This code is to produce all figures and results in the paper "Neural Networks as Geometric Chaotic Maps", submitted to IEEE Transactions of Neural Networks and Learning Systems. The current version of the paper is available at https://arxiv.org/abs/1912.05081v3
The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding, but the theoretical understanding of how neural networks learn chaos remains lacking. Here, we employ a geometric perspective to show that neural networks can efficaciously model chaotic dynamics by themselves becoming structurally chaotic. First, we confirm the efficacy of neural networks in emulating chaos by showing that parsimonious neural networks trained only on few data points suffice to reconstruct strange attractors, extrapolate outside training data boundaries, and accurately predict local divergence rates. Second, we show that the trained network’s map comprises a series of geometric stretching, rotation, and compression operations. These geometric operations indicate topological mixing and chaos, explaining why neural networks are naturally suitable to emulate chaotic dynamics.
The code performs the generation of training data, training neural networks on different basis functions from scratch. It may take 1.5 hours to perform the whole computation. The produced analyses and figures showcase the efficacy of traditional feedforward neural networks in modeling chaotic dynamics. It further uses the example of the Hénon map to illustrate the geometric property of the neural network, proving the structural similarity between neural networks and dissipative chaotic maps.
Open a Matlab prompt in the root directory, and run the following code in Matlab's window:
run
Or call Matlab directly from command line:
matlab -r "addpath(genpath('.')); run"
This code is distributed under the MIT license