Self-learning Monte Carlo method with Behler-Parrinello neural networks

We propose a general way to construct an effective Hamiltonian in the self-learning Monte Carlo method (SLMC), which speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. Its applications are, however, limited. This is because it is not obvious to find the explicit form of the effective Hamiltonians. Particularly, it is difficult to make trainable effective Hamiltonians including many-body interactions. In order to overcome this critical difficulty, we introduce the Behler-Parrinello neural networks (BPNNs) as effective Hamiltonian without any prior knowledge, which is used to construct the potential-energy surfaces in interacting many particle systems for molecular dynamics. We combine SLMC with BPNN by focusing on a divisibility of Hamiltonian and propose how to construct the elementwise configurations. We apply it to quantum impurity models. We observed significant improvement of the acceptance ratio from 0.01 (the effective Hamiltonian with the explicit form) to 0.76 (BPNN). This drastic improvement implies that the BPNN effective Hamiltonian includes many-body interaction, which is omitted in the effective Hamiltonian with the explicit forms. The BPNNs make SLMC more promising.

Figure 1

Schematic figure for the Markov chains of the original Monte Carlo and SLMC. The SLMC can bypass links of the computational costly weight estimations. Large blue (small red) circles denote the computational complexity $u_{\mathrm{original}}$ ($u_{\mathrm{SLMC}}$) for the calculating the weight of the original model $w\left(\mathcal{C}\right)$ [$w\left(\mathcal{C}\right)$]. ${\tau }_{\mathrm{original}}$ is the autocorrelation time of the original MC simulation. Since a complexity of a matrix determinant for Fermion systems is quite large, $u_{\mathrm{SLMC}}<u_{\mathrm{original}}$ is satisfied even if the neural networks are adopted. If the average acceptance ratio $\langle A\rangle$ is not too small, the autocorrelation time calculated as the number of the large blue circles is much shorter than that of the original simulation.

Figure 2

Schematic figures of Behler-Parrinello neural networks for the self-learning continuous-time interaction-expansion quantum Monte Carlo method. The configuration with $N$ vertices on the imaginary-time axis $\mathcal{C}=\{{\tau }_1,{\tau }_2,...,{\tau }_N\}$ is mapped to the set of the elementwise configurations ${\mathcal{C}}^{\mathrm{ele}}\left({\tau }_i-{\tau }_j\right)$ around a vertex $j$: $\mathcal{C}\to \{{\mathcal{C}}^{\mathrm{ele}}\left({\tau }_1-{\tau }_j\right),{\mathcal{C}}^{\mathrm{ele}}\left({\tau }_2-{\tau }_j\right),...,{\mathcal{C}}^{\mathrm{ele}}\left({\tau }_N-{\tau }_j\right)\}$. The elementwise configuration is expressed as a distribution function with $\delta$ functions. The total potential $E$ of the original calculation is expressed as $E=logw\left(\mathcal{C}\right)/(-\beta )$. $E$ obtained by the neural networks is expressed as $E=log{w}^{\mathrm{eff}}\left(\mathcal{C}\right)/(-\beta )=H_{\mathrm{eff}}\left(\mathcal{C}\right)$. The partial energy $E_i$ is defined as $E_i=h_{\mathrm{eff}}^{{\alpha }_i}\left({\mathit{c}}^j\right)/(-\beta )$. The color of the circle denotes a kind of a vertex. Same neural networks are used if the kind of a vertex is same.

Figure 3

Schematic figures of input and output data in Behler-Parrinello neural networks. The weight is expressed as a sum of the elementwise effective Hamiltonians. If we neglect hidden layers, the effective Hamiltonian is constructed by linear combinations.

Figure 4

Inverse-temperature dependence of the acceptance ratio in the self-learning continuous-time interaction expansion quantum Monte Carlo simulation in the Anderson impurity model. We consider $U=3D, \delta =0.5, V=1D$, and 500 SLMC steps.

Figure 5

Autocorrelation function of the auxiliary-spin magnetization for a quantum impurity model with $\beta =40, U=3D, \delta =0.5$, and $V=1D$. Unit time is defined in the main text.