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REVIEW 4 major objections 6 minor 29 references

Diffusion prior as a direct regularization term for FWI

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A pretrained diffusion denoiser can be added to FWI as a direct regularization term.

desk verdict A sensible plug-and-play diffusion-prior recipe for FWI, but Algorithm 1 as printed has a sign error that makes the reported misfit curves impossible, and the paper lacks code or data to resolve it. read the letter →

arxiv 2506.10141 v1 pith:KVPEXWSQ submitted 2025-06-11 physics.geo-ph cs.LGmath.OC

Yuke Xie , Hervé Chauris , Nicolas Desassis This is my paper · ORCID
classification physics.geo-phcs.LGmath.OC MSC 86A22
keywords fullwaveforminversiondiffusionpriorscorerematchingdenoisingprobabilisticmodelgenerativeregularizationseismicimaginginverseproblems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that a pretrained denoising diffusion probabilistic model can serve as a plug-in regularization term for full waveform inversion, without running the reverse diffusion process and without injecting noise into velocity models. The prior is derived from a score-rematching objective, and its gradient is inserted directly into the standard FWI update, balancing data misfit and learned geological structure through a single weight. If this works, seismic inversion would gain a low-cost way to inject learned structural knowledge into existing FWI codes, and the same recipe would extend to other inverse problems governed by differentiable forward operators.

What carries the argument

The central object is the score-rematching prior built from a pretrained DDPM denoiser. Starting from a point-mass variational approximation $q(m)=\delta(m-\hat m)$, the prior term in the variational bound reduces to matching the noise $\epsilon$ added to the current model with the denoiser's prediction $\epsilon_\theta(\sqrt{\bar\alpha_t}m+\sqrt{1-\bar\alpha_t}\epsilon,t)$, and differentiating that mismatch updates the clean model toward the learned manifold. The U-Net denoiser, trained on 40,000 Overthrust-derived 64-by-64 images, supplies the learned score, while a prescribed annealing schedule for $t$ and a linear weight schedule control how strongly the prior acts as the inversion proceeds.

What would settle it

Run the same 100-iteration FWI benchmark with the score-rematching term replaced by a fixed random gradient of equal magnitude; if convergence and final misfit stay the same, the learned prior is not doing the work. Alternatively, invert a target whose geological features are absent from the 40,000-image training set and check whether the regularizer lowers or raises the final data misfit relative to conventional FWI.

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Extended reading notes

Core claim

The paper's central claim is that the pretrained DDPM denoiser $\epsilon_\theta$ defines a usable prior through the score-rematching loss $\mathbb{E}_\epsilon\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}m+\sqrt{1-\bar\alpha_t}\epsilon, t)\|^2$, whose gradient with respect to the clean model $m$ acts as a direct regularizer for FWI. The inversion runs entirely in clean velocity-image space, with the diffusion time step $t$ annealed from large to small over iterations. On synthetic Overthrust-based benchmarks the authors report that this regularization converges faster, reaches a lower final data misfit, and produces more geologically plausible velocity models than conventional FWI, while a prior retrained on diverse synthetic images also improves recovery on the Marmousi2 dataset.

Load-bearing premise

The pretrained denoiser must approximate the score of the target velocity-model distribution well enough, and the hand-tuned annealing and weighting schedules used in the synthetic tests must keep working for other targets; the main benchmark stays inside the training distribution.

