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

arXiv: https://arxiv.org/abs/2502.03236 · HTML: https://mlsanity.com/paper/2502.03236/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2025.

## Built on by (most cited first, 4 of 4)

- [Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models](https://mlsanity.com/paper/2510.17457/) (2025)
- [OKG-LLM: Aligning Ocean Knowledge Graph with Observation Data via LLMs for Global Sea Surface Temperature Prediction](https://mlsanity.com/paper/2508.00933/) (2025)
- [Riemannian Geometry Speaks Louder Than Words: From Graph Foundation Model to Next-Generation Graph Intelligence](https://mlsanity.com/paper/2603.21601/) (2026)
- [LLM+Graph@VLDB'2025 Workshop Summary](https://mlsanity.com/paper/2604.02861/) (2026)

## Abstract (arXiv)

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.