mlsanityData to 14 Oct 2026

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

arXiv 2509.21659 ↗First posted 25 Sep 2025cs.LG physics.geo-phcited by 5

On the map: Diffusion models → Flow matching, time series and science

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.

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.

Builds on

34 of 79 references.

Closest ancestors · 5

Same direction, newest first

  1. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling9 Jun
  2. 2024PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction12 Apr
  3. 2024Controllable seismic velocity synthesis using generative diffusion models9 Feb
  4. 2023A prior regularized full waveform inversion using generative diffusion models22 Jun
  5. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov

Other references · 29

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Denoising Diffusion Implicit Modelscited by 9,781
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2022Denoising Diffusion Restoration Modelscited by 963
  6. 2021ILVR: Conditioning Method for Denoising Diffusion Probabilistic Modelscited by 635
  7. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  8. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  9. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 307
  10. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  11. 2023A Variational Perspective on Solving Inverse Problems with Diffusion Modelscited by 222
  12. 2022Scalable Adaptive Computation for Iterative Generationcited by 195
17 more
  1. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  2. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  3. 2018Regularization by Denoising: Clarifications and New Interpretationscited by 103
  4. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  5. 2018InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFscited by 61
  6. 2023Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustnesscited by 60
  7. 2023Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestrationcited by 46
  8. 2018Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalizationcited by 30
  9. 2021One-shot learning for solution operators of partial differential equationscited by 30
  10. 2023Efficient physics-informed neural networks using hash encodingcited by 21
  11. 2021PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splittingcited by 20
  12. 2016Total-variation regularization strategies in full-waveform inversioncited by 18
  13. 2024Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestrationcited by 11
  14. 2023Deep Learning for Solving and Estimating Dynamic Macro-Finance Modelscited by 6
  15. 2024Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networkscited by 5
  16. 2025Physics-informed waveform inversion using pretrained wavefield neural operatorscited by 5
  17. 2024Disk2Planet: A Robust and Automated Machine Learning Tool for Parameter Inference in Disk-Planet Systemscited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 2.

5 papers.

  1. 2025PI-MFM: Physics-informed multimodal foundation model for solving partial differential equationscited by 7
  2. 2026D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matchingcited by 6
  3. 2025MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamicscited by 1
  4. 2025Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascadecited by 0
  5. 2026pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learningcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Decoupled Latent Optimization of Diffusion Models for Full Waveform Inversion
  2. 2026Structure-aware Bayesian full waveform inversion via diffusion-regularized Stein variational gradient descent
  3. 2026Physical-State-Guided Diffusion Sampling for Full-Waveform Inversion
  4. 2025Diffusion prior as a direct regularization term for FWI
  5. 2026Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems
  6. 2026Full waveform inversion method based on diffusion model
  7. 2025Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields
  8. 2024Geological and Well prior assisted full waveform inversion using conditional diffusion models