Dynamical scaling method improved by a deep learning approach

TL;DR

A dynamical scaling analysis improved by a deep learning approach that employs a neural network, which significantly reduces the computational cost and enables the use of the entire dataset that was inaccessible with Gaussian process regression.

Abstract

We propose a dynamical scaling analysis improved by a deep learning approach. While Gaussian process regression has been widely employed for estimating scaling parameters, its computational cost for parameter optimization becomes a limitation in dynamical scaling analysis, where large datasets are involved. In contrast, the present method employs a neural network, which significantly reduces the computational cost and enables the use of the entire dataset that was inaccessible with Gaussian process regression. We applied the method to the 2D Ising model and the 2D 3-state Potts model, achieving higher accuracy and computational efficiency than conventional approaches.

Figures (7)

• Figure 1: The architecture of the neural network used in this study. FNN denotes a fully connected neural network. The input and output layers consist of a single neuron, while the hidden layer contains 20 neurons. The activation function is the softplus function.
• Figure 2: Size dependence for the 2D Ising model. It shows that the dependence becomes negligible for $L \geq 1001$. The data are plotted every 100 MCS. The legends indicate the number of samples used in each simulation.
• Figure 3: Relaxation data and dynamical scaling plot for the 2D Ising model used in the analysis. Both panels are plotted with 101 data points per temperature for visualization. (a) shows the relaxation data after discarding the initial relaxation regime, which was used for parameter estimation using a neural network. (b) shows the corresponding scaling plot. The estimated parameters are $T_{\mathrm{c}} = 2.269186(1)$, $\lambda = 0.0577(1)$, and $b = 2.1555(3)$.
• Figure 4: Estimated values of $T_{\mathrm{c}}$ as a function of batch size in the optimization of dynamical scaling parameters for the 2D Ising model. The horizontal blue line indicates the exact solution for the critical temperature. The error bars were estimated using the bootstrap method with 128 resamplings
• Figure 5: Optimization process of the dynamical scaling parameters in the 2D Ising model at batch size $256$. Each panel shows the optimization history of an individual parameter. The horizontal axis represents the number of epochs and is shared across all panels. In the second panel, which corresponds to the critical temperature $T_{\mathrm{c}}$, the exact solution is indicated by a reference line.