We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.

翻译：我们开发了融合几何与拓扑信息的数据驱动方法，以从观测数据中学习非线性动力学的简约表示。该方法利用与变分自编码器（VAEs）相关的训练策略，为一般流形潜在空间学习动力学的非线性状态空间模型。我们的方法被称为几何动态（GD）变分自编码器（GD-VAEs）。我们基于深度神经网络架构学习系统状态与演化的编码器与解码器，这些架构包括通用多层感知机（MLPs）、卷积神经网络（CNNs）及其他架构。受参数化偏微分方程和物理问题启发，我们评估了该方法在非线性Burgers方程、约束力学系统以及反应-扩散系统空间场的降维表示学习任务上的性能。GD-VAEs提供的方法可用于在流形潜在空间中获取表示，以应对涉及动力学的多样化学习任务。