Two-Parameter Flows for Learning Population Dynamics of Physical Systems

2026-05-25

AI Summary

This study addresses the challenge of learning the time-evolving probability density dynamics of high-dimensional systems from unlabeled snapshot data alone, without access to true trajectory information. The authors propose a two-stage flow-based method: first, they construct time-parameterized transport maps from a reference distribution to the marginal distributions at each observed time point using conditional flow matching; second, they estimate the physical velocity field by regressing on synthetic trajectories generated from these maps, thereby recovering the full dynamical system. This approach uniquely identifies non-gradient physical dynamics—including rotational and cyclic behaviors—without requiring trajectory data, overcoming limitations of classical optimal transport frameworks. The method accurately reconstructs complex collective dynamics in high-dimensional settings, demonstrating its capability to capture intricate, non-equilibrium processes.

Abstract

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.

Problem

• population dynamics
• probability density
• trajectory-free learning
• high-dimensional systems
• physics-time dynamics

Innovation

• two-parameter flows
• conditional flow matching
• population dynamics
• non-gradient dynamics
• optimal transport