Self-learning Monte Carlo with deep neural networks

The self-learning Monte Carlo (SLMC) method is a general algorithm to speedup MC simulations. Its efficiency has been demonstrated in various systems by introducing an effective model to propose global moves in the configuration space. In this paper, we show that deep neural networks can be naturally incorporated into SLMC, and without any prior knowledge can learn the original model accurately and efficiently. Demonstrated in quantum impurity models, we reduce the complexity for a local update from $O\left({\beta }^2\right)$ in Hirsch-Fye algorithm to $O(\beta ln\beta )$, which is a significant speedup especially for systems at low temperatures.

Figure 1

The performance of the effective model at different chemical potentials. The dashed blue line is the estimation of acceptance rates through MSE computed in a test data set [38]: $e^{-\sqrt{\mathrm{MSE}}}$. The discrepancy between the solid and the dashed line is due to the fact that in general $\langle p\rangle \ne e^{-\sqrt{\langle {(lnp)}^2\rangle }}=e^{-\sqrt{\mathrm{MSE}}}$ and the effective model is biased. Here $\beta =20,U=3.0,V=1.0$, and $L=120$. The number of neurons in the first and second hidden layers are set as $N_1=100$ and $N_2=50$. The activation function is the rectified linear unit $f\left(x\right)=max\{x,0\}$. Inset: A schematic show of the fully connected neural network. The red circles represent neurons in the input layer with $L=4$, and the blue circles represent neurons in two hidden layers with $N_1=6$ and $N_2=3$. The last layer is a linear output layer.

Figure 2

Top: Weight matrix $W_1$ taken from the fully connected neural network of $\mu =0$ in Fig. 1. Bottom: Average magnitude of nonzero matrix elements in $W_1$ of fully connected neural networks in Fig. 1. The nonzero matrix element is defined as the element that is greater than 10% of the maximum element in all seven weight matrices $W_1$ from networks trained at seven different chemical potentials. Only around 3% elements are nonzero in these weight matrices.

Figure 3

The structure of the convolutional neural network. The first layer is a convolutional layer with two kernels (dark and light blue) of size 3. The stride of the sliding inner product is 2. The second layer is a fully connected layer and the last layer is a linear layer.

Figure 4

(a) The performance of the convolutional network compared with the fully connected network in Fig. 1. The number of trainable parameters is 211 (two kernels in the first convolutional layer) or 291 (six kernels in the first convolutional layer). Here $\beta =20,U=3.0,V=1.0$, and $L=120$. (b) The performance of the convolutional network at different temperatures. Here $U=3.0,\mu =-1.0,V=1.0$, and $L=2\beta U$. The conventional neural network details are described in Supplemental Material [38].