Bernstein--von Mises theorems for Bayesian probabilistic numerics
Abstract
We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein--von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a √N-scale.
Related
- Semiparametric Bernstein-von Mises theorems from Stein's method
- A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics
- Bernstein-von Mises theorem for sparse generalized linear models
- A semi-parametric Bernstein-von Mises theorem at rate n−3/2 for frequency estimation
- Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints
BibTeX
@misc{gaudlitz2026bernsteinvon,
title = {{Bernstein--von Mises theorems for Bayesian probabilistic numerics}},
author = {Sascha Gaudlitz and Sven Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04124},
eprint = {2609.04124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04124},
}
RIS
TY - UNPB
TI - Bernstein--von Mises theorems for Bayesian probabilistic numerics
AU - Sascha Gaudlitz
AU - Sven Wang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04124
UR - https://arxiv.org/abs/2609.04124
AB - We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein--von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a √N-scale.
ER -