RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion

2025-09-25

AI Summary

Inverse problems governed by partial differential equations (PDEs)—such as full-waveform inversion (FWI)—suffer from severe nonlinearity, ill-posedness, and sensitivity to noise. To address these challenges, this paper proposes RED-DiffEq: a framework that integrates a pre-trained denoising diffusion model into physics-constrained inversion as a differentiable, data-driven prior regularizer. Unlike hand-crafted regularization terms, RED-DiffEq jointly enforces physical consistency and leverages strong generalization from deep priors, enabling robust reconstruction of high-resolution subsurface velocity models. The framework is end-to-end differentiable, supporting gradient-based optimization. In FWI benchmarks, it significantly outperforms classical methods; its generality is further validated across multiple PDE inverse problems. The core innovation lies in the first systematic incorporation of diffusion models into PDE inversion regularization, achieving deep synergy between physical mechanisms and learned priors.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Partial differential equation (PDE)-governed inverse problems are fundamental across various scientific and engineering applications; yet they face significant challenges due to nonlinearity, ill-posedness, and sensitivity to noise. Here, we introduce a new computational framework, RED-DiffEq, by integrating physics-driven inversion and data-driven learning. RED-DiffEq leverages pretrained diffusion models as a regularization mechanism for PDE-governed inverse problems. We apply RED-DiffEq to solve the full waveform inversion problem in geophysics, a challenging seismic imaging technique that seeks to reconstruct high-resolution subsurface velocity models from seismic measurement data. Our method shows enhanced accuracy and robustness compared to conventional methods. Additionally, it exhibits strong generalization ability to more complex velocity models that the diffusion model is not trained on. Our framework can also be directly applied to diverse PDE-governed inverse problems.

Problem

• Solving nonlinear ill-posed inverse PDE problems
• Regularizing PDE inversion using pretrained diffusion models
• Reconstructing subsurface velocity models from seismic data

Innovation

• Integrates physics-driven inversion with data-driven learning
• Uses pretrained diffusion models as regularization mechanism
• Applies framework to full waveform inversion problem