Ricci flow regularization in latent spaces for the forward learning of partial differential equations

TL;DR

This paper introduces a novel machine learning approach that uses Ricci flow to regularize latent spaces for better learning and representation of PDE dynamics, enhancing robustness and out-of-distribution generalization.

Contribution

The paper proposes a manifold-based encoder-decoder method that incorporates Ricci flow in latent spaces to improve PDE learning and representation, including extensions to higher-dimensional flows.

Findings

Ricci flow regularization improves out-of-distribution learning.

The method enhances adversarial robustness on PDE data.

Extensions to higher-dimensional geometric flows are demonstrated.

Abstract

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.