Space Is Intelligence: Neural Semigroup Superposition for Riemannian Metric Generation

2026-06-17

AI Summary

This work addresses the challenge of motion planning in complex environments, where traditional agents relying on internal planning and search mechanisms often struggle with efficiency. The authors propose embedding intelligence directly into the environment by constructing a Riemannian metric on the scene-induced configuration manifold, thereby reducing action execution to geodesic traversal without requiring explicit planners or collision detection. To achieve this, they introduce a neural semigroup superposition mechanism that integrates frame parameters, modulation parameters, and base coefficients to generate a unified Riemannian metric field. Coupled with an Encoder-Router network architecture, the model jointly optimizes geometric modeling and parameter modulation. Remarkably, after training solely on a single double-obstacle scenario, the model demonstrates zero-shot generalization to unseen obstacle configurations, producing paths whose collision-free cost is orders of magnitude lower than those of penetrating trajectories.

Abstract

Traditional approaches place intelligence in the agent, whether as a learned policy or a search procedure. We instead place intelligence in the space itself: a scene induces a Riemannian metric on the configuration manifold, and action reduces to following the geodesics of that metric rather than invoking a separate planner or collision checker. A single Encoder-Router network realizes this idea through three complementary parameter groups -- frame parameters that orient the generators, modulation parameters that govern their spatial propagation, and basic coefficients that determine their strength. These groups combine through a shared semigroup-superposition mechanism to produce a single Riemannian metric field, yielding a compact architecture whose geometry scales naturally with scene complexity. Trained on a single two-obstacle scene, the model demonstrates robust zero-shot generalization across unseen obstacle configurations, with orders-of-magnitude separation between collision-free and obstacle-penetrating path costs.

Problem

• Riemannian metric
• motion planning
• geometric intelligence
• zero-shot generalization
• configuration manifold

Innovation

• Riemannian metric generation
• neural semigroup superposition
• geodesic-based planning
• zero-shot generalization
• Encoder-Router architecture