitions (44)

  • Theorem 1
  • Remark 1: Priors on subspaces
  • Remark 2: Condition \ref{['gemol']} and parabolic PDE
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 4
  • ...and 34 more