Bayesian Operator Learning: Posterior Existence and Convergence of Point Estimates for Gaussian Priors
Abstract
We develop a Bayesian framework for learning nonlinear operators between infinite-dimensional spaces. Given a map g0:π³βπ΄ between separable Hilbert spaces, we study the recovery of g0 from nββ noisy input-output pairs (X,Z)=(Xi,Zi)i=1n with Zi= g0 (Xi ) + Ei. Here the Xiβπ³ are randomly drawn 'design' points in a compact subset of π³, and the Ei are assumed to be i.i.d. draws from a Gaussian white noise process indexed by π΄. For any 'operator-valued' prior βG supported on the space of continuous operators, we show existence of the posterior βG|(X,Z) as a regular conditional distribution, and provide a characterization of its Radon-Nikodym derivative. For Gaussian priors, we establish algebraic (in the sample size n) convergence rates for the posterior mean towards the ground truth; this corresponds to a ridge regularized kernel estimator. Moreover, we show that the posterior mean is minimax optimal (up to logarithmic factors) over hyperrectangles when the smoothness of the prior matches that of the ground truth. To illustrate the applicability of our analysis, we derive explicit learning rates for the Darcy flow solution operator.
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BibTeX
@misc{reinhardt2026bayesian,
title = {{Bayesian Operator Learning: Posterior Existence and Convergence of Point Estimates for Gaussian Priors}},
author = {Niklas Reinhardt and Jakob Zech},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03666},
eprint = {2610.03666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03666},
}
RIS
TY - UNPB
TI - Bayesian Operator Learning: Posterior Existence and Convergence of Point Estimates for Gaussian Priors
AU - Niklas Reinhardt
AU - Jakob Zech
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03666
UR - https://arxiv.org/abs/2610.03666
AB - We develop a Bayesian framework for learning nonlinear operators between infinite-dimensional spaces. Given a map g_0:π³βπ΄ between separable Hilbert spaces, we study the recovery of g_0 from nββ noisy input-output pairs (X,Z)=(X_i,Z_i)_i=1^n with Z_i= g_0 (X_i ) + E_i. Here the X_iβπ³ are randomly drawn 'design' points in a compact subset of π³, and the E_i are assumed to be i.i.d. draws from a Gaussian white noise process indexed by π΄. For any 'operator-valued' prior β_G supported on the space of continuous operators, we show existence of the posterior β_G|(X,Z) as a regular conditional distribution, and provide a characterization of its Radon-Nikodym derivative. For Gaussian priors, we establish algebraic (in the sample size n) convergence rates for the posterior mean towards the ground truth; this corresponds to a ridge regularized kernel estimator. Moreover, we show that the posterior mean is minimax optimal (up to logarithmic factors) over hyperrectangles when the smoothness of the prior matches that of the ground truth. To illustrate the applicability of our analysis, we derive explicit learning rates for the Darcy flow solution operator.
ER -