Flynn Andrew, Herteux Joschka, Tsachouridis Vassilios A, Räth Christoph, Amann Andreas
School of Mathematical Sciences, University College Cork, Cork T12 XF62, Ireland.
Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft-und Raumfahrt, Münchner Str. 20, 82234 Wessling, Germany.
Chaos. 2021 Jul;31(7):073122. doi: 10.1063/5.0055699.
The learning capabilities of a reservoir computer (RC) can be stifled due to symmetry in its design. Including quadratic terms in the training of a RC produces a "square readout matrix" that breaks the symmetry to quell the influence of "mirror-attractors," which are inverted copies of the RC's solutions in state space. In this paper, we prove analytically that certain symmetries in the training data forbid the square readout matrix to exist. These analytical results are explored numerically from the perspective of "multifunctionality," by training the RC to specifically reconstruct a coexistence of the Lorenz attractor and its mirror-attractor. We demonstrate that the square readout matrix emerges when the position of one attractor is slightly altered, even if there are overlapping regions between the attractors or if there is a second pair of attractors. We also find that at large spectral radius values of the RC's internal connections, the square readout matrix reappears prior to the RC crossing the edge of chaos.
由于储层计算机（RC）设计中的对称性，其学习能力可能会受到抑制。在RC的训练中包含二次项会产生一个“平方读出矩阵”，该矩阵打破对称性以抑制“镜像吸引子”的影响，“镜像吸引子”是RC在状态空间中的解的反转副本。在本文中，我们通过分析证明训练数据中的某些对称性禁止平方读出矩阵的存在。从“多功能性”的角度对这些分析结果进行了数值探索，通过训练RC来专门重建洛伦兹吸引子及其镜像吸引子的共存。我们证明，即使吸引子之间存在重叠区域或存在第二对吸引子，当一个吸引子的位置略有改变时，平方读出矩阵就会出现。我们还发现，在RC内部连接的大谱半径值处，平方读出矩阵在RC越过混沌边缘之前重新出现。
