Hyperparameter Selection via Early Stopping for Bayesian Semilinear PDEs

Abstract

We provide a data-driven method for choosing the scale of a Gaussian prior for non-linear Bayesian inverse problems arising from semilinear PDEs. Following \cite{koers2024}, the non-linear model for the parameter $f$ is reparametrized as a linear model for $v = \mathbb{L}u_f$, in which the prior scale can be selected by early stopping at the discrepancy principle \cite{tienstra2025}. We extend the transfer of frequentist guarantees from the linearised problem to the original one beyond Lipschitz solution maps, to maps admitting an arbitrary modulus of continuity, and we provide a general condition on the non-linearity under which such a map exists and is Lipschitz near the truth. The resulting posterior for $f$ contracts adaptively over a range of Sobolev smoothness, and its credible sets have asymptotic frequentist coverage one. We demonstrate our theory in detail for the stationary Schrödinger equation, for which we also provide numerical experiments. We further show how the results apply to the stationary Allen-Cahn equation and to a parabolic Allen-Cahn equation, and we discuss Darcy flow as an edge case outside the semilinear setting. The proposed method thus provides a data-driven way to tune Gaussian priors via early stopping, which is computationally efficient and statistically near-optimal for non-linear problems.