# Geometric framework for biological evolution

arXiv: https://arxiv.org/abs/2603.15198 · HTML: https://mlsanity.com/paper/2603.15198/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [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)
- [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)
- [Covariant Gradient Descent](https://mlsanity.com/paper/2504.05279/) (2025)
- [Geometric Learning Dynamics](https://mlsanity.com/paper/2504.14728/) (2025)

## Abstract (arXiv)

We develop a generally covariant description of evolutionary dynamics that operates consistently in both genotype and phenotype spaces. We show that the maximum entropy principle yields a fundamental identification between the inverse metric tensor and the covariance matrix, revealing the Lande equation as a covariant gradient ascent equation. This demonstrates that evolution can be modeled as a learning process on the fitness landscape, with the specific learning algorithm determined by the functional relation between the metric tensor and the noise covariance arising from microscopic dynamics. While the metric (or the inverse genotypic covariance matrix) has been extensively characterized empirically, the noise covariance and its associated observable (the covariance of evolutionary changes) have never been directly measured. This poses the experimental challenge of determining the functional form relating metric to noise covariance.