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 simulating Ricci flow in a physics-informed setting, and manifold quantities can be matched so that Ricci flow is empirically achieved. With our method, the manifold is discerned through the training procedure, while the latent evolution due to Ricci flow induces a more accommodating representation over static methods. We present our method on a range of experiments consisting of PDE data that encompasses desirable characteristics such as periodicity and randomness. By incorporating latent dynamics, we sustain a manifold latent representation for all values in the ambient PDE time interval. Furthermore, the dynamical manifold latent space facilitates qualities such as learning for out-of-distribution data, and robustness. We showcase our method by demonstrating these features.

翻译：我们提出了一种基于流形的自编码器方法，用于学习时间上的动力学，特别是偏微分方程（PDEs），其中流形潜在空间根据里奇流演化。这可以通过在物理信息约束下模拟里奇流来实现，并且可以匹配流形上的量，从而在经验上实现里奇流。通过我们的方法，流形在训练过程中被识别，而由里奇流引起的潜在演化相比于静态方法能诱导出更具适应性的表示。我们在包含周期性、随机性等理想特性的一系列偏微分方程数据实验中展示了该方法。通过引入潜在动力学，我们在环境偏微分方程时间区间内的所有值上维持了一个流形潜在表示。此外，动态的流形潜在空间促进了诸如分布外数据学习和鲁棒性等特性。我们通过展示这些特性来验证我们的方法。