We present a manifold-based autoencoder method for learning dynamics in time, notably partial differential equations (PDEs), in which the manifold latent space evolves according to Ricci flow. This can be accomplished by parameterizing the latent manifold stage and subsequently simulating Ricci flow in a physics-informed setting, matching manifold quantities so that Ricci flow is empirically achieved. We emphasize dynamics that admit low-dimensional representations. With our method, the manifold, induced by the metric, is discerned through the training procedure, while the latent evolution due to Ricci flow provides an accommodating representation. By use of this flow, we sustain a canonical manifold latent representation for all values in the ambient PDE time interval continuum. We showcase that the Ricci flow facilitates qualities such as learning for out-of-distribution data and adversarial robustness on select PDE data. Moreover, we provide a thorough expansion of our methods in regard to special cases, such as neural discovery of non-parametric geometric flows based on conformally flat metrics with entropic strategies from Ricci flow theory.

翻译：我们提出了一种基于流形的自编码器方法，用于学习时间相关的动力学，特别是偏微分方程（PDEs），其中流形潜在空间根据里奇流演化。这可以通过参数化潜在流形阶段，随后在物理信息约束下模拟里奇流来实现，通过匹配流形量使得里奇流在经验上得以达成。我们重点关注那些允许低维表示的动力学系统。通过我们的方法，由度量诱导的流形在训练过程中被识别，而由里奇流驱动的潜在演化则提供了一个适应性强的表示。利用这种流，我们在环境PDE时间连续区间内为所有值维持了一个规范的流形潜在表示。我们证明里奇流能够促进诸如分布外数据学习以及在选定PDE数据上的对抗鲁棒性等特性。此外，我们针对特殊情形对方法进行了全面拓展，例如基于共形平坦度量的非参数几何流的神经发现，其中运用了来自里奇流理论的熵策略。