Self-Flow-Matching assisted Full Waveform Inversion

Xinquan Huang; Paris Perdikaris

cs.LGMar 13, 2026

Self-Flow-Matching assisted Full Waveform Inversion

Authors: Xinquan Huang, Paris Perdikaris

Organizations: University of Pennsylvania

Abstract

Full-waveform inversion (FWI) is a high-resolution seismic imaging method that estimates subsurface velocity by matching simulated and recorded waveforms. However, FWI is highly nonlinear, prone to cycle skipping, and sensitive to noise, particularly when low frequencies are missing or the initial model is poor, leading to failures under imperfect acquisition. Diffusion-regularized FWI introduces generative priors to encourage geologically realistic models, but these priors typically require costly offline pretraining and can deteriorate under distribution shift. Moreover, they assume Gaussian initialization and a fixed noise schedule, in which it is unclear how to map a deterministic FWI iterate and its starting model to a well-defined diffusion time or noise level. To address these limitations, we introduce Self-Flow-Matching assisted Full-Waveform Inversion (SFM-FWI), a physics-driven framework that eliminates the need for large-scale offline pretraining while avoiding the noise-level alignment ambiguity. SFM-FWI leverages flow matching to learn a transport field without assuming Gaussian initialization or a predefined noise schedule, so the initial model can be used directly as the starting point of the dynamics. Our approach trains a single flow network online using the governing physics and observed data. At each outer iteration, we build an interpolated model and update the flow by backpropagating the FWI data misfit, providing self-supervision without external training pairs. Experiments on challenging synthetic benchmarks show that SFM-FWI delivers more accurate reconstructions, greater noise robustness, and more stable convergence than standard FWI and pretraining-free regularization methods.

Figures & tables

Figure 1: SFM-FWI schematic. The method alternates between outer sampling and inner misfit-driven online training of a flow network to update $\hat{\mathbf{m}}_{1}$ . This produces a stable coarse-to-fine inversion without pretrained priors.

Figure 2: Convergence curves on the Marmousi sub-region benchmark. Left: data-misfit versus iteration. Middle: relative L2 error of the inverted velocity model with respect to the ground truth. Right: SSIM versus iteration.

Figure 3: Marmousi sub-region inversion results (clean data). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-FWI, and the proposed SFM-FWI.

Approaches	Relative L2 error $\downarrow$	SSIM $\uparrow$
Conventional FWI	0.0213	0.853
Conventional FWI with TV	0.0217	0.862
DIP-FWI	0.0177	0.866
SFM-FWI	0.0130	0.907

Table 1: Quantitative comparison of different approaches on Marmousi sub-region.

Figure 4: Marmousi inversion results (clean data). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Approaches	Relative L2 error $\downarrow$	SSIM $\uparrow$
Conventional FWI	0.0933	0.762
Conventional FWI with TV	0.0934	0.786
DIP-FWI	0.0765	0.779
SFM-FWI	0.0682	0.791

Table 2: Quantitative comparison of different approaches on Marmousi.

Figure 5: Overthrust inversion results (clean data). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Approaches	Relative L2 error $\downarrow$	SSIM $\uparrow$
Conventional FWI	0.0962	0.664
Conventional FWI with TV	0.0959	0.742
DIP-FWI	0.0443	0.851
SFM-FWI	0.0390	0.869

Table 3: Quantitative comparison of different approaches on Overthrust.

Figure 6: Salt body inversion results (clean data). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Approaches	Relative L2 error $\downarrow$	SSIM $\uparrow$
Conventional FWI	0.1983	0.667
Conventional FWI with TV	0.1534	0.811
DIP-FWI	0.4736	0.531
SFM-FWI	0.0607	0.926

Table 4: Quantitative comparison of different approaches on Salt body.

Figure 7: Convergence curves on the salt-body benchmark. Left: data-misfit versus iteration. Middle: relative L2 error of the inverted velocity model with respect to the ground truth. Right: SSIM versus iteration.

Figure 8: Marmousi inversion results (poor initial model). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Figure 9: Simulated synthetic shot gathers recorded at 2.5 km (left) before adding noise, and corresponding noisy data (right).

Figure 10: Marmousi inversion results (noisy data). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Figure 11: Marmousi inversion results (sparse shots). Top row: true velocity model, Gaussian-smoothed initial model, and conventional FWI result. Bottom row: TV-regularized FWI, DIP-assisted FWI, and the proposed SFM-FWI.

Approach	Poor initialization		Noisy observations		Sparse shots
Approach	Relative L2 error $\downarrow$	SSIM $\uparrow$	Relative L2 error $\downarrow$	SSIM $\uparrow$	Relative L2 error $\downarrow$	SSIM $\uparrow$
FWI	0.4234	0.494	0.2219	0.299	0.1292	0.639
FWI + TV	0.2766	0.525	0.1298	0.663	0.1249	0.688
DIP-FWI	0.1813	0.596	0.0943	0.711	0.1160	0.692
SFM-FWI	0.0835	0.732	0.0857	0.731	0.1059	0.712

Table 5: Quantitative comparison of different approaches on Marmousi under three scenarios.

Figure 12: The rank evolution of inverted velocity models with different approaches.

Figure 13: The inversion evolution at different iteration steps.

Figure 14: Potential deblurring (inverse-Hessian-like) effect of SFM-FWI. The left panel represents the corrupted velocity models, and the right panel represents the corresponding corrected velocity models after the flow network prediction.

