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arXiv:2011.10719 · Physical review. B./Physical review. B

Predicting impurity spectral functions using machine learning

Erica J. Sturm, Matthew R. Carbone, Deyu Lu, Andreas Weichselbaum, Robert Konik

Published Jun 14, 2021• 21 citations
arXivHTMLPDF

Abstract

The Anderson Impurity Model (AIM) is a canonical model of quantum many-body physics. Here we investigate whether machine learning models, both neural networks (NN) and kernel ridge regression (KRR), can accurately predict the AIM spectral function in all of its regimes, from empty orbital, to mixed valence, to Kondo. To tackle this question, we construct two large spectral databases containing approximately 410 000 and 600 000 spectral functions of the single-channel impurity problem. We show that the NN models can accurately predict the AIM spectral function in all of its regimes, with pointwise mean absolute errors down to 0.003 in normalized units. We find that the trained NN models outperform models based on KRR and enjoy a speedup on the order of ${10}^{5}$ over traditional AIM solvers. The required size of the training set of our model can be significantly reduced using farthest point sampling in the AIM parameter space, which is important for generalizing our method to more complicated multichannel impurity problems of relevance to predicting the properties of real materials.

DOI
10.1103/physrevb.103.245118

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The Anderson Impurity Model (AIM) is a canonical model of quantum many-body physics. Here we investigate whether machine learning models, both neural networks (NN) and kernel ridge regression (KRR), can accurately predict the AIM spectral function in all of its regimes, from empty orbital, to mixed valence, to Kondo. To tackle this question, we construct two large spectral databases containing approximately 410 000 and 600 000 spectral functions of the single-channel impurity problem. We show that the NN models can accurately predict the AIM spectral function in all of its regimes, with pointwise mean absolute errors down to 0.003 in normalized units. We find that the trained NN models outperform models based on KRR and enjoy a speedup on the order of 105{10}^{5}105 over traditional AIM solvers. The required size of the training set of our model can be significantly reduced using farthest point sampling in the AIM parameter space, which is important for generalizing our method to more complicated multichannel impurity problems of relevance to predicting the properties of real materials.
Abstract Using numerical renormalization group calculation, we construct a dataset with 100 K samples, and train six different neural networks for the prediction of spectral functions from density of states (DOS) of the host material. We find that a combination of gated recurrent unit (GRU) network and bidirectional GRU (BiGRU) performances the best among all the six neural networks. The mean absolute error of the GRU + BiGRU network can reach 0.052 and 0.043 when this network is evaluated on the original dataset and two other independent datasets. The average time of spectral function predictions from machine learning is on the scale of 10 −5 –10 −6 that of traditional impurity solvers for Anderson impurity model. This investigation pave the way for the application of recurrent neural network and convolutional neural network in the prediction of spectral functions from DOSs in machine learning solvers of magnetic impurity problems.
Machine learning approaches have recently been applied to the study of various problems in physics. Most of these studies are focused on interpreting the data generated by conventional numerical methods or the data on an existing experimental database. An interesting question is whether it is possible to use a machine learning approach, in particular a neural network, for solving the many-body problem. In this paper, we present a neural network solver for the single impurity Anderson model, the paradigm of an interacting quantum problem in small clusters. We demonstrate that the neural-network-based solver provides quantitative accurate results for the spectral function as compared to the exact diagonalization method. This opens the possibility of utilizing the neural network approach as an impurity solver for other many-body numerical approaches, such as the dynamical mean field theory.
