# Optimal Machine Intelligence at the Edge of Chaos

arXiv: https://arxiv.org/abs/1909.05176 · HTML: https://mlsanity.com/paper/1909.05176/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2019.

## Built on by (most cited first, 7 of 7)

- [Edge of chaos as a guiding principle for modern neural network training](https://mlsanity.com/paper/2107.09437/) (2021)
- [Over the Edge of Chaos? Excess Complexity as a Roadblock to Artificial General Intelligence](https://mlsanity.com/paper/2407.03652/) (2024)
- [Multiple Descents in Deep Learning as a Sequence of Order-Chaos Transitions in LSTM Networks](https://mlsanity.com/paper/2505.20030/) (2025)
- [Adversarial Data Augmentation for Single Domain Generalization via Lyapunov Exponent-Guided Optimization](https://mlsanity.com/paper/2507.04302/) (2025)
- [Toward a Physics of Deep Learning and Brains](https://mlsanity.com/paper/2509.22649/) (2025)
- [Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks](https://mlsanity.com/paper/2201.07934/) (2022)
- [Towards Critical Branching Mechanism in Recurrent Neural Networks](https://mlsanity.com/paper/2606.10384/) (2026)

## Abstract (arXiv)

It has long been suggested that the biological brain operates at some critical point between two different phases, possibly order and chaos. Despite many indirect empirical evidence from the brain and analytical indication on simple neural networks, the foundation of this hypothesis on generic non-linear systems remains unclear. Here we develop a general theory that reveals the exact edge of chaos is the boundary between the chaotic phase and the (pseudo)periodic phase arising from Neimark-Sacker bifurcation. This edge is analytically determined by the asymptotic Jacobian norm values of the non-linear operator and influenced by the dimensionality of the system. The optimality at the edge of chaos is associated with the highest information transfer between input and output at this point similar to that of the logistic map. As empirical validations, our experiments on the various deep learning models in computer vision demonstrate the optimality of the models near the edge of chaos, and we observe that the state-of-art training algorithms push the models towards such edge as they become more accurate. We further establishes the theoretical understanding of deep learning model generalization through asymptotic stability.