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arxiv: 2606.31288 · v1 · pith:SFZB5CGKnew · submitted 2026-06-30 · 💻 cs.LG · math.PR· physics.geo-ph

Probabilistic Inversion with Flow Matching

Baldur Paulwitz , Stefan Buske This is my paper

Pith reviewed 2026-07-01 06:19 UTC · model grok-4.3

classification 💻 cs.LG math.PRphysics.geo-ph
keywords flow matchingprobabilistic inversionseismic inversionfull-waveform inversionvelocity modelsposterior distributiongenerative models
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The pith

Flow Matching from generative AI adapts directly to produce posterior distributions over velocity models in geophysical inversion.

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

The paper establishes that Flow Matching can be adapted from its generative AI origins to sample posterior distributions in probabilistic inversion problems. The authors transfer the core mathematical framework with minor changes and test it on a basic 2D velocity model plus the OpenFWI dataset for more realistic seismic cases. A sympathetic reader would care because traditional inversion often yields single best-fit models while this approach supplies full distributions that quantify uncertainty. If the adaptation holds, it supplies an efficient generative route to ensembles of velocity models consistent with observed data.

Core claim

We adapt the well-established mathematical theory of Flow Matching from generative Artificial Intelligence to the context of probabilistic inversion. We evaluate the approach with two case studies: a simple 2D velocity model to illustrate the general features of the method, and the OpenFWI dataset to show its capabilities for probabilistic inversion of more complex seismic velocity models.

What carries the argument

Flow Matching, which learns a time-dependent vector field to transport samples from a simple base distribution to a target data distribution, here repurposed to map noise or prior samples onto posterior velocity models conditioned on seismic observations.

If this is right

  • The method yields multiple velocity models drawn from the posterior rather than a single deterministic solution.
  • Uncertainty quantification becomes available through the spread of the generated ensemble.
  • The same adapted framework scales from toy 2D cases to the more complex velocity structures in the OpenFWI benchmark.
  • Only small modifications to the original AI training procedure are required for the geophysical setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transport mechanism might be applied to other inverse problems that admit a probabilistic formulation, such as electromagnetic or gravity data inversion.
  • Combining the learned flow with explicit wave-equation constraints during training could reduce the number of samples needed for accurate posteriors.
  • Efficiency comparisons against existing sampling methods on identical datasets would quantify any computational advantage.

Load-bearing premise

The Flow Matching framework transfers directly to geophysical probabilistic inversion settings with only minor adaptations and produces meaningful posterior distributions on velocity models.

What would settle it

Generate samples with the adapted Flow Matching on a test case whose true posterior is known from exhaustive MCMC sampling and observe whether the generated velocity-model ensemble matches the reference distribution in statistical moments and coverage.

Figures

Figures reproduced from arXiv: 2606.31288 by Baldur Paulwitz, Stefan Buske.

Figure 1
Figure 1. Figure 1: Simple 2D velocity model. known initial distribution, such as the d-dimensional Normal distribution, into either an unconditioned target distribution, like p(m), or a conditioned target distribution, like p(m|dsyn). In the literature, the first case is referred to as unguided Flow Matching and the second case as guided Flow Matching. This transformation of probability distribution is achieved by learning C… view at source ↗
Figure 2
Figure 2. Figure 2: Conditional Flow Matching for a Dirac distribution [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Marginal) Flow Matching. Blue dots are samples from the Normal distribution (after projecting [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Probabilistic inversion result for given traveltime t = 0.06 s [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Guided Flow Matching. Blue dots are samples from the Normal distribution (after projecting them [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Guided Flow Matching with Classifier-Free Guidance for [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: On the left: the absolute error of the predicted samples in the target distribution without CFG [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A representative velocity model from the FlatFaultB dataset. Source positions are marked as red [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Our Flow Matching approach. First the observed data (seismic wavefield) is preprocessed to obtain [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Probabilistic inversion of a velocity model in the validation dataset. Left: target velocity model. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Probabilistic inversion results as a function of time along the trajectory. [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Assessing the uncertainty of a probabilistic inversion result. Upper row: target velocity model (left) [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Probabilistic inversion results of a isotropic velocity model in the validation dataset. Generated with [PITH_FULL_IMAGE:figures/full_fig_p027_22.png] view at source ↗
read the original abstract

We demonstrate the application of Flow Matching, a technique originating from generative Artificial Intelligence, to probabilistic inversion in geophysical settings, such as seismic Full-Waveform inversion. We adapt the well-established mathematical theory of Flow Matching from generative Artificial Intelligence to the context of probabilistic inversion. We evaluate the approach with two case studies: a simple 2D velocity model to illustrate the general features of the method, and the OpenFWI dataset to show its capabilities for probabilistic inversion of more complex seismic velocity models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper adapts the Flow Matching framework from generative AI to the task of probabilistic inversion in geophysical settings such as seismic full-waveform inversion. It presents the mathematical adaptation and demonstrates the method on two case studies: a simple 2D velocity model to illustrate general features, and the OpenFWI dataset to show applicability to more complex velocity models.

Significance. If the adapted Flow Matching procedure produces samples from the true posterior p(m|d), the work could provide a scalable alternative to MCMC for uncertainty quantification in geophysical inversions. The significance is currently limited by the absence of any quantitative validation that the generated distributions match an independent reference sampler on the same forward and noise model.

major comments (1)
  1. [Abstract / Case studies] The central claim that the adapted Flow Matching yields samples from the Bayesian posterior requires verification against an independent sampler (e.g., MCMC) on identical forward and noise models; the two case studies are described only as illustrations and contain no such quantitative checks (moments, KL divergence, coverage, or posterior predictive tests).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and the constructive suggestion to strengthen the validation of the method. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / Case studies] The central claim that the adapted Flow Matching yields samples from the Bayesian posterior requires verification against an independent sampler (e.g., MCMC) on identical forward and noise models; the two case studies are described only as illustrations and contain no such quantitative checks (moments, KL divergence, coverage, or posterior predictive tests).

    Authors: We agree that the current case studies are presented as illustrations rather than as quantitative benchmarks, and that direct comparison against MCMC (or equivalent) on the same forward operator and noise model would provide stronger evidence that the generated samples match the target posterior. The manuscript's mathematical section derives the flow-matching objective from the conditional probability path that targets p(m|d), so that, under the assumption of exact training, the procedure samples from the posterior by construction. In practice, however, training is approximate and no error metrics are reported. We will revise the abstract, introduction, and conclusions to explicitly state that the experiments are illustrative and to qualify the central claim accordingly. For the simple 2D velocity model we will add a limited quantitative check (e.g., comparison of first and second moments or a posterior-predictive test) against a reference MCMC run performed with identical forward and noise models; such a comparison is computationally feasible in 2-D. For the OpenFWI example, MCMC remains prohibitive, so we will note this limitation and leave a full benchmark for future work. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation chain not visible in provided text

full rationale

The abstract and description adapt an established Flow Matching framework to geophysical inversion without presenting any equations, derivations, or load-bearing steps. No self-definitional mappings, fitted inputs renamed as predictions, or self-citation chains appear. The central claim is an application of prior theory to new settings, with case studies described only illustratively. This is the common honest finding when no mathematical reduction to inputs is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities used in the adaptation.

pith-pipeline@v0.9.1-grok · 5595 in / 928 out tokens · 26923 ms · 2026-07-01T06:19:10.427432+00:00 · methodology

discussion (0)

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Reference graph

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