A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots

The population dynamics of molecules, cells, and organisms are governed by a number of unknown internal and external forces. In the last decade, population dynamics have predominately been modeled with Wasserstein gradient flows. However, since gradient flows minimize free energy, they fail to capture important dynamical properties, such as periodicity. In this work, we propose a change in perspective by considering population dynamics that minimize Wasserstein Lagrangian action, rather than free energy. As our main theoretical contributions, we derive the Hamiltonian equations of motion from the principle of least population-level action and we show that these mechanics encompass classical mechanics, quantum mechanics, and gradient flows. We further leverage the Hamiltonian perspective to propose an algorithm that learns the population mechanics from observed marginals, without specifying the Lagrangian. We demonstrate that by directly learning the population mechanics, our method forecasts and interpolates unseen marginals without a reference process, and outperforms gradient flow and flow matching methods across a wide range of real and simulated experiments.