Eigen analysis of neural autoregressive models of multi-scale chaotic systems: Stability and error propagation

Recent years have seen unprecedented success in the development of data-driven autoregressive (AR) models for predicting high-dimensional multi-scale chaotic systems, e.g., weather, climate, and ocean. These models have proven to be more accurate than state-of-the-art numerical models at orders of magnitude lower computational cost. Despite their success, these models suffer from instabilities when integrated for short periods of time and show unphysical predictions. A cause of this instability is spectral bias, wherein the small scales of the system are poorly represented resulting in the consequent nontrivial error propagation through the neural AR models. In this work, for the first time, we present a rigorous theoretical dynamical systems analysis of such error propagation on state-of-the-art neural networks and operators to close this gap between the theory and practice of the application of neural AR models for scientific applications.