Editorial extensions

If this is right

  • Existing FWI pipelines can incorporate a learned prior by adding one gradient term, with no reverse diffusion sampling and no noisy intermediate velocity states.
  • Because wave propagation always sees clean velocity fields, the method avoids non-physical scattering and numerical dispersion that can arise when noise is injected into the model.
  • The soft weighting of the diffusion prior provides a continuum between purely data-driven updates and strongly regularized solutions, which is more flexible than a hard GAN-manifold constraint.
  • The same score-rematching regularization applies to any differentiable forward operator, not only the acoustic wave equation used in the experiments.
  • Retraining the DDPM on a more diverse set of synthetic velocity models improves transfer to unseen geological settings such as Marmousi2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the prior term only needs a differentiable denoiser, the same score-rematching update could be attached to other grid-based inversions, such as electromagnetic or medical imaging, without altering the underlying forward solver.
  • An adaptive schedule that sets the diffusion time step and prior weight from the current data misfit could replace the hand-tuned cosine-modulated linear schedule and might make the method transfer more reliably to out-of-distribution targets.
  • The score-rematching gradient points along the denoiser's learned score, so the update behaves like an annealed score ascent; this suggests the regularizer may implicitly encode local uncertainty information even though the paper does not sample the posterior.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes to regularize full waveform inversion by adding a score-rematching term based on a pretrained DDPM denoiser to the standard FWI update. The method avoids running the full reverse diffusion process and operates directly on clean velocity images. The authors present a variational derivation leading to an objective that combines a denoising-score-matching regularizer with the FWI data misfit, give pseudocode for training the DDPM and for the regularized FWI loop, and report synthetic acoustic experiments on Overthrust-derived and Marmousi2 models, claiming faster convergence and improved inversion quality relative to conventional FWI.

Significance. If the central algorithm is corrected, the proposal is a potentially useful and computationally light way to inject learned geological priors into FWI, and the choice to avoid noisy intermediate states is sensible for wave-propagation stability. The manuscript usefully provides training pseudocode, explicit schedules, and evidence on synthetic data. However, as printed, the FWI algorithm line has a sign inconsistency with the stated objective, the variational derivation contains algebraic errors, the main benchmark is in-distribution with respect to the prior training set, and the claimed GAN comparison is not present in the experiments. These issues are fixable but currently block acceptance.

major comments (4)
  1. [Algorithm 1, line 6; Eq. (23)] In Algorithm 1, line 6, the update is written as m <- m - lambda_t grad_m[ E_epsilon ||epsilon - epsilon_theta(sqrt(alpha_bar_t) m + sqrt(1 - alpha_bar_t) epsilon, t)||^2 - w_t ||F(m)-d_obs||^2 ]. Because the data-misfit term enters with a minus sign, the update ascends on the data misfit; this contradicts Eq. (23), where the data-misfit term is added, and it cannot produce the monotonically decreasing misfit in Fig. 9. The paper does not provide code or data to verify which objective was actually optimized, so the central claim that the method is a minimal modification of standard FWI is not testable as printed. Please correct the pseudocode, clarify that w_t modulates the prior term, and make the exact objective explicit.
  2. [Section 3.1, Eqs. (15)-(16)] The variational derivation is algebraically incorrect as written: in Eq. (15), the equality E_q[log q - log p(m|d)] = E_q[log q - log p(m) - log p(m|d)] + c does not follow from Bayes' rule, since log p(m|d) is proportional to log p(d|m) + log p(m), not to log p(m) + log p(m|d). Eq. (16) also uses p(x|d) where the likelihood p(d|x) is intended and mixes the Dirac variational distribution with the DDPM forward-chain notation. The final objective in Eq. (23) is plausible, but the derivation does not support it in its current form; please rewrite this section with a correct Bayesian expansion.
  3. [Sections 4 and 5] The main numerical benchmark is in-distribution: the DDPM is trained on 40,000 Overthrust-derived 64x64 images (Section 4), and the target model used in Section 5 is a crop of the Overthrust model. The observed improvement therefore partly reflects the prior having already encoded the target geological style. Section 7.3 itself acknowledges that an unrepresentative training set can bias the inversion toward unrealistic solutions. To support the claimed robustness, the qualitative Marmousi2 experiment in Section 6 should be supplemented with acquisition details, quantitative misfit or similarity metrics, and ideally a genuinely out-of-distribution test, or the claims should be restricted to the in-distribution setting.
  4. [Abstract and Conclusions; Section 7.2] The abstract and conclusion state that the proposed method offers enhanced fidelity and robustness 'compared to ... GAN-based FWI approaches,' but no GAN-based FWI experiment is reported in the paper; Section 7.2 discusses GAN priors only in qualitative terms. Please either add the promised comparison or remove the claim from the abstract and conclusion.
minor comments (6)
  1. [Section 4] The training description first says the model was trained for 80 epochs and then says it was trained for a total of 120 epochs; please clarify the correct number.
  2. [Section 5] The grid spacing is reported twice with different values (9.5 m and 8.71 m for dz and dx); please reconcile or explain the discrepancy.
  3. [Section 5] The initial velocity model shown in Fig. 5c is not described; please state how the starting model was generated (for example, by smoothing the true model).
  4. [Section 3.2, Eq. (22)] The phrase 'w(βt) is a weighting function that can be negated' is unclear; if 'neglected' is intended, the role and eventual dropping of this weighting factor should be stated explicitly.
  5. [Section 6] The Marmousi2 experiment reports no quantitative result; please include the acquisition setup and at least one metric (e.g., final data misfit or structural similarity) so that the generalization claim can be assessed.
  6. [References] Several references are incomplete as given: the Marmousi2 citation (Martin) lacks a publication venue, and the Aminzadeh et al. entry would benefit from complete pagination.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the diffusion-prior regularizer is an independently trained model and the FWI update combines it with the standard data misfit, so the central claim does not reduce to its inputs.

