Few-shot machine learning in the three-dimensional Ising model

We investigate theoretically the phase transition in a three-dimensional cubic Ising model utilizing state-of-the-art machine learning algorithms. Supervised machine learning models show high accuracies (99%) in phase classification and very small relative errors ($<{10}^{-4}$) of the energies in different spin configurations. Unsupervised machine learning models are introduced to study the spin configuration reconstructions and reductions, and the phases of reconstructed spin configurations can be accurately classified by a linear logistic algorithm. Based on the comparison between various machine learning models, we develop a few-shot strategy to predict phase transitions in larger lattices from a trained sample in smaller lattices. The few-shot machine learning strategy for a three-dimensional (3D) Ising model enables us to study the 3D Ising model efficiently and provides an integrated and highly accurate approach to other spin models.

Figure 1

(Upper panel) Standard Monte Carlo procedure to capture the key physical parameters $T_c$ and $v$; (lower panel) Monte Carlo–based machine learning procedure. The relationship between the Monte Carlo and the machine learning procedures are indicated by the arrows in the middle.

Figure 2

(a) The average outputs of different machine learning models SVM (red dot), NN (green rectangle), and 3D CNN (blue star) on the test set with $L=16$. (b) The average energies predicted by the 3D CNN model. The black solid line and the red dots are obtained from the Monte Carlo simulations and 3D CNN model, respectively.

Figure 3

(a) The output of the first principal component as a function of the magnetization ($M$) in a cubic lattice with $L=16$. The explained variance, i.e., the ratio between the variance of the principal components and the total variance, is shown in the inset of (a). (b) Absolute value of the first PCA output as a function of temperature. (c) Finite-size scaling analysis to determine critical exponent $v$ based on the first principal component and energy; the critical exponent is fitted to be 0.629. (d) Finite-size scaling analysis to determine transition $K_c$.

Figure 4

(a) The average outputs of different RBM combined with a linear logistic classifier and PCA combined with a NN classifier, both with $L=16$. (b) Accuracies as a function of $L$ for different supervised and combined supervised models.

Figure 5

The performance of the model trained by mixing data with different $L$. (a) The average output of the mixed trained model as a function of temperature. (b) The overall accuracies of the mixed trained model tested on different $L$.

Figure 6

The performance of the mixed trained model on the unexplored size $L=28$. (a) The average output of the mixed trained model as a function of temperature. (b) The accuracies of the model at different temperatures $T$.