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

Shaokang Zu, Junxiong Jia, Deyu Meng

TL;DR

This work establishes a general, provable contraction theory for variational posteriors in nonlinear PDE inverse problems. By recasting inference through a reparametrized forward map and Gaussian priors, the authors show that the variational posterior concentrates around the true parameter at rate $\varepsilon_N$ with a controllable variational error $\gamma_N$, and for common variational families $\gamma_N^2$ is dominated by $\varepsilon_N^2$ up to logarithms. The theory is demonstrated on Darcy flow and inverse potential problems for subdiffusion, achieving minimax-optimal rates (up to logs), and extended to severely ill-posed cases like inverse medium scattering with log-type stability, as well as problems with unknown model parameters via model-selection-like approaches. Overall, variational Bayes remains statistically competitive with full posterior inference while offering substantial computational advantages for statistically-inverse PDE problems.

Abstract

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.

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

TL;DR

This work establishes a general, provable contraction theory for variational posteriors in nonlinear PDE inverse problems. By recasting inference through a reparametrized forward map and Gaussian priors, the authors show that the variational posterior concentrates around the true parameter at rate with a controllable variational error , and for common variational families is dominated by up to logarithms. The theory is demonstrated on Darcy flow and inverse potential problems for subdiffusion, achieving minimax-optimal rates (up to logs), and extended to severely ill-posed cases like inverse medium scattering with log-type stability, as well as problems with unknown model parameters via model-selection-like approaches. Overall, variational Bayes remains statistically competitive with full posterior inference while offering substantial computational advantages for statistically-inverse PDE problems.

Abstract

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.
Paper Structure (28 sections, 35 theorems, 477 equations)

This paper contains 28 sections, 35 theorems, 477 equations.

Table of Contents

  1. Introduction
  2. General settings
  3. Basic notations
  4. Statistical settings
  5. Variational posterior consistency theorem
  6. Conditions for the forward map
  7. Rescaled Gaussian prior
  8. High-dimensional Gaussian sieve priors
  9. Contraction rate for two typical inverse problems
  10. Darcy flow problem
  11. Inverse potential problem for a subdiffusion equation
  12. Contraction rate for severely ill-posed problem
  13. Construction of the prior distribution
  14. Contraction theorem with weak conditions
  15. Inverse medium scattering problem
  16. ...and 13 more sections

Key Result

Proposition 2.1

Suppose that for a subset $\Theta \subset L^2(\mathcal{Z},\mathbb{R})$ and some finite constant $U = U_{\mathcal{G},\Theta} > 0$, we have For the model density from modeldensity, we have for every $\theta_1,\theta_2 \in \Theta$, and for $C_U = \frac{1-e^{-U^2/2}}{2U^2}$,

Theorems & Definitions (70)

  • Proposition 2.1
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Theorem 3.9
  • Remark 3.10
  • Theorem 4.1
  • Remark 4.2
  • ...and 60 more