Geometric Learning Dynamics

Clinical Snapshot

Key Findings

• P Value: Not applicable for theoretical study

• Effect Size: Power-law relationship g ∝ κ^α with three distinct regimes

• Primary Outcome: Three fundamental learning regimes identified based on power-law exponent α

• Nnt Or Sensitivity: α = 1/2 identified as key mechanism for biological complexity emergence

• Confidence Interval: Not provided

Clinical Application

Theoretical framework requires significant computational expertise for implementation No direct relevance to PBS, TGA, or RACGP guidelines; may inform future research directions in computational biology Limited direct clinical application; may inform understanding of biological learning processes

Abstract

We present a unified geometric framework for modeling learning dynamics in physical, biological, and machine learning systems. The theory reveals three fundamental regimes, each emerging from the power-law relationship g ∝ κ α between the metric tensor g in the space of trainable variables and the noise covariance matrix κ . The quantum regime corresponds to α = 1 and describes Schrödinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to α = 1 2 and describes very fast machine learning algorithms. The equilibration regime corresponds to α = 0 and describes classical models of biological evolution. We argue that the emergence of the intermediate regime α = 1 2 is a key mechanism underlying the emergence of biological complexity.

References

1. 1.Vanchurin, V. (2026). Geometric learning dynamics. Biological Cybernetics. https://doi.org/10.3390/e24010007