# A duality connecting neural network and cosmological dynamics

arXiv: https://arxiv.org/abs/2202.11104 · HTML: https://mlsanity.com/paper/2202.11104/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 22 Feb 2022.

## Builds on (most cited first, 22 of 22)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Asymptotics of Wide Networks from Feynman Diagrams](https://mlsanity.com/paper/1909.11304/) (2019)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Tensor Programs III: Neural Matrix Laws](https://mlsanity.com/paper/2009.10685/) (2020)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [The edge of chaos: quantum field theory and deep neural networks](https://mlsanity.com/paper/2109.13247/) (2021)
- [Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimization](https://mlsanity.com/paper/2201.11137/) (2022)

## Built on by (most cited first, 10 of 10)

- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [On learning higher-order cumulants in diffusion models](https://mlsanity.com/paper/2410.21212/) (2024)
- [Towards a Phenomenological Understanding of Neural Networks: Data](https://mlsanity.com/paper/2305.00995/) (2023)
- [Environmental effects on emergent strategy in micro-scale multi-agent reinforcement learning](https://mlsanity.com/paper/2307.00994/) (2023)
- [Collective variables of neural networks: empirical time evolution and scaling laws](https://mlsanity.com/paper/2410.07451/) (2024)
- [Beyond Scaling Curves: Internal Dynamics of Neural Networks Through the NTK Lens](https://mlsanity.com/paper/2507.05035/) (2025)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Spectral Reach: Understanding Neural Scaling as Progress into the Spectral Tail](https://mlsanity.com/paper/2605.31244/) (2026)
- [Bulk-boundary decomposition of neural networks](https://mlsanity.com/paper/2511.02003/) (2025)

## Abstract (arXiv)

We demonstrate that the dynamics of neural networks trained with gradient descent and the dynamics of scalar fields in a flat, vacuum energy dominated Universe are structurally profoundly related. This duality provides the framework for synergies between these systems, to understand and explain neural network dynamics and new ways of simulating and describing early Universe models. Working in the continuous-time limit of neural networks, we analytically match the dynamics of the mean background and the dynamics of small perturbations around the mean field, highlighting potential differences in separate limits. We perform empirical tests of this analytic description and quantitatively show the dependence of the effective field theory parameters on hyperparameters of the neural network. As a result of this duality, the cosmological constant is matched inversely to the learning rate in the gradient descent update.