Ricci flow regularization in latent spaces for the forward learning of partial differential equations
This addresses the challenge of representing time-evolving PDE dynamics in machine learning, though it appears incremental as it builds on existing manifold learning and physics-informed approaches.
The authors tackled the problem of learning partial differential equation dynamics by developing an encoder-decoder method where the latent space evolves via Ricci flow, which enables learning from out-of-distribution data and provides adversarial robustness on PDE datasets.
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 from Ricci flow theory.