Iter.	$R_{\mathrm{band}}\uparrow$	HF gain	$\\|\nabla m\\|_{2}^{2}$ gain	$p90(\\|\nabla m\\|)$ gain
0	1.33	0.74	0.46	2.07
200	1.22	0.91	1.15	3.14
400	1.45	1.07	1.43	5.02
600	1.60	1.12	2.33	4.43
800	1.73	1.16	3.03	3.78

Table 6: Quantitative evidence of the deblurring-like effect.

Benchmark	FWI	FWI+TV	DIP-FWI	SFM-FWI
Marmousi Sub-region	20.85	20.89	23.36	23.38
Marmousi	9.01	8.99	10.96	10.93
Overthrust	35.50	35.18	37.39	37.57
Salt body	47.17	47.09	50.31	50.33

Table 7: Computational cost comparison (wall-clock time) per inversion run (minutes).

Figure 15: Ablation study on the sensitivity of SFM-FWI to the number of outer steps

T

and inner steps

K

on the Marmousi sub-region benchmark under a fixed budget

T\times K=3000

. Left: relative L2 error versus iteration. Right: SSIM versus iteration.

Figure 16: Comparison with a pretrained diffusion-prior baseline on the Overthrust benchmark. From left to right: true velocity model, smoothed initial model, DiffusionFWI using pretrained weights from ( Wang et al., 2023 ) (trained on OpenFWI), a pretraining-free self-diffusion variant of our approach (Self-diffusion FWI), and the proposed SFM-FWI. The relative L2 errors of those approaches are 0.0530, 0.0445, and 0.0365, while the corresponding SSIMs are 0.7806, 0.8336, and 0.8730. SFM-FWI better matches both structure and amplitudes, yielding the lowest model error and the highest SSIM.

Method	Relative L2 error $\downarrow$	SSIM $\uparrow$
DiffusionFWI (pretrained on OpenFWI)	0.0530	0.7806
Self-diffusion FWI (random-noise start)	0.0445	0.8336
SFM-FWI (online, physics-driven)	0.0365	0.8730

Table 8: Quantitative comparison under prior mismatch on the Overthrust sub-region.

Inverse Problems

Inverse problems infer hidden causes, parameters, or signals from indirect, noisy measurements using physical models, priors, and statistical methods.

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Objectives: Full-waveform inversion (FWI) is a high-resolution geophysical imaging technique that reconstructs subsurface velocity models by iteratively minimizing the misfit between predicted and observed seismic data. However, under complex geological conditions, conventional FWI suffers from strong dependence on the initial model and tends to produce unstable results when the data are sparse or contaminated by noise. Methods: To address these limitations, this paper proposes a physics-driven generative adversarial network-based full-waveform inversion method. The proposed approach integrates the data-driven capability of deep neural networks with the physical constraints imposed by the seismic wave equation, and employs adversarial training through a discriminator to enhance the stability and robustness of the inversion results. Results: Experimental results on two representative benchmark geological models demonstrate that the proposed method can effectively recover complex velocity structures and achieves superior performance in terms of structural similarity (SSIM) and signal-to-noise ratio (SNR). Conclusions: This method provides a promising solution for alleviating the initial-model dependence in full-waveform inversion and shows strong potential for practical applications.

Xinyi Zhang, Caiyun Liu, Jie Xiong +1

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Seismic Imaging Inverse Problems Image Inverse Problems Generative Adversarial Networks Physics-Informed Generative Modeling

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Full waveform inversion (FWI) recovers subsurface velocity from seismic recordings by solving a severely ill-posed, nonconvex PDE-constrained optimization. Classical regularizers stabilize the inversion but fail to reproduce realistic geological structures; recent diffusion-prior methods improve realism at the cost of a fragile trade-off between data fidelity and prior consistency. We propose Decoupled Latent Optimization (DLO), which relaxes the standard latent-optimization formulation into a quadratic-penalty objective over an auxiliary physical variable and a latent variable. The data-fidelity gradient acts in physical space, the diffusion sampler contributes only through a decoded prior sample, and the standard smoothed-velocity initialization of classical FWI is preserved. On the OpenFWI benchmark, DLO outperforms classical regularizers and existing diffusion-based methods under clean, noisy, and missing-trace acquisitions. The prior, trained on 70*70 OpenFWI models, transfers directly to the Marmousi and Overthrust benchmarks, where DLO recovers intricate fault structures and remains robust to initialization smoothing and measurement noise.

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Diffusion-Based Inverse Problems PDE Inverse Problems Inverse Problems

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Improving Full Waveform Inversion in Large Model Era

Full Waveform Inversion (FWI) is a highly nonlinear and ill-posed problem that aims to recover subsurface velocity maps from surface-recorded seismic waveforms data. Existing data-driven FWI typically uses small models, as available datasets have limited volume, geological diversity, and spatial extent, leading to substantial concerns about overfitting. Although they perform well on synthetic datasets, current methods fail to generalize to more realistic geological structures. In this work, we show that a model trained entirely on simulated and relatively simple data can generalize remarkably well to challenging and unseen geological benchmarks. We provide a working recipe that tames a billion-parameter model for FWI through coordinated scaling across three axes: model capacity, data diversity, and training strategy. Our model achieves state-of-the-art performance on OpenFWI and significantly narrows the generalization gap in data-driven FWI. Across six challenging geophysical benchmarks, including Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP, Sigsbee, and SEAM Phase I, it infers complex structures absent from the training set and delivers significant performance improvements (SSIM from 0.5844 to 0.7669). Overall, our results demonstrate that with an appropriate scaling strategy, large models trained on simple synthetic data can achieve substantial generalization to more complex and realistic geological structures.

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Abstract

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