Title: Identifying Systems with Symmetries using Equivariant Autoregressive Reservoir Computers Show Chinese title

Title: 利用等变自回归水库计算机识别具有对称性的系统

Abstract: 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. Show Chinese abstract

Abstract: 本文中报告的研究重点是使用等变自回归水库计算机识别具有对称性的系统。 结构化矩阵逼近理论的通用结果被提出，探讨了两种方法。 首先，对通用保持对称性的非线性时间延迟嵌入进行了全面检查。 这涉及分析从正在研究的等变系统中采样的时间序列数据。 其次，应用稀疏最小二乘法来辨别输出耦合矩阵的近似表示。 这些矩阵在确定等变系统的非线性自回归表示中起着关键作用。 这些矩阵的结构特性由系统固有的对称集决定。 本文概述了从所述技术中派生出的典型算法，提供了它们实际应用的见解。 强调了与经典水库计算方法相比，在模拟等变动力系统时，结构化识别精度有显著提高。