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

TL;DR

This work applies a new computational framework, RED-DiffEq, by integrating physics-driven inversion and data-driven learning 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.

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.

Figures (11)

• Figure 1: Forward modeling and full waveform inversion. The velocity model (top) with five sources (A--E) generates seismic data under different conditions: clean data (noise-free), noisy data, and data with missing traces.
• Figure 2: Diffusion model architecture and generated samples of velocity map. (a) Schematic of the U-Net denoising network used in the diffusion model. A noisy velocity model is processed through an encoder–decoder U-Net with ResBlocks, downsampling and upsampling layers. Sinusoidal time embeddings are passed through a multilayer perceptron and injected into each ResBlock to condition the network on the diffusion timestep. The network predicts the added noise. (b) Overview of the complete diffusion process, showing both the forward noising process and the learned reverse denoising process. In this study, we set the maximum diffusion time step $T = 1000$ (c) Comparison between velocity maps from the training dataset (top row) and unconditionally generated velocity maps from a single diffusion model pretrained with four velocity model families altogether (bottom row).
• Figure 3: Schematic illustration of RED-DiffEq for full waveform inversion. (a) An overview of the inversion process that iteratively updates the velocity model. (b) An illustration of each iteration step of RED-DiffEq. (c) Calculation of the diffusion-based regularization term. (d) Optional post-processing refinement step using the pretrained diffusion model to further refine the velocity map after the main inversion process is finished. (e) Domain Decomposed Regularization (DDR) strategy for RED-DiffEq. We can train the diffusion model on a small domain and apply on a larger domain during inversion using sliding windows to compute the regularization.
• Figure 4: Results on the OpenFWI dataset. (a) Qualitative comparison among different methods. (b) Quantitative comparison among different methods. (c) Vertical profile results of different methods. (d) Performance of different methods under different Gaussian noise levels in the seismic data. (e) Performance of different methods under different Laplacian noise levels in the seismic data. (f) Performance of different methods under different number of missing traces in the seismic data.
• Figure 5: Examples of uncertainty quantification of RED-DiffEq. The ensemble mean and standard deviation are computed using an ensemble size of 20. We illustrate the correlation between pixel-wise absolute error and predicted uncertainty, where $R$ represents the Spearman rank, and $\rho$ represents the Pearson correlation. Vertical profiles display the mean prediction with 95% confidence intervals at the midpoint (350 m) of each velocity model.