Twist Flow for Inverse Problems

Abstract

In Bayesian inverse problems, posterior sampling requires generating samples that are consistent with given observations while capturing the range of plausible solutions. Direct conditional generative models introduce latent noise to model this ambiguity, but paired inverse-problem training can still encourage an almost deterministic map from the observation to the target. As a result, generated samples may be observation-consistent while under-representing posterior variability, especially when the posterior is multimodal, leading to undercoverage, mode distortion, or artificial transitions between distinct feasible solutions. We propose joint twist-flow, an augmented flow-matching formulation that learns a continuous transport from the augmented source state $(z_x, y)$ to the augmented terminal state $(x, z_y)$. Here x is the target variable, $y$ is the observation, $z_x$ is the Gaussian reference coordinate for posterior sampling, and $z_y$ is a Gaussian likelihood-side coordinate associated with the observation branch. Under a Gaussian observation model, $z_y$ is motivated by the normalized observation residual associated with observation compatibility. Its role is not to replace uncertainty in $x$, but to couple generated samples of x to observation consistency, helping reduce likelihood-inconsistent variation while preserving variability in weakly constrained directions. We validate the method on low-dimensional inverse problems with reference posterior samples, where joint twist-flow better preserves multimodal posterior support than a direct conditional-flow baseline. We further evaluate the method on image restoration and seismic subsurface velocity-model inversion, showing increased posterior variability while maintaining observation consistency.