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

2025-02-05

AI Summary

Existing dynamic interaction systems are predominantly modeled in Euclidean space, neglecting intrinsic geometric structure and thermodynamic constraints—such as the second law of thermodynamics (entropy non-decrease)—leading to physical inconsistencies. To address this, we propose Physics-Informed Riemannian Graph Ordinary Differential Equations (Riemannian Graph ODEs), the first framework integrating constrained Ricci flow and geometry-aware Gyro-transformers into the graph ODE paradigm. It jointly models structural evolution and entropy dynamics on Riemannian manifolds. Theoretically, we prove that our model strictly satisfies entropy non-decrease, unifying differential-geometric modeling with physical interpretability. Empirical evaluation across multiple real-world dynamic graph datasets demonstrates significant improvements over Euclidean graph ODE baselines, achieving superior predictive accuracy and enhanced physical consistency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Model dynamic systems with Riemannian geometry
• Incorporate physics laws in graph ODE
• Ensure entropy non-decreasing in dynamics

Innovation

• Riemannian geometry
• Physics-informed ODE
• Entropy-increasing dynamics