  We present a manifold-based autoencoder 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 simulating
Ricci flow in a physics-informed setting, and manifold quantities can be
matched so that Ricci flow is empirically achieved. With our method, the
manifold is discerned through the training procedure, while the latent
evolution due to Ricci flow induces a more accommodating representation over
static methods. We present our method on a range of experiments consisting of
PDE data that encompasses desirable characteristics such as periodicity and
randomness. By incorporating latent dynamics, we sustain a manifold latent
representation for all values in the ambient PDE time interval. Furthermore,
the dynamical manifold latent space facilitates qualities such as learning for
out-of-distribution data, and robustness. We showcase our method by
demonstrating these features.