By using the numerical renormalization group (NRG) method, we construct a large dataset with about one million spectral functions of the Anderson quantum impurity model. The dataset contains the density of states (DOS) of the host material, the strength of Coulomb interaction between on-site electrons (U), and the hybridization between the host material and the impurity site (Γ). The continued DOS and spectral functions are stored with Chebyshev coefficients and wavelet functions, respectively. From this dataset, we build seven different machine learning networks to predict the spectral function from the input data, DOS, U, and Γ. Three different evaluation indexes, mean absolute error (MAE), relative error (RE) and root mean square error (RMSE), are used to analyze the prediction abilities of different network models. Detailed analysis shows that, for the two kinds of widely used recurrent neural networks (RNNs), gate recurrent unit (GRU) has better performance than the long short term memory (LSTM) network. A combination of bidirectional GRU (BiGRU) and GRU has the best performance among GRU, BiGRU, LSTM, and BiLSTM. The MAE peak of BiGRU+GRU reaches 0.00037. We have also tested a one-dimensional convolutional neural network (1DCNN) with 20 hidden layers and a residual neural network (ResNet), we find that the 1DCNN has almost the same performance of the BiGRU+GRU network for the original dataset, while the robustness testing seems to be a little weak than BiGRU+GRU when we test all these models on two other independent datasets. The ResNet has the worst performance among all the seven network models. The datasets presented in this paper, including the large data set of the spectral function of Anderson quantum impurity model, are openly available at https://doi.org/10.57760/sciencedb.j00113.00192 .
Understanding charge motion in a background of interacting quantum spins is a fundamental problem in quantum many-body physics. The most extensively studied model for this problem is the so-called $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{'}\ensuremath{'}}\text{\ensuremath{-}}J$ model, where the determination of the parameter ${t}^{\ensuremath{'}}$ in the context of cuprate superconductors is challenging. Here we present a theoretical study of the spectral functions of a mobile hole in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{'}\ensuremath{'}}\text{\ensuremath{-}}J$ model using two machine-learning techniques: KKK-nearest neighbor regression (KNN) and a feed-forward neural network (FFNN). We employ the self-consistent Born approximation to generate a dataset of about $1.3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ spectral functions. We show that, for the forward problem, both methods allow for the accurate and efficient prediction of spectral functions, allowing, e.g., rapid searches through parameter space. Furthermore, we find that for the inverse problem (inferring Hamiltonian parameters from spectra), the FFNN can, but the KNN cannot, accurately predict the model parameters using merely the density of states. Our results suggest that it may be possible to use deep-learning methods to predict materials parameters from experimentally measured spectral functions.
We present a machine learning (ML) framework for predicting Green’s functions of molecular systems, from which photoemission spectra and quasiparticle energies at quantum many-body level can be obtained. Kernel ridge regression is adopted to predict self-energy matrix elements on compact imaginary frequency grids from static and dynamical mean-field electronic features, which gives direct access to real-frequency many-body Green’s functions through analytic continuation and Dyson’s equation. Feature and self-energy matrices are represented in a symmetry-adapted intrinsic atomic orbital plus projected atomic orbital basis to enforce rotational invariance. We demonstrate good transferability and high data efficiency of the proposed ML method across molecular sizes and chemical species by showing accurate predictions of density of states (DOS) and quasiparticle energies at the level of many-body perturbation theory ( GW ) or full configuration interaction. For the ML model trained on 48 out of 1995 molecules randomly sampled from the QM7 and QM9 data sets, we report the mean absolute errors of ML-predicted highest occupied and lowest unoccupied molecular orbital energies to be 0.13 and 0.10 eV, respectively, compared to GW @PBE0. We further showcase the capability of this method by applying the same ML model to predict DOS for significantly larger organic molecules with up to 44 heavy atoms.
We employ variational autoencoders to extract physical insight from a dataset of one-particle Anderson impurity model spectral functions. Autoencoders are trained to find a low-dimensional, latent space representation that faithfully characterizes each element of the training set, as measured by a reconstruction error. Variational autoencoders, a probabilistic generalization of standard autoencoders, further condition the learned latent space to promote highly interpretable features. In our study, we find that the learned latent space components strongly correlate with well known, but nontrivial, parameters that characterize emergent behaviors in the Anderson impurity model. In particular, one latent space component correlates with particle-hole asymmetry, while another is in near one-to-one correspondence with the Kondo temperature, a dynamically generated low-energy scale in the impurity model. With symbolic regression, we model this component as a function of bare physical input parameters and rediscover the non-perturbative formula for the Kondo temperature. The machine learning pipeline we develop opens opportunities to discover new domain knowledge in other physical systems.