Twist Flow for Inverse Problems

Shiqin Zeng, Zijun Deng, Felix J. Herrmann

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…