full rationale

The proposed objective in Eq. 23 is the DDPM denoising/score-rematching loss evaluated at the current model m plus the standard FWI data-misfit term. The DDPM is trained independently in Section 4 on 40,000 Overthrust-derived images, and the prior is not defined in terms of the seismic data or the inversion output. The variational derivation follows Graikos et al. (2022), which is cited as external prior work, so the score-matching regularization is not imported from a self-citation chain. The only self-citations (Xie et al., 2024) are used for dataset provenance and as prior GAN-work context; neither is a uniqueness theorem, a fitted parameter, or a load-bearing premise. The main benchmark target is cropped from the same Overthrust family used to train the prior, which makes the experiment in-distribution and limits the demonstrated generalization, but that is a benchmark-design limitation, not a circular derivation: the inversion still solves the acoustic wave equation, and the reported misfit reduction is not imposed by construction. The hand-tuned time-step and weighting schedules are hyperparameters, not fitted predictions. Algorithm 1 contains a sign inconsistency in the data-misfit term, which is a correctness and reproducibility issue independent of circularity. No equation in the paper reduces to its own input, so no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a pretrained DDPM whose score accuracy is assumed, on a point-estimate variational approximation, and on hand-tuned annealing and weighting schedules. The algebraic path from the variational bound to the final objective (Eqs. 15-23) is not clean, which weakens the axiomatic footing. No new physical entities are introduced.

free parameters (4)
  • Diffusion time-step annealing schedule t_i = linear decay with cosine perturbation over 100 iterations (exact formula not given)
    Hand-designed schedule decides the noise level at which the score rematching prior is evaluated in each FWI iteration; no sensitivity study is provided.
  • Prior weighting schedule w_t = linearly increasing from 0.1 to 0.2 over iterations
    Hand-chosen balance between data misfit and prior; Section 7.2 says a suboptimal weight under- or over-regularizes, so the result depends on this choice.
  • Gradient normalization factor between FWI and prior gradients = unstated
    The method normalizes the FWI gradient relative to the diffusion prior gradient, but the exact normalization constant or procedure is not reported.
  • Observation noise variance sigma_noise in data likelihood = unstated; negated by weighting schedule
    The Gaussian likelihood variance is mentioned but not assigned a value; it is folded into the hand-chosen weighting schedule.
assumptions (6)
  • standard math The DDPM forward process q(xt|x0) adds Gaussian noise with a fixed variance schedule beta_t, and the reverse process is approximated by Gaussian transitions.
    Standard assumption from Ho et al. 2020, used in Section 2.1.
  • domain assumption The pretrained U-Net denoiser epsilon_theta accurately approximates the score function of the target velocity-image distribution at all annealed timesteps.
    The paper does not measure denoiser score error on the target; the regularization quality depends on this approximation holding.
  • domain assumption A single-point variational approximation q(m) = delta(m - m0) is adequate for the posterior inference problem.
    Equation 19 collapses the variational distribution to a point, which is a strong approximation for a multi-modal nonlinear inverse problem.
  • domain assumption Seismic observations are generated by the acoustic wave equation with known parameters and Gaussian noise.
    The data misfit term assumes an exact forward model; experiments use synthetic noise-free data, so the Gaussian noise model is not validated.
  • ad hoc to paper The hand-crafted annealing time schedule and weighting schedule are valid for the target inversion task.
    These schedules are introduced in Section 5 without derivation, ablation, or robustness analysis.
  • domain assumption The velocity model can be represented as a 64x64 image belonging to the same distribution as the DDPM training set.
    The Overthrust test model is cropped from the same dataset family used to train the prior, so the in-distribution assumption is strong.

