Neural Networks as Geometric Chaotic Maps
This provides a theoretical foundation for using neural networks in chaotic systems modeling, which is incremental as it builds on existing empirical observations.
The paper tackled the problem of theoretically understanding how neural networks learn chaotic dynamics by employing a geometric perspective, showing that neural networks can efficiently model chaos by becoming structurally chaotic themselves, with results including accurate reconstruction of strange attractors and prediction of local divergence rates using few data points.
The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how neural networks learn chaos is lacking. Here, we employ a geometric perspective to show that neural networks can efficiently model chaotic dynamics by becoming structurally chaotic themselves. We first confirm neural network's efficiency in emulating chaos by showing that a parsimonious neural network trained only on few data points can reconstruct strange attractors, extrapolate outside training data boundaries, and accurately predict local divergence rates. We then posit that the trained network's map comprises sequential 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.