Data-Driven Transport Geometry for Rare Events in Metastable Systems

Rare-event transitions in metastable systems are expensive to sample because direct simulation spends most of its time in long-lived basins, and only rarely undergo transitions. We study whether Implicit Manifold-valued Diffusions (IMDs), learned directly from trajectory point clouds, can act as data-driven surrogates of the transport geometry underlying such transitions. Our central hypothesis is that when the training cloud is drawn from a biased dynamical ensemble (such as successful reactive trajectories or steady-state Langevin samples) the learned operator inherits both the support of the data and its sampling bias. On a curved transition tube and on the Müller-Brown landscape, IMDs trained on reactive data recover the dominant transition corridor and yield substantially higher short-horizon crossing probabilities than unbiased reference simulations, without access to the ambient drift. When trained on steady-state data, IMDs reproduce the coarse basin occupancy structure of the reference process. These results support IMDs as lightweight surrogates for transport geometry in metastable systems, while clarifying that they approximate coarse transport structure rather than exact kinetics or invariant laws.