# Emergent field theories from neural networks

arXiv: https://arxiv.org/abs/2411.08138 · HTML: https://mlsanity.com/paper/2411.08138/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 12 Nov 2024.

## Builds on (most cited first, 9 of 9)

- [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)
- [Autonomous particles](https://mlsanity.com/paper/2301.10077/) (2023)
- [Bio-inspired Machine Learning: programmed death and replication](https://mlsanity.com/paper/2207.04886/) (2022)

## Built on by (most cited first, 4 of 4)

- [Covariant Gradient Descent](https://mlsanity.com/paper/2504.05279/) (2025)
- [Geometric Learning Dynamics](https://mlsanity.com/paper/2504.14728/) (2025)
- [Molecular Learning Dynamics](https://mlsanity.com/paper/2504.10560/) (2025)
- [Bulk-boundary decomposition of neural networks](https://mlsanity.com/paper/2511.02003/) (2025)

## Abstract (arXiv)

We establish a duality relation between Hamiltonian systems and neural network-based learning systems. We show that the Hamilton's equations for position and momentum variables correspond to the equations governing the activation dynamics of non-trainable variables and the learning dynamics of trainable variables. The duality is then applied to model various field theories using the activation and learning dynamics of neural networks. For Klein-Gordon fields, the corresponding weight tensor is symmetric, while for Dirac fields, the weight tensor must contain an anti-symmetric tensor factor. The dynamical components of the weight and bias tensors correspond, respectively, to the temporal and spatial components of the gauge field.