Training biases in machine learning for the analytic continuation of quantum many-body Green's functions

TL;DR

This paper develops a machine learning approach using residual neural networks and uncertainty estimation to improve the analytic continuation of Green's functions in quantum many-body physics, addressing biases and noise effects.

Contribution

It introduces a neural network method with uncertainty estimation for analytic continuation, highlighting bias issues and benchmarking against traditional methods.

Findings

Achieves high-quality spectral predictions comparable to maximum entropy methods.

Identifies regions of uncertainty in spectral predictions using Monte Carlo dropout.

Demonstrates effectiveness on real material data, SrVO3.

Abstract

We address the problem of analytic continuation of imaginary-frequency Green's functions, which is crucial in many-body physics, using machine learning based on a multi-level residual neural network. We specifically address potential biases that can be introduced due to the use of artificially created spectral functions that are employed to train the neural network. We also implement an uncertainty estimation of the predicted spectral function, based on Monte Carlo dropout, which allows to identify frequency regions where the prediction might not be accurate, and we study the effect of noise, in particular also for situations where the noise level during training is different from that in the actual data. Our analysis demonstrates that this method can indeed achieve a high quality of prediction, comparable or better than the widely used maximum entropy method, but that further improvement is currently limited by the lack of true data that can be used for training. We also benchmark our approach by applying it to the case of SrVO$_3$, where an accurate spectral function has been obtained from dynamical mean-field theory using a solver that works directly on the real frequency axis.