Ricci flow-guided autoencoders in learning time-dependent dynamics

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.