On gradient stability in nonlinear PDE models and inference in interacting particle systems

TL;DR

This paper develops a framework using the implicit function theorem in Banach spaces to analyze gradient stability and identifiability in nonlinear PDE models and interacting particle systems, with applications to sampling algorithms.

Contribution

It introduces a novel approach to verify gradient stability and injectivity for nonlinear PDE inverse problems, with concrete examples and convergence results for sampling algorithms.

Findings

Verified gradient stability for nonlinear PDE inverse problems.

Established injectivity and identifiability results.

Proved polynomial time convergence of a Langevin-type sampling algorithm.

Abstract

We consider general parameter to solution maps $\theta \mapsto \mathcal G(\theta)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system.