  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.
