An echo state network (ESN) is a type of reservoir computer that uses a recurrent neural network with a sparsely connected hidden layer. Compared with other recurrent neural networks, one great advantage of ESN is the simplicity of its training process. Yet, despite the seemingly restricted learnable parameters, ESN has been shown to successfully capture the spatial-temporal dynamics of complex patterns. Here we build an ESN to model the coarsening dynamics of charge-density waves (CDW) in a semi-classical Holstein model, which exhibits a checkerboard electron density modulation at half-filling stabilized by a commensurate lattice distortion. The inputs to the ESN are local CDW order-parameters in a finite neighborhood centered around a given site, while the output is the predicted CDW order of the center site at the next time step. Special care is taken in the design of couplings between hidden layer and input nodes to ensure lattice symmetries are properly incorporated into the ESN model. Since the model predictions depend only on CDW configurations of a finite domain, the ESN is scalable and transferrable in the sense that a model trained on dataset from a small system can be directly applied to dynamical simulations on larger lattices. Our work opens a new avenue for efficient dynamical modeling of pattern formations in functional electron materials.

翻译：回声状态网络（ESN）是一种储层计算机，它使用具有稀疏连接隐藏层的循环神经网络。与其他循环神经网络相比，ESN的一大优势在于其训练过程简单。然而，尽管可学习参数看似受限，ESN已被证明能够成功捕捉复杂模式的时空动力学。本文构建了一个ESN来模拟半经典Holstein模型中电荷密度波（CDW）的粗化动力学，该模型在半填充条件下通过共格晶格畸变稳定，呈现出棋盘状的电子密度调制。ESN的输入是围绕给定格点的有限邻域内的局部CDW序参量，而输出是下一时间步中心格点的预测CDW序。在隐藏层与输入节点之间耦合的设计中特别考虑了晶格对称性，以确保其被恰当地纳入ESN模型。由于模型预测仅依赖于有限区域的CDW构型，因此ESN具有可扩展性和可迁移性，即在小型系统数据集上训练的模型可直接应用于更大晶格的动力学模拟。我们的工作为功能电子材料中图案形成的高效动力学建模开辟了一条新途径。