Pioneer: Physics-informed Riemannian Graph ODE for Entropy-increasing Dynamics

TL;DR

Pioneer introduces a physics-informed Riemannian graph ODE framework that models entropy-increasing dynamics by incorporating system geometry and physics laws, outperforming traditional Euclidean-based methods.

Contribution

It is the first to integrate Riemannian geometry and physics laws into graph ODEs for entropy-increasing systems, providing a more realistic modeling approach.

Findings

Proves entropy non-decreasing property of the model

Demonstrates superior performance on real datasets

Incorporates Riemannian geometry and physics constraints

Abstract

Dynamic interacting system modeling is important for understanding and simulating real world systems. The system is typically described as a graph, where multiple objects dynamically interact with each other and evolve over time. In recent years, graph Ordinary Differential Equations (ODE) receive increasing research attentions. While achieving encouraging results, existing solutions prioritize the traditional Euclidean space, and neglect the intrinsic geometry of the system and physics laws, e.g., the principle of entropy increasing. The limitations above motivate us to rethink the system dynamics from a fresh perspective of Riemannian geometry, and pose a more realistic problem of physics-informed dynamic system modeling, considering the underlying geometry and physics law for the first time. In this paper, we present a novel physics-informed Riemannian graph ODE for a wide range of entropy-increasing dynamic systems (termed as Pioneer). In particular, we formulate a differential system on the Riemannian manifold, where a manifold-valued graph ODE is governed by the proposed constrained Ricci flow, and a manifold preserving Gyro-transform aware of system geometry. Theoretically, we report the provable entropy non-decreasing of our formulation, obeying the physics laws. Empirical results show the superiority of Pioneer on real datasets.