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：我们提出了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，特别是偏微分方程（PDEs），其中流形潜空间根据里奇流演化。通过在物理信息框架中模拟里奇流，并匹配流形量以实现经验性的里奇流，该方法得以实现。利用我们的方法，流形作为训练过程的一部分被学习，从而可以识别出理想的几何结构，同时演化过程相较于静态方法诱导出更具适应性的潜在表示。我们在一系列包含PDEs的数值实验中展示了该方法，这些PDEs涵盖了周期性和随机性等理想特征，并评估了在分布内和外推场景下的误差。