Identifying Systems with Symmetries using Equivariant Autoregressive Reservoir Computers

The investigation reported in this document focuses on identifying systems with symmetries using equivariant autoregressive reservoir computers. General results in structured matrix approximation theory are presented, exploring a two-fold approach. Firstly, a comprehensive examination of generic symmetry-preserving nonlinear time delay embedding is conducted. This involves analyzing time series data sampled from an equivariant system under study. Secondly, sparse least-squares methods are applied to discern approximate representations of the output coupling matrices. These matrices play a critical role in determining the nonlinear autoregressive representation of an equivariant system. The structural characteristics of these matrices are dictated by the set of symmetries inherent in the system. The document outlines prototypical algorithms derived from the described techniques, offering insight into their practical applications. Emphasis is placed on the significant improvement on structured identification precision when compared to classical reservoir computing methods for the simulation of equivariant dynamical systems.