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Pioneer: Physics-informed Riemannian Graph ODE for Entropy-increasing Dynamics
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Proposes Pioneer, a physics-informed Riemannian graph ODE that couples a constrained Ricci flow (claimed entropy non-decreasing) with a manifold-preserving Gyro-transform to predict trajectories on curved manifolds for dynamic interacting systems.
Problem
Graph ODE models for dynamic interacting systems ignore the intrinsic geometry of the system and physics laws such as entropy increase, producing geometry-unaware or anti-physical trajectories.
Approach
Formulate the graph dynamics on a kappa-stereographic Riemannian manifold, constrain the canonical Ricci flow with a learned sigmoid-MLP term so that von Neumann entropy is non-decreasing, and replace exp/log maps with a Gyro-transform that keeps features on the manifold via matrix-vector multiplication.
Key results
- On Social dataset with prediction length 40, Pioneer achieves MAPE 44.07 vs 47.82 for the runner-up HOPE.
- On CReSIS with prediction length 1, Pioneer reports MAPE 3.61 and RMSE 47.78, best among compared methods.
- Pioneer achieves best results on nearly all settings across Social, Weather and CReSIS, except one Weather case (RMSE at length 10, where HOPE is best).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add structural evolution to the graph (rather than keep it fixed) (ablation) | Pioneer · Social, prediction length 10 | MAPE: 11.61 -> 9.15 | Helped |
| use Ricci flow rather than only a parameterized graph-dynamics model (ablation) | Pioneer · Social, prediction length 10 | MAPE: 10.49 -> 9.15 | Helped |
| constrain the Ricci flow rather than use canonical Ricci flow (ablation) | Pioneer · Social, prediction length 10 | MAPE: 10.76 -> 9.15 | Helped |
| use Gyro-transform rather than exponential/logarithmic maps (ablation) | Pioneer · Social, prediction length 10 | MAPE: 10.63 -> 9.15 | Helped |
| use the constrained Ricci flow to enforce non-decreasing entropy | Pioneer · Social | von Neumann entropy | Helped |
| model on Riemannian geometry rather than its Euclidean version | Pioneer · CReSIS | MAPE | Helped |
Surprising findings
- Canonical Ricci flow alone does not guarantee entropy increase, and CG-ODE violates entropy increase on the Social dataset.
Limitations
- Entropy non-decreasing theorem relies on the condition m >= n (edges at least nodes), which is assumed to hold for most real graphs.
- Complexity is quadratic in the number of objects, same as existing graph ODEs.
Experimental setup
Datasets: Social Weather CReSIS
Benchmarks: Social Weather CReSIS
Metrics: MAPE RMSE
Baselines: LSTM NRI VGRNN DGCRN NODE HBNODE LG-ODE CG-ODE HOPE
Builds on: graph attention network canonical Ricci flow Forman-Ricci curvature kappa-stereographic model CG-ODE
Closest papers by meaning
- A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics (ICLR 2025 · similarity 0.83)
- Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models (NeurIPS 2025 · similarity 0.83)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.82)
- Curvature-aware Graph Attention for PDEs on Manifolds (ICML 2025 · similarity 0.82)
- Neural Persistence Dynamics (NeurIPS 2024 · similarity 0.82)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.82)
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