Consistency of Variational Inference for Nonlinear Inverse Problems of Partial Differential Equations

We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing from exact Bayesian inference, variational inference transforms the inference problem into an optimization problem by introducing variational sets. Based on a modified ``prior mass and testing’’ framework, we propose general conditions for three categories of inverse problems: mildly ill-posed, severely ill-posed, and those with unknown model parameters. Concentrating on the widely utilized variational sets comprising the truncated Gaussian or the mean-field family, we demonstrate that for all three categories, the convergence rate can be decomposed into a true distribution term and a variational approximation term. Moreover, we illustrate that the true distribution term dominates the convergence rates, thereby substantiating the effectiveness of the variational inference method for inverse problems of PDEs. As specific examples, we examine a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.

💡 Research Summary

This paper establishes rigorous convergence guarantees for variational Bayesian inference (VBI) when applied to statistical inverse problems governed by nonlinear partial differential equations (PDEs). Traditional Bayesian analysis of inverse problems has focused on the asymptotic behavior of exact posterior distributions, often relying on conjugate or Gaussian‑process priors and assuming linear forward operators. In contrast, VBI replaces the posterior by the solution of an optimization problem that minimizes the Kullback–Leibler (KL) divergence over a prescribed variational family. While VBI is computationally attractive—especially for high‑dimensional PDE models where Markov chain Monte Carlo (MCMC) becomes prohibitive—its theoretical accuracy has been largely unexplored, particularly in the nonlinear setting.

The authors adapt the “prior mass and testing” framework, originally developed for exact Bayesian contraction, to the variational context. They introduce three categories of ill‑posedness: mildly ill‑posed, severely ill‑posed, and problems with additional unknown model parameters (e.g., fractional orders). For each category they formulate regularity and conditional stability assumptions on the forward map (G). The forward map is allowed to grow polynomially in the norm of the parameter and to satisfy a conditional stability estimate that links perturbations in the data to perturbations in the parameter.

Two widely used variational families are considered:

1. Truncated Gaussian measures – Gaussian processes whose expansion coefficients are truncated to a finite number of basis functions, thereby controlling dimensionality while preserving the prior’s smoothness properties.
2. Mean‑field families – Product measures where each coordinate of the parameter is approximated by an independent Gaussian, leading to a fully factorized KL minimization.

The main theoretical result shows that the variational posterior (\widehat Q) satisfies a contraction inequality of the form

\