Neural Network Solver for Small Quantum Clusters

Round 1

Reviewer 1 Report

This is an interesting paper suggesting application of machine learning to solve the single impurity Anderson model. This introduces the possibility of future use of the neural network approach as an impurity solver for many-body problems. The overall approach and the conclusions are interesting. 

Author Response

We would like to thank the referee for the positive review of our manuscript. We agree that this is an interesting work for using neural networks as a solver for many body problems.

Reviewer 2 Report

In this manuscript, the authors present a neural network solver for single impurity Anderson model. In order to convert the many-body problem to a problem can be solved by supervised learning method, they map the continuous Green function into a finite cluster using energies and hoppings as feature vectors, and represent spectral function in terms of Chebyshev polynomials with a damping kernel using coefficients of the Chebyshev polynomials expansion as output labels. Their neural network solver achieved good agreement with direct numerical calculation in terms of accuracy. It's a very interesting idea to use machine learning to solve the many-body problem. I am favorable to the publication of this manuscript, and I have several questions/comments:

1. I understand the computational complexity grows exponentially with the number of sites Nb, so the smaller Nb is favorable. I hope the authors can provide more details about the choice of Nb=6, like how to reach this decision. I guess it's a result of balancing the accuracy and computational complexity.

2. There is some ambiguity about the size of train-test set. At line 208, the authors "generated 5,000 samples"; at line 247, they use 1,000 samples to compare neural network solver with direct numerical calculation. Is the test set (1,000 samples) in those 5,000 samples, i.e. the size of train set is 4000 samples? Or test set (1,000 samples) is not in those 5,000 samples, i.e. the size of train set is 5000 samples.

3. The range of input parameters is provided in Eq. (13) without much explanation. I hope the authors can provide more details about the choice of these ranges. Also, if there is a particular problem with input parameters out of these ranges (the Green function can't be mapped into 6 sites accurately with parameters in range), will the neural network solver still give accurate results? Or have to train the solver again with range including the new parameters?

Author Response

We would like to thank the referee for the positive review of our manuscript. We appreciate the questions and comments posted by the referee. We address them in the following.

1. We thank the referee for the comment. The Nb=6 is a somewhat arbitrary choice here. We could have performed the calculations for a larger number of bath sites. It would require more computational time to generate the database for training the neural networks. The time required scales as 4^Nb if no symmetry is imposed in the calculation. However, the computational time required for training the neural networks would change relatively modestly as it scales as Nb^3. 

The main purpose of the present manuscript is to test the idea of using neural networks to approximately map the impurity problem to a representation suitable for neural networks. We choose Nb=6, so that we do not need to spend too much computational time to build the database. 

We added a paragraph in the revised manuscript to discuss the choice of Nb. 

2. We thank the referee for pointing out this confusing point. 

We used 5000 samples for training the neural networks. 1000 samples were then generated by the trained neural network. 

We added a sentence in the revised manuscript to emphasize this point.

3.We thank the referee for this question. The range of the parameters are chosen as a realistic range of parameters used in most studies for the dynamical mean field theory. In particular, it covers the value of U which corresponds to the phase transition, at around U approximately equal to 2.5 times of the bandwidth, in the dynamical mean field solution of the Hubbard model .

We do not expect the neural networks to provide accurate results if the parameters are far away from this range. It is possible that additional training data are required. 

We added a discussion of this problem in the revised manuscript to talk about the valid range of parameters.