Extrapolation-Aware Predictions with Flow Matching

Flow Matching (FM) is a leading approach for probabilistic prediction in scientific and control applications. When conditioned on inputs outside the training support, however, FM models extrapolate, yielding untrustworthy predictions that cannot be distinguished from valid ones. We trace this to the Lipschitz regularity of neural vector fields, which keeps transport energy nearly constant around the data manifold, so off-manifold inputs follow regular transport paths that produce plausible predictions. We introduce Diverging Flows, a training objective that preserves near-optimal transport for on-manifold conditions while making transport deliberately inefficient for off-manifold ones. We show that the resulting energy margin lower-bounds trajectory curvature, so extrapolations can be detected from the sampling trajectory itself, without auxiliary models, with a threshold calibrated by conformal prediction. Across low-dimensional synthetic manifolds, scientific regression on $64\times 64$ and $128\times 128$ fields, and disjoint-domain generation on $32\times 32$ images, a single model reliably detects extrapolations while retaining FM's predictive fidelity, whereas on the same data FM likelihood is near chance.