Echo State network for coarsening dynamics of charge density waves

TL;DR

This work builds an ESN to model the coarsening dynamics of charge-density waves (CDWs) in a semiclassical Holstein model, which exhibits a checkerboard electron density modulation at half-filling stabilized by a commensurate lattice distortion.

Abstract

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.

Figures (8)

• Figure 1: Schematic diagram of a scalable ML framework based on echo state network (ESN) for coarsening dynamics a scalar order-parameter field $\phi(\mathbf r, t)$. The middle block shows the architecture of an ESN. The three main components are the input layer $\mathbf x(t)$, the reservoir neural net $\mathbf u(t)$, and the output neurons $\mathbf y(t)$. The reservoir is fully connected to both the input and output layers with weight matrices $\mathbb{W}_{\rm in}$ and $\mathbb{W}_{\rm out}$, respectively. The ESN is designed to predict the order parameter $\phi(\mathbf r_i, t+\Delta t)$ of a given lattice site-$i$ at the next time-step, which corresponds to the signal neuron at the output layer. The ESN prediction is based on the current local order-parameters $\phi(\mathbf r_j, t)$ within a finite neighborhood $\mathcal{N}_i$ of site-$i$. The neighborhood configuration is flattened into an array $\mathbf x(t)$ in the input layer of the ESN.
• Figure 2: Coarsening of Ising domains simulated by cell dynamics (left) and ESN (right). The color intensity shows the value of local order parameter $\phi(\mathbf r, t) \in [-1, +1]$ at different times after a thermal quench.
• Figure 3: Scaled correlation functions $C(r, t)$ versus $r / t^{1/2}$ based snapshots obtained from (a) cell dynamics simulations and (b) ESN predictions.
• Figure 4: (a) a snapshot of the on-site electron number $n_i = n(\mathbf r_i)$ obtained from ED-Langevin dynamics simulations. The corresponding local Ising-type CDW order parameter $\phi_i$ is shown in panel (b).
• Figure 5: Panels $(a)$ and $(b)$ represent the benchmark of ESN predictions for time series of local CDW order parameter $\phi_i(t)$ at two randomly selected sites.