On Neural Learnability of Chaotic Dynamics

Earth Signals and Systems Group, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 91106, USA (Dated: December 12, 2019) In modeling nonlinear dynamics, neural networks are of interest for prediction and uncertainty quantification. The "learnability" of chaotic dynamics by neural networks, however, remains poorly understood. In this work, we show that a parsimonious network trained on few data points suffices for accurate prediction of local divergence rates on the whole attractor. To understand neural learnability, we decompose the mappings in the neural network into a series of geometric stretching and compressing operations that indicate topological mixing and, therefore, chaos. This reveals that neural networks and chaotic dynamical systems are structurally similar, which yields excellent reproduction of local divergence rates. To build parsimonious networks, we employ an approach that matches the spectral features of the dynamics of deep learning those of polynomial regression.