用拉格朗日力学建模群体动态，能捕捉周期性等复杂行为。
A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots
- 基于耗散的沃瑟斯坦拉格朗日作用量，构建二阶动力学框架。
- 直接从观测分布中学习，可预测和插值未知状态，性能优于现有方法。
- 适用于分子、细胞到群体的复杂动态，尤其擅长处理周期运动。
分子、细胞和生物体的群体动态由一系列未知力驱动。过去十年，这类动态主要通过沃尔沙泰因梯度流建模，但梯度流仅最小化自由能，无法捕捉周期性等关键动力学特征。本文提出新视角：在阻尼沃尔沙泰因拉格朗日作用量下，最小化群体层面的作用量。通过推导对应的哈密顿方程，我们形式化了沃尔沙泰因拉格朗日力学，这是一种包含经典力学、量子力学和梯度流的结构化二阶动力学类。随后，我们提出WLM算法，首次无需指定拉格朗日函数即可从观测边缘分布中学习这些二阶动态。通过直接学习群体力学，WLM不仅能预测还能插值未观测的边缘分布，在涡旋动力学、胚胎发育和群体聚集等多种动态中均优于现有的梯度流与流匹配方法。
原文摘要 · Abstract (English)
The population dynamics of molecules, cells, and organisms are governed by a number of unknown forces. In the last decade, population dynamics have predominantly 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 dynamics that minimize a population-level action under a damped Wasserstein Lagrangian. By deriving the corresponding Hamiltonian equations of motion, we formalize Wasserstein Lagrangian Mechanics, a structured class of second-order dynamics that encompasses classical mechanics, quantum mechanics, and gradient flows. We then propose WLM as the first algorithm that learns these second-order dynamics from observed marginals, without specifying the Lagrangian. By directly learning the population mechanics, WLM can both forecast and interpolate unseen marginals, and outperforms existing gradient flow and flow matching methods across a wide range of dynamics, including vortex dynamics, embryonic development, and flocking.
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