Neural Persistence Dynamics

TL;DR

This paper introduces Neural Persistence Dynamics, a novel topological approach to modeling and predicting the dynamics of evolving point cloud systems by learning from topological features at each time point.

Contribution

It proposes a new method that leverages topological features and latent ODEs to model system dynamics without requiring individual trajectory data.

Findings

Outperforms state-of-the-art in parameter regression tasks

Validates stability of topological features for dynamic modeling

Demonstrates effectiveness on diverse collective behavior datasets

Abstract

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.