Neural network-based approach to analytic continuation

While most quantum Monte Carlo methods provide Green's function of imaginary time, the dynamical quantities relevant to experiments need to be expressed in real frequency. The standard approach that allows one to obtain these functions is based on the MaxEnt method. Unfortunately, this is an ill-conditioned problem where small errors in the imaginary time Green's functions lead to large errors in the real frequency quantities. And since the Monte Carlo results always have some statistical errors, in order to obtain reliable results this procedure must be iteratively repeated multiple times and then averaged.

It has already been proposed to replace this procedure by using an artificial neural network to perform the analytic continuation [1-3]. Such a possibility follows from the universal approximation theorem, which states that any piecewise continuous function can be approximated to any degree of accuracy by a sufficiently large neural network. In these approaches, at the first stage a large number of spectral functions A(ω) are modeled in a physically meaningful way, usually as sums of random Gaussian or Lorentzian peaks. Then, corresponding imaginary time Green's functions G(τ) are calculated. Pairs [G(τ),A(ω)] are then used to train the network to be able to reconstruct A(ω) from a given G(τ). Here, we propose an alternative approach, where we train the network with the help of the actual Green's function generated in QMC, not the ones calculated from postulated A(ω). Then, it is more general in that there is no need to postulate the functional form of A(ω). An additional advantage is that this method can be easily applied also to other inverse problems, like the deconvolution of ARPES data.

1. Louis-François Arsenault et al., Inverse Problems 33, 115007 (2017)

2. R. Fournier, et al., Phys. Rev. Lett. 124, 056401 (2020)

3. Lukas Kades, et al., Phys. Rev. D 102, 096001 (2020)