Geometric Learning Dynamics

doi:10.1007/s00422-026-01041-9

Geometric Learning Dynamics

2026-04-18

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PRE

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V

Vitaly Vanchurin *

DOI：10.1007/s00422-026-01041-9

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Abstract

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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 proportional to kappa alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g \propto \kappa <^>\alpha $$\end{document} between the metric tensor g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g $$\end{document} in the space of trainable variables and the noise covariance matrix kappa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \kappa $$\end{document}. The quantum regime corresponds to alpha=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha = 1 $$\end{document} and describes Schr & ouml;dinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to alpha=12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha = frac{1}{2} $$\end{document} and describes very fast machine learning algorithms. The equilibration regime corresponds to alpha=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha = 0 $$\end{document} and describes classical models of biological evolution. We argue that the emergence of the intermediate regime alpha=12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha = frac{1}{2} $$\end{document} is a key mechanism underlying the emergence of biological complexity.

Keywords:

Geometric learning

Efficient learning

Emergent quantumness

Biological evolution

Geometric Learning Dynamics

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