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.

通过学习拓扑特征的隐藏动态模型，我们提出的神经持久动态模型在各种参数回归任务中实质性地优于当前最先进的模型。

神经持续动力学

Neural Persistence Dynamics

We propose a method for the unsupervised reconstruction of a
temporally-coherent sequence of surfaces from a sequence of time-evolving point
clouds, yielding dense, semantically meaningful correspondences between all
keyframes. We represent the reconstructed surface as an atlas, using a neural
network. Using canonical correspondences defined via the atlas, we encourage
the reconstruction to be as isometric as possible across frames, leading to
semantically-meaningful reconstruction. Through experiments and comparisons, we
empirically show that our method achieves results that exceed that state of the
art in the accuracy of unsupervised correspondences and accuracy of surface
reconstruction.

本研究提出了一种无监督的重建方法，从时间演变的点云序列中重建出具有时间一致性的表面序列，并使用神经网络表示重建的表面，通过构建规范对应关系，实现了跨帧的等距重建，达到了超过现有技术的无监督对应关系和表面重建精度的效果。