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arXiv:2004.04711 · New Journal of Physics

Phase detection with neural networks: interpreting the black box

Anna Dawid, Patrick Huembeli, Michał Tomza, Maciej Lewenstein, Alexandre Dauphin

Published Oct 23, 2020• 35 citations
arXivHTMLPDF

Abstract

Abstract Neural networks (NNs) usually hinder any insight into the reasoning behind their predictions. We demonstrate how influence functions can unravel the black box of NN when trained to predict the phases of the one-dimensional extended spinless Fermi–Hubbard model at half-filling. Results provide strong evidence that the NN correctly learns an order parameter describing the quantum transition in this model. We demonstrate that influence functions allow to check that the network, trained to recognize known quantum phases, can predict new unknown ones within the data set. Moreover, we show they can guide physicists in understanding patterns responsible for the phase transition. This method requires no a priori knowledge on the order parameter, has no dependence on the NN’s architecture or the underlying physical model, and is therefore applicable to a broad class of physical models or experimental data.

DOI
10.1088/1367-2630/abc463

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Abstract Neural networks (NNs) usually hinder any insight into the reasoning behind their predictions. We demonstrate how influence functions can unravel the black box of NN when trained to predict the phases of the one-dimensional extended spinless Fermi–Hubbard model at half-filling. Results provide strong evidence that the NN correctly learns an order parameter describing the quantum transition in this model. We demonstrate that influence functions allow to check that the network, trained to recognize known quantum phases, can predict new unknown ones within the data set. Moreover, we show they can guide physicists in understanding patterns responsible for the phase transition. This method requires no a priori knowledge on the order parameter, has no dependence on the NN’s architecture or the underlying physical model, and is therefore applicable to a broad class of physical models or experimental data.
Identifying phase transitions and classifying phases of matter is central to understanding the properties and behavior of a broad range of material systems. In recent years, machine-learning (ML) techniques have been successfully applied to perform such tasks in a data-driven manner. However, the success of this approach notwithstanding, we still lack a clear understanding of ML methods for detecting phase transitions, particularly of those that utilize neural networks (NNs). In this work, we derive analytical expressions for the optimal output of three widely used NN-based methods for detecting phase transitions. These optimal predictions correspond to the results obtained in the limit of high model capacity. Therefore, in practice they can, for example, be recovered using sufficiently large, well-trained NNs. The inner workings of the considered methods are revealed through the explicit dependence of the optimal output on the input data. By evaluating the analytical expressions, we can identify phase transitions directly from experimentally accessible data without training NNs, which makes this procedure favorable in terms of computation time. Our theoretical results are supported by extensive numerical simulations covering, e.g., topological, quantum, and many-body localization phase transitions. We expect similar analyses to provide a deeper understanding of other classification tasks in condensed matter physics.
Recently, neural networks (NNs) have become a powerful tool for detecting quantum phases of matter. Unfortunately, NNs are black boxes and only identify phases without elucidating their properties. Novel physics benefits most from insights about phases, traditionally extracted in spin systems using spin correlators. Here, we combine two approaches and design TetrisCNN, a convolutional NN with parallel branches using different kernels that detects the phases of spin systems and expresses their essential descriptors, called order parameters, in a symbolic form based on spin correlators. We demonstrate this on the example of snapshots of the one-dimensional transverse-field Ising model taken in various bases. We show also that TetrisCNN can detect more complex order parameters using the example of two-dimensional Ising gauge theory. This work can lead to the integration of NNs with quantum simulators to study new exotic phases of matter.
Conventionally, the training of a neural network for learning phases of matter uses real physical quantities as the training set. However, it has been demonstrated in several studies that this may not be required. Here we investigate the phase transitions of the two-dimensional (2D) q-state Potts models on the square lattice using a pre-trained neural network (NN). The employed NN was trained previously using two artifically made configurations as the training set. Hence no training is conducted for the present study. Remarkably, the used NN not only calculates the critical points of the considered phase transitions precisely, but also determines the nature of these phase transition definitely without ambiguity. Our results as well as the outcomes found previously suggest that training a NN using real physical quantities as the training set is not required to obtain a working NN. These unconventional studies also imply that if the configuration space of a system can be classified into two categories, then the NN considered here is likely applicable to study the phase transition of that system. Comparison between our approach and other known NN methods for studying the 2D q-state Potts models is briefly discussed as well.
The identification of phases of matter is a challenging task, especially in quantum mechanics, where the complexity of the ground state appears to grow exponentially with the size of the system. Traditionally, physicists have to identify the relevant order parameters for the classification of the different phases. We here follow a radically different approach: we address this problem with a state-of-the-art deep learning technique, adversarial domain adaptation. We derive the phase diagram of the whole parameter space starting from a fixed and known subspace using unsupervised learning. This method has the advantage that the input of the algorithm can be directly the ground state without any ad hoc feature engineering. Furthermore, the dimension of the parameter space is unrestricted. More specifically, the input data set contains both labeled and unlabeled data instances. The first kind is a system that admits an accurate analytical or numerical solution, and one can recover its phase diagram. The second type is the physical system with an unknown phase diagram. Adversarial domain adaptation uses both types of data to create invariant feature extracting layers in a deep learning architecture. Once these layers are trained, we can attach an unsupervised learner to the network to find phase transitions. We show the success of this technique by applying it on several paradigmatic models: the Ising model with different temperatures, the Bose-Hubbard model, and the Su-Schrieffer-Heeger model with disorder. The method finds unknown transitions successfully and predicts transition points in close agreement with standard methods. This study opens the door to the classification of physical systems where the phase boundaries are complex such as the many-body localization problem or the Bose glass phase.
Abstract We discuss an aspect of neural networks for the purpose of phase transition detection. To this end, we first train the neural network by feeding Ising/Potts configurations with labels of temperature so that it can predict the temperature of the input. We do not explicitly supervise whether the configurations are in the ordered/disordered phase. Nevertheless, we can identify the critical temperature from the parameters (weights and biases) of the trained neural network. We attempt to understand how temperature-supervised neural networks capture information on the phase transition by paying attention to what quantities they learn. Our detailed analyses reveal that they learn different physical quantities depending on how well they are trained. The main observation in this study is how the weights in the trained neural network can have information on the phase transition in addition to temperature.
Abstract The phase transition of the two-dimensional U(1) quantum link model on the triangular lattice is investigated by employing a supervised neural network (NN) consisting of only one input layer, one hidden layer of two neurons, and one output layer. No information on the studied model is used when the NN training is conducted. Instead, two artificially made configurations are considered as the training set. Interestingly, the obtained NN not only estimates the critical point accurately but also uncovers the physics correctly. The results presented here imply that a supervised NN, which has a very simple architecture and is trained without any input from the investigated model, can identify the targeted phase structure with high precision.