# Geometric Learning Dynamics

arXiv: https://arxiv.org/abs/2504.14728 · HTML: https://mlsanity.com/paper/2504.14728/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 20 Apr 2025.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [The world as a neural network](https://mlsanity.com/paper/2008.01540/) (2020)
- [Towards a Theory of Evolution as Multilevel Learning](https://mlsanity.com/paper/2110.14602/) (2021)
- [Towards a theory of machine learning](https://mlsanity.com/paper/2004.09280/) (2020)
- [Emergent Quantumness in Neural Networks](https://mlsanity.com/paper/2012.05082/) (2020)
- [Towards a theory of quantum gravity from neural networks](https://mlsanity.com/paper/2111.00903/) (2021)
- [Thermodynamics of Evolution and the Origin of Life](https://mlsanity.com/paper/2110.15066/) (2021)
- [Dataset-learning duality and emergent criticality](https://mlsanity.com/paper/2405.17391/) (2024)
- [Emergent field theories from neural networks](https://mlsanity.com/paper/2411.08138/) (2024)
- [Covariant Gradient Descent](https://mlsanity.com/paper/2504.05279/) (2025)
- [Molecular Learning Dynamics](https://mlsanity.com/paper/2504.10560/) (2025)

## Built on by (most cited first, 2 of 2)

- [Verifying Good Regulator Conditions for Hypergraph Observers: Natural Gradient Learning from Causal Invariance via Established Theorems](https://mlsanity.com/paper/2603.09067/) (2026)
- [Geometric framework for biological evolution](https://mlsanity.com/paper/2603.15198/) (2026)

## Abstract (arXiv)

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 propto kappaα between the metric tensor g in the space of trainable variables and the noise covariance matrix kappa. The quantum regime corresponds to α = 1 and describes Schr"odinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to α = tfrac12 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 = tfrac12 is a key mechanism underlying the emergence of biological complexity.