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This research explores a novel paradigm, Geometric Flow Networks (GFN), for enhancing neural computation by evolving internal representations over time.

April 17, 2026Open Access

Geometric Flow Networks: A Physics-Informed Paradigm for Sequential Intelligence

Key Points

This research explores a novel paradigm, Geometric Flow Networks (GFN), for enhancing neural computation by evolving internal representations over time.
Introduced the Geodesic State Space Model (G-SSM) leveraging Riemannian manifolds and symplectic integration.
Developed the Inertial State Network (ISN) to maintain persistent state and manage context through deep stateful operations.
Conducted empirical evaluations to benchmark performance against traditional Transformer models.
The ISN achieves character-level language generation with a perplexity of 2.48 using only 363,329 parameters.
In long-context scenarios, ISN maintains a throughput of 302 tokens per second, outperforming Transformers with 231 TPS.
GFN demonstrates constant memory complexity and significant parameter reduction compared to existing models.

Abstract

Geometric Flow Networks (GFN) constitute a paradigm of neural computation that formalizes intelligence as the continuous evolution of a persistent internal world governed by structural invariants. Unlike the stateless correlation computed by self-attention mechanisms, GFN treats computation as the trajectory of a state vector flowing through a geometric manifold where inputs act as external perturbations that curve the trajectory without replacing the state. This architecture ensures that information is transformed according to structural conservation laws rather than being transiently buffered or destroyed, thereby enabling a demonstrated constant-memory footprint for the Recursive State Memory regardless of context length. Two distinct realizations demonstrate the paradigm's scope and generality. The Geodesic State Space Model (G-SSM) formalizes representation as a continuous flow on a learned Riemannian manifold, evolving phase-space variables (x, v) through symplectic integration. In parallel, the Inertial State Network (ISN) implements a deep stateful pipeline where both semantic scanning and world evolution maintain persistent momentum. Our empirical evaluations establish a rigorous efficiency benchmark: the ISN achieves character-level language generation with a perplexity of 2. 48 using only 363, 329 parameters. In long-context inference (L = 2000), the ISN maintains a constant throughput of 302 TPS, significantly outperforming Transformer baselines (231 TPS) which exhibit O (N²) degradation. By shifting from statistical correlation to structure-preserving dynamics, GFN enables a theoretical constant state memory complexity and significant parameter reduction. While the paradigm provides structural grounding against hallucinations through deterministic invariants, we observe that in open-domain generative tasks, the effectiveness of this resistance is coupled to the scale and precision of the learned manifold metrics. Our findings indicate that while topological constraints offer absolute guarantees in logic, linguistic domains exhibit a "soft" resistance subject to metric resolution. Code and models are available at https: //github. com/DepthMuun/gfn and https: //huggingface. co/DepthMuun.

Geometric Flow Networks: A Physics-Informed Paradigm for Sequential Intelligence

Key Points

Abstract

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