We present a manifold-based autoencoder method for learning nonlinear 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 methodology, the manifold is learned as part of the training procedure, so ideal geometries may be discerned, while the evolution simultaneously induces a more accommodating latent representation over static methods. We present our method on a range of numerical experiments consisting of PDEs that encompass desirable characteristics such as periodicity and randomness, remarking error on in-distribution and extrapolation scenarios.

翻译：我们提出一种基于流形的自编码器方法，用于学习非线性时变动力学系统（特别是偏微分方程（PDEs）），其中流形潜空间根据Ricci流演化。该方法可在物理信息驱动框架下模拟Ricci流，并通过匹配流形参量实现经验性Ricci流。通过本方法，流形作为训练过程的一部分被学习，从而可识别理想几何结构，同时该演化过程相比于静态方法能诱导出更具适应性的潜在表示。我们在涵盖周期性与随机性等理想特征的一系列偏微分方程数值实验中展示了该方法，并给出了分布内与外推场景下的误差分析。