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，用于学习时间维度上的非线性动力学（特别是偏微分方程体系），其中流形潜空间遵循Ricci流演化。该方法可通过在物理信息引导框架下模拟Ricci流实现，通过匹配流形几何量使Ricci流经验性地达成。在本方法中，流形作为训练流程的组成部分被同步学习，从而可自动识别理想几何结构，同时其演化过程相较于静态方法能生成更具适应性的潜空间表征。我们在一系列涵盖周期性与随机性等典型特征的偏微分方程数值实验中验证了该方法，并讨论了在分布内预测与外推场景下的误差特征。