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，用于学习非线性时间动力学问题（特别是偏微分方程），其中流形隐空间随Ricci流演变。该方法通过在物理信息框架中模拟Ricci流实现，通过匹配流形量使Ricci流经验性地达成。利用我们的方法，流形在训练过程中被学习，从而能够识别出理想的几何结构，同时这种演化过程相较于静态方法诱导出更具适应性的隐空间表示。我们在包含周期性和随机性等理想特性的偏微分方程数值实验上验证了该方法，并报告了分布内和推演场景下的误差分析。