# Acoustic Waveform Inversion with Image-to-Image Schr"odinger Bridges

arXiv: https://arxiv.org/abs/2506.15346 · HTML: https://mlsanity.com/paper/2506.15346/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 25 of its 44 references that are in the map.
- First posted on arXiv 18 Jun 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, 25 of 25)

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- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.14951/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Deep-learning inversion: a next generation seismic velocity-model building method](https://mlsanity.com/paper/1902.06267/) (2019)
- [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)
- [Controllable seismic velocity synthesis using generative diffusion models](https://mlsanity.com/paper/2402.06277/) (2024)
- [An Empirical Study of Large-Scale Data-Driven Full Waveform Inversion](https://mlsanity.com/paper/2307.15388/) (2023)

## Abstract (arXiv)

Recent developments in application of deep learning models to acoustic Full Waveform Inversion (FWI) are marked by the use of diffusion models as prior distributions for Bayesian-like inference procedures. The advantage of these methods is the ability to generate high-resolution samples, which are otherwise unattainable with classical inversion methods or other deep learning-based solutions. However, the iterative and stochastic nature of sampling from diffusion models along with heuristic nature of output control remain limiting factors for their applicability. For instance, an optimal way to include the approximate velocity model into diffusion-based inversion scheme remains unclear, even though it is considered an essential part of FWI pipeline. We address the issue by employing a Schr"odinger Bridge that interpolates between the distributions of ground truth and smoothed velocity models. To facilitate the learning of nonlinear drifts that transfer samples between distributions we extend the concept of Image-to-Image Schr"odinger Bridge (I2SB) to conditional sampling, resulting in a conditional Image-to-Image Schr"odinger Bridge (cI2SB) framework. To validate our method, we assess its effectiveness in reconstructing the reference velocity model from its smoothed approximation, coupled with the observed seismic signal of fixed shape. Our experiments demonstrate that the proposed solution outperforms our reimplementation of conditional diffusion model suggested in earlier works, while requiring only a few neural function evaluations (NFEs) to achieve sample fidelity superior to that attained with supervised learning-based approach. The supplementary code implementing the algorithms described in this paper can be found in the repository https://github.com/stankevich-mipt/seismicinversionviaI2SB.