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Learning the continuous dynamics of a system from snapshots of its temporalmarginals is a problem which appears throughout natural sciences and machinelearning, including in quantum systems, single-cell biological data, andgenerative modeling. In these settings, we assume access to cross-sectionalsamples that are uncorrelated over time, rather than full trajectories ofsamples. In order to better understand the systems under observation, we wouldlike to learn a model of the underlying process that allows us to propagatesamples in time and thereby simulate entire individual trajectories. In thiswork, we propose Action Matching, a method for learning a rich family ofdynamics using only independent samples from its time evolution. We derive atractable training objective, which does not rely on explicit assumptions aboutthe underlying dynamics and does not require back-propagation throughdifferential equations or optimal transport solvers. Inspired by connectionswith optimal transport, we derive extensions of Action Matching to learnstochastic differential equations and dynamics involving creation anddestruction of probability mass. Finally, we showcase applications of ActionMatching by achieving competitive performance in a diverse set of experimentsfrom biology, physics, and generative modeling.

Action Matching: Learning Stochastic Dynamics from Samples