Unifying and extending Diffusion Models through PDEs for solving Inverse Problems

• URL: http://arxiv.org/abs/2504.07437v1
• Date: Thu, 10 Apr 2025 04:07:36 GMT
• Title: Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
• Authors: Agnimitra Dasgupta, Alexsander Marciano da Cunha, Ali Fardisi, Mehrnegar Aminy, Brianna Binder, Bryan Shaddy, Assad A Oberai,
• Abstract summary: Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML)<n>Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and differential equations.<n>In this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits.
• Score: 3.1225172236361165
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and stochastic differential equations. In contrast to the conventional presentation, in this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits that include a constructive derivation of the forward and reverse processes, a unified derivation of multiple formulations and sampling strategies, and the discovery of a new class of models. We also apply the conditional version of these models to solving canonical conditional density estimation problems and challenging inverse problems. These problems help establish benchmarks for systematically quantifying the performance of different formulations and sampling strategies in this study, and for future studies. Finally, we identify and implement a mechanism through which a single diffusion model can be applied to measurements obtained from multiple measurement operators. Taken together, the contents of this manuscript provide a new understanding and several new directions in the application of diffusion models to solving physics-based inverse problems.

Related papers

• A Survey on Diffusion Models for Inverse Problems [110.6628926886398]
  We provide an overview of methods that utilize pre-trained diffusion models to solve inverse problems without requiring further training. We discuss specific challenges and potential solutions associated with using latent diffusion models for inverse problems.
  arXiv  Detail & Related papers  (2024-09-30T17:34:01Z)
• Conditional score-based diffusion models for solving inverse problems in mechanics [6.319616423658121]
  We propose a framework to perform Bayesian inference using conditional score-based diffusion models. Conditional score-based diffusion models are generative models that learn to approximate the score function of a conditional distribution. We demonstrate the efficacy of the proposed approach on a suite of high-dimensional inverse problems in mechanics.
  arXiv  Detail & Related papers  (2024-06-19T02:09:15Z)
• Diffeomorphic Measure Matching with Kernels for Generative Modeling [1.2058600649065618]
  This article presents a framework for transport of probability measures towards minimum divergence generative modeling and sampling using ordinary differential equations (ODEs) and Reproducing Kernel Hilbert Spaces (RKHSs) A theoretical analysis of the proposed method is presented, giving a priori error bounds in terms of the complexity of the model, the number of samples in the training set, and model misspecification.
  arXiv  Detail & Related papers  (2024-02-12T21:44:20Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• A Variational Perspective on Solving Inverse Problems with Diffusion Models [101.831766524264]
  Inverse tasks can be formulated as inferring a posterior distribution over data. This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. We propose a variational approach that by design seeks to approximate the true posterior distribution.
  arXiv  Detail & Related papers  (2023-05-07T23:00:47Z)
• Reflected Diffusion Models [93.26107023470979]
  We present Reflected Diffusion Models, which reverse a reflected differential equation evolving on the support of the data. Our approach learns the score function through a generalized score matching loss and extends key components of standard diffusion models.
  arXiv  Detail & Related papers  (2023-04-10T17:54:38Z)
• Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC [102.64648158034568]
  diffusion models have quickly become the prevailing approach to generative modeling in many domains. We propose an energy-based parameterization of diffusion models which enables the use of new compositional operators. We find these samplers lead to notable improvements in compositional generation across a wide set of problems.
  arXiv  Detail & Related papers  (2023-02-22T18:48:46Z)
• Approximate Latent Force Model Inference [1.3927943269211591]
  latent force models offer an interpretable alternative to purely data driven tools for inference in dynamical systems. We show that a neural operator approach can scale our model to thousands of instances, enabling fast, distributed computation.
  arXiv  Detail & Related papers  (2021-09-24T09:55:00Z)
• Low-rank statistical finite elements for scalable model-data synthesis [0.8602553195689513]
  statFEM acknowledges a priori model misspecification, by embedding forcing within the governing equations. The method reconstructs the observed data-generating processes with minimal loss of information. This article overcomes this hurdle by embedding a low-rank approximation of the underlying dense covariance matrix.
  arXiv  Detail & Related papers  (2021-09-10T09:51:43Z)
• Accounting for Unobserved Confounding in Domain Generalization [107.0464488046289]
  This paper investigates the problem of learning robust, generalizable prediction models from a combination of datasets. Part of the challenge of learning robust models lies in the influence of unobserved confounders. We demonstrate the empirical performance of our approach on healthcare data from different modalities.
  arXiv  Detail & Related papers  (2020-07-21T08:18:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.