Two-Parameter Flows for Learning Population Dynamics of Physical Systems

This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.

Scientists often want to predict how complex systems, such as fluids or clouds of charged particles, change over time. But in many cases, tracking each individual path is unreliable or impossible. This paper introduces a method that learns how these clouds are connected, so it can predict how the overall collection of possible points in a cloud evolves. The method is especially useful for systems with swirling motion, where some existing approaches are too restrictive or expensive. In experiments, it accurately captured important behavior in particle and fluid simulations while running much faster than detailed physical models.