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

arXiv: https://arxiv.org/abs/2509.21659 · HTML: https://mlsanity.com/paper/2509.21659/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 34 of its 79 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Scalable Adaptive Computation for Iterative Generation](https://mlsanity.com/paper/2212.11972/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [A prior regularized full waveform inversion using generative diffusion models](https://mlsanity.com/paper/2306.12776/) (2023)
- [Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization](https://mlsanity.com/paper/1809.10262/) (2018)
- [One-shot learning for solution operators of partial differential equations](https://mlsanity.com/paper/2104.05512/) (2021)
- [Efficient physics-informed neural networks using hash encoding](https://mlsanity.com/paper/2302.13397/) (2023)
- [PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting](https://mlsanity.com/paper/2109.14536/) (2021)
- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [Total-variation regularization strategies in full-waveform inversion](https://mlsanity.com/paper/1608.06159/) (2016)
- [Controllable seismic velocity synthesis using generative diffusion models](https://mlsanity.com/paper/2402.06277/) (2024)
- [Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestration](https://mlsanity.com/paper/2409.16572/) (2024)
- [PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction](https://mlsanity.com/paper/2404.08412/) (2024)

## Built on by (most cited first, 5 of 5)

- [PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations](https://mlsanity.com/paper/2512.23056/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics](https://mlsanity.com/paper/2512.17273/) (2025)
- [Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade](https://mlsanity.com/paper/2512.01572/) (2025)
- [pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning](https://mlsanity.com/paper/2603.16757/) (2026)

## Abstract (arXiv)

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.