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arXiv:1801.01127 · Physical review. B./Physical review. B

Self-learning Monte Carlo with deep neural networks

Huitao Shen, Junwei Liu, Liang Fu

Published May 29, 2018• 78 citations
arXivHTMLPDF

Abstract

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 $O({\ensuremath{\beta}}^{2})$ in Hirsch-Fye algorithm to $O(\ensuremath{\beta}ln\ensuremath{\beta})$, which is a significant speedup especially for systems at low temperatures.

DOI
10.1103/physrevb.97.205140

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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(\ensuremathβ2)O({\ensuremath{\beta}}^{2})O(\ensuremathβ2) in Hirsch-Fye algorithm to O(\ensuremathβln\ensuremathβ)O(\ensuremath{\beta}ln\ensuremath{\beta})O(\ensuremathβln\ensuremathβ), which is a significant speedup especially for systems at low temperatures.
The Self-Learning Monte Carlo (SLMC) method is a Monte Carlo approach that has emerged in recent years by integrating concepts from machine learning with conventional Monte Carlo techniques. Designed to accelerate the numerical study of interacting many-body systems, SLMC significantly improves sampling efficiency by constructing an effective model -- via machine learning methods -- based on configurations generated by conventional Monte Carlo methods and then proposes global updates based on the effective model. This enhancement leads to a substantial reduction in autocorrelation time, especially near the critical region, where traditional methods typically suffer from critical slowing down and increased computation complexity. Moreover, SLMC maintains statistical accuracy by implementing a cumulative update scheme that rigorously satisfies the detailed balance condition. And more recent applications have extended the SLMC to convolutional neural networks with applications not only in condensed matter physics but also high-energy physics, quantum chemistry, and quantum simulations. The generic applicability and high computational efficiency make SLMC a powerful and scalable framework for quantum Monte Carlo simulations of strongly correlated electron systems, extending the reach of numerical investigations beyond the limitations of conventional techniques.
As a new general-purpose Monte Carlo method, self-learning Monte Carlo (SLMC) has been numerically demonstrated in boson systems. In this work, we propose a new type of update algorithm, dubbed cumulative update, which could be naturally integrated into SLMC. Based on many local updates in the self-learned effective model, cumulative update efficiently proposes a global move with a high acceptance probability in the original model. Cumulative update can reduce the conventional computational cost of a full sweep O(N4) to O(N3), and also effectively reduce the auto-correlation time to be 1. We numerically show its efficiency through the well-known double-exchange model. With cumulative update, the SLMC could be several hundreds of times faster than the conventional local update method. By employing the cumulative update, SLMC can be generally used in any sign-problem-free Fermion systems and maximize the performance of Monte Carlo simulation.
The self-learning Monte Carlo (SLMC) method speeds up the Monte Carlo simulation by designing and training an effective model to propose efficient global updates. We implement the continuous-time quantum Monte Carlo algorithm for quantum impurity models in the framework of SLMC. We introduce and train a diagram generating function (DGF) to model the probability distribution of field configurations in the continuous imaginary time at all orders of diagrammatic expansion. By using the DGF to propose global updates, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.
Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10--20 times speedup.
Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10–20 times speedup.
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.