Consider a heterogeneous population of points evolving with time. While the population evolves, both in size and nature, we can observe it periodically, through snapshots taken at different timestamps. Each of these snapshots is formed by sampling points from the population at that time, and then creating features to recover point clouds. While these snapshots describe the population's evolution on aggregate, they do not provide directly insights on individual trajectories. This scenario is encountered in several applications, notably single-cell genomics experiments, tracking of particles, or when studying crowd motion. In this paper, we propose to model that dynamic as resulting from the celebrated Jordan-Kinderlehrer-Otto (JKO) proximal scheme. The JKO scheme posits that the configuration taken by a population at time $t$ is one that trades off a decrease w.r.t. an energy (the model we seek to learn) penalized by an optimal transport distance w.r.t. the previous configuration. To that end, we propose JKOnet, a neural architecture that combines an energy model on measures, with (small) optimal displacements solved with input convex neural networks (ICNN). We demonstrate the applicability of our model to explain and predict population dynamics.

该论文提出了一种新的方法来建模一个随时间演化的粒子种群的集体动力学，该方法中使用了 JKOnet 来在只有样本快照的情况下计算 JKO 流，并证明其相较于直接前向方法具有更好的性能和稳健性。

人口动态的近端优化运输建模