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. 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：我们提出一种基于流形的自编码器方法，用于学习时变动力学（特别是偏微分方程描述的系统），其中流形潜空间根据里奇流演化。该方法通过在物理信息框架中模拟里奇流实现，并通过匹配流形量使得里奇流得以经验性地达成。借助我们的方法，流形通过训练过程自动识别，而里奇流驱动的潜空间演化相较于静态方法能产生更具适应性的表示。我们在包含周期性与随机性等理想特征的偏微分方程数据上开展系列实验，验证了该方法。动态流形潜空间有助于实现诸如对分布外数据的学习与鲁棒性等特性，我们通过实验展示了这些优势。