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

Li Sun, Ziheng Zhang, Zixi Wang, Yujie Wang, Qiqi Wan, Hao Li, Hao Peng, Philip S. Yu

TL;DR

Pioneer tackles the problem of modeling entropy-increasing dynamics in dynamic interacting systems by introducing a physics-informed, Riemannian graph ODE framework. It couples a Constrained Ricci Flow to govern time-evolving edge correlations with a manifold-preserving Gyro-transform for encoding/decoding on a κ-sterographical manifold, ensuring geometry- and physics-consistent predictions. The authors prove entropy non-decreasing behavior under mild graph-density conditions and demonstrate superior predictive performance and geometry-aware reasoning on real-world datasets, along with detailed ablations and entropy analyses. This approach enables more realistic, geometry-aware dynamic simulations with potential applications in climate, epidemiology, and social systems, where physics laws and intrinsic geometry are essential.

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.

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

TL;DR

Pioneer tackles the problem of modeling entropy-increasing dynamics in dynamic interacting systems by introducing a physics-informed, Riemannian graph ODE framework. It couples a Constrained Ricci Flow to govern time-evolving edge correlations with a manifold-preserving Gyro-transform for encoding/decoding on a κ-sterographical manifold, ensuring geometry- and physics-consistent predictions. The authors prove entropy non-decreasing behavior under mild graph-density conditions and demonstrate superior predictive performance and geometry-aware reasoning on real-world datasets, along with detailed ablations and entropy analyses. This approach enables more realistic, geometry-aware dynamic simulations with potential applications in climate, epidemiology, and social systems, where physics laws and intrinsic geometry are essential.

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.

Paper Structure

This paper contains 27 sections, 17 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Riemannian Manifold
  4. Thermodynamics and Ricci Flow
  5. Dynamic System and Graph ODEs
  6. Methodology: Pioneer
  7. $\boldsymbol \kappa-$sterographical model
  8. Physics-informed Differential System
  9. Theory
  10. Manifold-Preserving Transformation
  11. Remark
  12. Riemannian Encoder & Decoder
  13. Objective Function & Complexity Analysis.
  14. Experiment
  15. Datasets & Baselines
  16. ...and 12 more sections

Figures (5)

  • Figure 1: Illustrated example on Earth's system. Traditional Euclidean models expose to generate unrealistic trajectory as in (a), while we preserve the manifold geometry as in (b).
  • Figure 2: Overall architecture of Pioneer.
  • Figure 3: Results of geometric ablation.
  • Figure 4: The changes of entropy over time.
  • Figure 5: Prediction results of Glacier thickness.

Theorems & Definitions (2)

  • proof
  • proof