Active Diffusion-Based Inference for Ill-Posed Inverse Problems under Incomplete Priors

Jitao Xu; Nobuo Sato; Yaohang Li

stat.MLAug 27, 2026

Active Diffusion-Based Inference for Ill-Posed Inverse Problems under Incomplete Priors

Authors: Jitao Xu, Nobuo Sato, Yaohang Li

Organizations: Department of Computer Science, Old Dominion University, Norfolk, Virginia 23529, USA · Jefferson Lab, Newport News, Virginia 23606, USA

Abstract

Many scientific and engineering applications require estimating unknown parameters from experimentally observable data -- an inverse problem that is inherently challenging due to nonlinearity, noise, and ill-posedness. In this paper, we propose an active diffusion-based inverse problem solver. A DM is trained to learn the mapping between the parameter space and the observable space. By iteratively detecting and correcting model misspecification through posterior uncertainty, the method discovers and learns the correct region of parameter space, even when initial training bounds exclude the true parameters. This provides a principled, Bayesian justification for adaptive domain augmentation and ensures robust inference for inverse problems under incomplete prior knowledge. We demonstrate the effectiveness of our inverse solver for a toy inverse problem with infinite solutions, and for the parameterization of the quantum correlation functions to event observables in a Quantum Chromodynamics analysis of nucleon structure.

Figures & tables

Figure 1: Overview of the active diffusion-based inference framework. Left (Training Phase): parameters are sampled from the prior domain, passed through the forward model to generate training data, and used to train the DM. Right (Inference Phase): given a target observable $y^{*}$ , the trained model performs inference; if the estimated uncertainty exceeds the threshold $\tau$ , the domain is adaptively expanded and the model is retrained until convergence.

Figure 2: Ensemble results (4 independent runs) for the toy inverse problem. Top: posterior samples with true solution (solid circle) and marginal distributions. Bottom: pairwise Wasserstein distance matrices between runs (lower values indicate better agreement). (a) In-domain: consistent predictions with low pairwise Wasserstein distances ( $\bar{W}=0.12$ ). (b) Out-of-domain: inconsistent predictions with high pairwise Wasserstein distances ( $\bar{W}=3.00$ ). (c) After active learning: predictions converge with low pairwise Wasserstein distances ( $\bar{W}=0.19$ ).

Figure 3: In-domain inference with true parameters in the training domain. The model accurately recovers all parameters with low uncertainty. The shaded orange region denotes the prior support.

Figure 4: Out-of-domain inference without active learning. Four independently trained models (shown in different colors) produce inconsistent predictions with high ensemble disagreement ( $\sigma_{ens}$ ), indicating model misspecification.

Figure 5: Out-of-domain inference with active learning. After adaptive domain expansion, the model successfully recovers all parameters, where all ground-truth values are within three standard deviations of the corresponding posterior means.

Method	In-domain	OOD fixed	OOD +AL
MDN	0.067	2.497	0.082
INN	0.003	1.679	0.025
NF	0.045	107.243	0.079
DM	0.015	1.279	0.032

Table 1: Comparison of various generative models on the toy circle inverse problem.

Diffusion-Based Inverse Problems

Diffusion-based inverse problems use generative priors to recover images and physical fields from incomplete, noisy, or indirect measurements.

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Estimation of instrument and noise parameters for inverse problem based on prior diffusion model

Jean-François Giovannelli

Groupe Signal-Image, IMS (Univ. Bordeaux, CNRS, BINP), Talence, France

Parameter Estimation Bayesian Inverse Problems Diffusion-Based Inverse Problems

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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.

Abstract

Diffusion-Based Inverse Problems

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