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Cite this review

Pith. "Pith review of Diffusion prior as a direct regularization term for FWI." pith.science (2026). https://pith.science/paper/KVPEXWSQ

@misc{pith2026250610141,
  author       = {Pith},
  title        = {Pith review of: Diffusion prior as a direct regularization term for FWI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KVPEXWSQ}},
  note         = {Machine review of arXiv:2506.10141}
}
read the original abstract

Diffusion models have recently shown promise as powerful generative priors for inverse problems. However, conventional applications require solving the full reverse diffusion process and operating on noisy intermediate states, which poses challenges for physics-constrained computational seismic imaging. In particular, such instability is pronounced in non-linear solvers like those used in Full Waveform Inversion (FWI), where wave propagation through noisy velocity fields can lead to numerical artifacts and poor inversion quality. In this work, we propose a simple yet effective framework that directly integrates a pretrained Denoising Diffusion Probabilistic Model (DDPM) as a score-based generative diffusion prior into FWI through a score rematching strategy. Unlike traditional diffusion approaches, our method avoids the reverse diffusion sampling and needs fewer iterations. We operate the image inversion entirely in the clean image space, eliminating the need to operate through noisy velocity models. The generative diffusion prior can be introduced as a simple regularization term in the standard FWI update rule, requiring minimal modification to existing FWI pipelines. This promotes stable wave propagation and can improve convergence behavior and inversion quality. Numerical experiments suggest that the proposed method offers enhanced fidelity and robustness compared to conventional and GAN-based FWI approaches, while remaining practical and computationally efficient for seismic imaging and other inverse problem tasks.

Figures

Figures reproduced from arXiv: 2506.10141 by the authors.

Figure 1
Figure 1. a Diagram: the model is trained to remove noise from given time step t. b Using pretrained DDPM to compute the gradient of the clean image m towards the data manifold by re-matching the score. So the first two terms in Equation 16 can be evaluated over time X t KL (q(xt−1|xt, m) ∥ pθ(xt−1|xt)) (21) At each time step, using the parameterization in Equation 7 X t KL (q(xt−1|xt, m) ∥ pθ(xt−1|xt)) = X t Eϵ∼N(0,I) h 1 2σ… view at source ↗
Figure 2
Figure 2. A diagram illustrating the optimization process with two gradient components. The orange arrow represents [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Generated samples using pretrained DDPM, the images represent the data manifold of the prior distribution. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Diagram of the U-Net architecture used for noise prediction in DDPM. The network consists of several down [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Visualization of the velocity model depicting the subsurface structure, with receivers distributed across every [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Predefined diffusion time-step t annealing schedule for evaluating the score-matching function across inversion iterations i = 0 to i = 99, constructed by superimposing a cosine perturbation on a linear decay function. To balance the influence of the diffusion-prior wi…
Figure 7
Figure 7. Figure 7: a Inversion result using conventional FWI (no prior) and b inversion result using our proposed diffusion-prior FWI with regularization term (with prior). We use the Adam optimizer [Kingma and Ba, 2014] with a learning rate of 1 × 10−2 for both conventional FWI and diff…
Figure 8
Figure 8. Figure 8: Intermediate inversion results across selected iterations, comparing conventional FWI (no prior) and diffusion [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: Data misfit history measured by the ℓ2 norm for conventional FWI (without prior) and diffusion-prior FWI (with prior). The diffusion-prior FWI demonstrates faster and more stable convergence, achieving a lower final data misfit. converge toward geologically plausible s…
Figure 10
Figure 10. Figure 10: Additional tests using different initial velocity models (left), evaluated under the same optimization [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: Samples generated by retraining the DDPM on 100,000 Gaussian process-based images, demonstrating its [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Inversion result on Marmousi2 dataset using the DDPM retrained on Gaussian process-generated images. [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

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