We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i.e., a requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from a $\textit{topological perspective}$, but instead of summarizing entire observation sequences (as done previously), we propose learning a latent dynamical model from topological features $\textit{per time point}$. The latter is then used to formulate a downstream regression task to predict the parametrization of some a priori specified governing equation. We implement this idea based on a latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this modeling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model - $\textit{Neural Persistence Dynamics}$ - substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

翻译：我们研究了从时变点云拓扑结构中学习动力学的问题，时变点云是展现集体行为（如昆虫群、鸟群或物理学中的粒子系统）的普遍时空模型。在此类系统中，模式产生于自驱动实体间的（局部）相互作用。尽管存在几种已被深入理解的运动与相互作用控制方程，但将其拟合到数据中 notoriously 困难，因为大多数先前工作需要关于个体运动轨迹的知识，即随着实体数量增加，这一要求愈发难以满足。为规避此类混杂因素，我们从 $\textit{拓扑视角}$ 研究集体行为，但不同于先前工作那样总结整个观测序列，我们提出从 $\textit{每个时间点}$ 的拓扑特征中学习一个潜在动力学模型。随后，该模型被用于构建下游回归任务，以预测某些先验指定的控制方程的参数化。我们基于从向量化（静态）持久性图中学习的潜在常微分方程来实现这一思想，并证明近期持久同调稳定性结果的组合为这一建模选择提供了依据。各种（消融）实验不仅证明了每个模型组件的相关性，而且提供了令人信服的经验证据，表明我们提出的模型——$\textit{神经持久性动力学}$——在多种参数回归任务中显著优于现有最先进方法。