We present a manifold-based machine learning encoder-decoder 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 parameterizing the latent manifold stage and subsequently simulating Ricci flow in a physics-informed setting, matching manifold quantities so that Ricci flow is empirically achieved. We emphasize dynamics that admit low-dimensional representations. With our method, the manifold, induced by the metric, is discerned through the training procedure, while the latent evolution due to Ricci flow provides an accommodating representation. By use of this flow, we sustain a canonical manifold latent representation for all values in the ambient PDE time interval continuum. We showcase that the Ricci flow facilitates qualities such as learning for out-of-distribution data and adversarial robustness on select PDE data. Moreover, we provide a thorough expansion of our methods in regard to special cases which allow higher-dimensional representations, such as Ricci flow on the hypersphere and neural discovery of non-parametric geometric flows with entropic strategies.

翻译：我们提出了一种基于流形的机器学习编码器-解码器方法，用于学习时间动力学，特别是偏微分方程，其中流形潜在空间根据里奇流演化。这可以通过参数化潜在流形阶段，随后在物理信息驱动的情境中模拟里奇流来实现，匹配流形量使得里奇流被经验性地达成。我们强调了具备低维表示特性的动力学。通过我们的方法，由度量诱导的流形在训练过程中被识别，而由里奇流驱动的潜在演化为其提供了适应性表示。通过使用这种流，我们在环境偏微分方程时间区间连续体中的所有数值上维持了正则化的流形潜在表示。我们展示了里奇流促进了诸如分布外数据的学习以及特定偏微分方程数据上的对抗鲁棒性等特性。此外，我们针对允许高维表示的特殊情形（如超球面上的里奇流以及通过熵策略进行非参数几何流的神发现）对我们的方法进行了全面扩展。