Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem

Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.

We propose latent-space diffusion posterior sampling (L-DPS), an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). The method addresses three challenges in PDE-constrained inversion: implicit sample-based priors without tractable densities, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations during posterior sampling. L-DPS combines a variational autoencoder, an unconditional latent diffusion model, diffusion posterior sampling, and a differentiable neural surrogate. The VAE maps the parameter field to a lower-dimensional latent space, the diffusion model learns an implicit prior score in this latent space, and DPS combines this learned prior with likelihood-based guidance. The likelihood gradient is evaluated through the decoder-surrogate composition, avoiding repeated calls to the full numerical PDE solver. We evaluate the method on an inverse Darcy flow problem with an unknown spatially distributed permeability field inferred from sparse and noisy pressure observations. L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines such as conditional latent diffusion and inverse FNO in sparse and noisy regimes. We further compare L-DPS with a KLE-MAP baseline and study mixed-prior generalization and the sensitivity of inversion accuracy to surrogate forward-model error.

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.