Probabilistic Inversion with Flow Matching

Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a separate data-consistency update. In these common formulations, the forward model is not explicitly enforced within the learned conditional velocity field. We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems. Under linear interpolation, we express the conditional velocity $v(x_t,t,y)$ in terms of the posterior mean $E[x_1 | x_t,y]$, and characterize that mean as the unique minimizer of a variational objective whose data-consistency term is explicit. We further prove that the velocity field defines a probability flow from the source distribution to the measurement-conditioned posterior. Splitting the variational objective yields a conditional velocity parameterization with operator-dependent data-consistency updates, which we train end-to-end under the flow-matching objective, with no additional guidance at inference. Our method achieves state-of-the-art PSNR with $50\times$ fewer function evaluations than the strongest flow baseline. Varying the sampling steps provides test-time control over the distortion-perception trade-off without retraining.

Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers assume linear forward models and/or make simplifying approximations in posterior sampling. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) to decompose the posterior into a likelihood step and a prior step. Specifically, we sample from the likelihood step using Langevin dynamics and leverage the Stochastic Interpolants (SI) framework to integrate a pretrained flow model into the prior step. We provide a form for the prior step that uses SI's reverse-time SDE, and show connections to previous PnP methods. Moreover, with the aid of the flow prior's straight probability paths and a novel timestep correction technique for the reverse-time SDE, FlowSGS requires fewer network evaluations in its prior step than plug-and-play diffusion samplers. Our experiments show state-of-the-art performance on a range of inverse problems. For the first time, we provide an experiment on a nonlinear inverse problem (Fourier phase retrieval) for flow-based inverse solvers.

Full waveform inversion (FWI) is the gold standard for subsurface imaging, with applications from carbon sequestration to energy and mineral exploration to earthquake hazard assessment. Machine learning approaches to FWI need field-scale, geologically diverse, and physically realistic training data, but existing resources such as Marmousi, SEAM, and OpenFWI fall short on spatial extent, temporal extent, geological diversity, and physical realism. We address these limitations with SubsurfaceGen, a GPU-accelerated generator for 3D velocity models and seismic data. Along with SubsurfaceGen, we release a paired dataset of 4,276 2D velocity slices, 5 s wavefields, and 8 s shot gathers drawn from 42 realistic, field-scale 3D velocity models, each spanning 10 km x 10 km laterally and 6.19 km deep at 10 m resolution. The dataset spans six geological settings -- four built with SubsurfaceGen and two drawn from prior sources -- relevant for carbon sequestration and hydrocarbon exploration. We use this dataset to evaluate neural operators on wavefield prediction and encoder-decoders on end-to-end velocity inversion, holding out one geological setting for out-of-distribution testing. These experiments surface failure modes at field-scale and demonstrate how SubsurfaceGen and the associated dataset can impact ML-based FWI.