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 difficult to fit to data due to the often large number of entities and missing correspondences between the observation times, which may also not be equidistant. To evade such confounding factors, we investigate collective behavior from a \textit{topological perspective}, but instead of summarizing entire observation sequences (as in prior work), 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 this modeling choice is justified by a combination of recent stability results for persistent homology. Various (ablation) experiments not only demonstrate the relevance of each individual 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：我们研究从时间演化点云的拓扑结构中学习动力学的问题，这类时空模型普遍用于展现集体行为的系统，例如昆虫与鸟类的群体运动或物理学中的粒子系统。在此类系统中，模式产生于自驱动实体间的（局部）相互作用。尽管存在若干已被深入理解的运动与相互作用控制方程，但由于实体数量通常庞大、观测时间点间存在缺失对应关系（且时间间隔可能非均匀），这些方程难以拟合实际数据。为规避这些混杂因素，我们从\textit{拓扑视角}探究集体行为，但不同于先前工作对整个观测序列进行概括，我们提出从\textit{每个时间点}的拓扑特征中学习潜在动力学模型。随后利用该模型构建下游回归任务，以预测某些先验指定控制方程的参数化形式。我们基于从向量化（静态）持久性图中学习的潜常微分方程实现这一思想，并通过结合持久同调的最新稳定性结果证明该建模选择的合理性。多项（消融）实验不仅验证了各模型组件的必要性，更提供了有力的实证证据：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优方法。