Ricci flow-guided autoencoders in learning time-dependent dynamics

2024-01-26

arXiv.org

AI Summary

This work addresses geometric mismatch in latent spaces when modeling time-varying partial differential equations (PDEs). We propose the Ricci Flow-Guided Manifold Autoencoder (Ricci-MAE), which models the latent space as a time-evolving Riemannian manifold whose metric evolves under Ricci flow—enabling geometry-aware, physics-constrained learning. Ricci-MAE integrates manifold autoencoding, physics-informed neural networks (PINNs), and entropy-driven conformally flat metric optimization, marking the first use of Ricci flow as an implicit dynamical mechanism for latent evolution. Evaluations across diverse PDE families demonstrate significant improvements in long-term extrapolation accuracy and out-of-distribution generalization, enhanced adversarial robustness, and the ability to invert unknown geometric evolution laws. The framework establishes a novel paradigm for nonparametric geometric flow discovery.

Technology Category

Machine Learning: Learning with ManifoldsSearch and Optimization: Learning to SearchKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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 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, such as neural discovery of non-parametric geometric flows based on conformally flat metrics with entropic strategies from Ricci flow theory.

Problem

• Learning time-dependent dynamics
• Ricci flow-guided autoencoders
• Low-dimensional manifold representations

Innovation

• Manifold-based autoencoder for PDEs
• Ricci flow-guided latent space evolution
• Low-dimensional dynamics representation learning