We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models by extending the diffusion-based framework of Mou et al. (2024) to the infinite-dimensional setting. The posterior is represented as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, which allows us to control posterior moments and obtain non-asymptotic concentration rates in Hilbert norms under various likelihood curvature and regularity conditions. We also establish a quantitative Laplace approximation for the posterior. The theory is illustrated in a nonparametric linear Gaussian inverse problem.

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Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys methodological and theoretical developments at the intersection of operator learning and inverse problems. It begins by summarizing the probabilistic and deterministic approaches to inverse problems, and pays special attention to emerging measure-centric formulations that treat observed data or unknown parameters as probability distributions. The discussion then turns to operator learning by covering essential components such as data generation, loss functions, and widely used architectures for representing function-to-function maps. The core of the chapter centers on the end-to-end inverse operator learning paradigm, which aims to directly map observed data to the solution of the inverse problem without requiring explicit knowledge of the forward map. It highlights the unique challenge that noise plays in this data-driven inversion setting, presents structure-aware architectures for both point predictions and posterior estimates, and surveys relevant theory for linear and nonlinear inverse problems. The chapter also discusses the estimation of priors and regularizers, where operator learning is used more selectively within classical inversion algorithms.

We study Bayesian inverse problems with mixed noise, modeled as a combination of additive and multiplicative Gaussian components. While traditional inference methods often assume fixed or known noise characteristics, real-world applications, particularly in physics and chemistry, frequently involve noise with unknown and heterogeneous structure. Motivated by recent advances in flow-based generative modeling, we propose a novel inference framework based on conditional flow matching embedded within an Expectation-Maximization (EM) algorithm to jointly estimate posterior samplers and noise parameters. To enable high-dimensional inference and improve scalability, we use simulation-free ODE-based flow matching as the generative model in the E-step of the EM algorithm. We prove that, under suitable assumptions, the EM updates converge to the true noise parameters in the population limit of infinite observations. Our numerical results illustrate the effectiveness of combining EM inference with flow matching for mixed-noise Bayesian inverse problems.

Recent advancements in solving Bayesian inverse problems have spotlighted de-noising diffusion models (DDMs) as effective priors. Although these have great potential, DDM priors yield complex posterior distributions that are challenging to sample.

Inverse problems governed by partial differential equations (PDEs) play a crucial role in various fields, including computational science, image processing, and engineering. Particularly, Darcy flow equation is a fundamental equation in fluid mechanics, which plays a crucial role in understanding fluid flow through porous media. Bayesian methods provide an effective approach for solving PDEs inverse problems, while their numerical implementation requires numerous evaluations of computationally expensive forward solvers. Therefore, the adoption of surrogate models with lower computational costs is essential. However, constructing a globally accurate surrogate model for high-dimensional complex problems demands high model capacity and large amounts of data. To address this challenge, this study proposes an efficient locally accurate surrogate that focuses on the high-probability regions of the true likelihood in inverse problems, with relatively low model complexity and few training data requirements. Additionally, we introduce a sequential Bayesian design strategy to acquire the proposed surrogate since the high-probability region of the likelihood is unknown. The strategy treats the posterior evolution process of sequential Bayesian design as a Gaussian process, enabling algorithmic acceleration through one-step ahead prior. The complete algorithmic framework is referred to as Sequential Bayesian design for locally accurate surrogate (SBD-LAS). Finally, three experiments based the Darcy flow equation demonstrate the advantages of the proposed method in terms of both inversion accuracy and computational speed.

A key challenge in maximizing the benefits of Magnetic Resonance Imaging (MRI) in clinical settings is to accelerate acquisition times without significantly degrading image quality. This objective requires a balance between under-sampling the raw k-space measurements for faster acquisitions and gathering sufficient raw information for high-fidelity image reconstruction and analysis tasks. To achieve this balance, we propose to use sequential Bayesian experimental design (BED) to provide an adaptive and task-dependent selection of the most informative measurements. Measurements are sequentially augmented with new samples selected to maximize information gain on a posterior distribution over target images. Selection is performed via a gradient-based optimization of a design parameter that defines a subsampling pattern. In this work, we introduce a new active BED procedure that leverages diffusion-based generative models to handle the high dimensionality of the images and employs stochastic optimization to select among a variety of patterns while meeting the acquisition process constraints and budget. So doing, we show how our setting can optimize, not only standard image reconstruction, but also any associated image analysis task. The versatility and performance of our approach are demonstrated on several MRI acquisitions.

This paper presents a practical application of Relative Trajectory Balance (RTB), a recently introduced off-policy reinforcement learning (RL) objective that can asymptotically solve Bayesian inverse problems optimally. We extend the original work by using RTB to train conditional diffusion model posteriors from pretrained unconditional priors for challenging linear and non-linear inverse problems in vision, and science. We use the objective alongside techniques such as off-policy backtracking exploration to improve training. Importantly, our results show that existing training-free diffusion posterior methods struggle to perform effective posterior inference in latent space due to inherent biases.