Reconstructing spectral functions via automatic differentiation

TL;DR

This paper introduces an automatic differentiation framework using neural networks to reconstruct spectral functions from noisy Green's functions, improving accuracy over traditional methods especially in high-noise scenarios.

Contribution

The novel approach employs neural networks as non-local regularizers within an AD framework for spectral reconstruction, avoiding explicit physical priors and enhancing performance.

Findings

Outperforms maximum entropy method in high-noise conditions

Uses neural networks for non-local regularization of spectral functions

Achieves better reconstruction accuracy measured by entropy and MSE

Abstract

Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in many-body physics. However, the inversion is proved to be ill-posed in the realistic systems with noisy Green's functions. In this Letter, we propose an automatic differentiation(AD) framework as a generic tool for the spectral reconstruction from propagator observable. Exploiting the neural networks' regularization as a non-local smoothness regulator of the spectral function, we represent spectral functions by neural networks and use the propagator's reconstruction error to optimize the network parameters unsupervisedly. In the training process, except for the positive-definite form for the spectral function, there are no other explicit physical priors embedded into the neural networks. The reconstruction performance is assessed through relative entropy and mean square error for two different network representations. Compared to the maximum entropy method, the AD framework achieves better performance in the large-noise situation. It is noted that the freedom of introducing non-local regularization is an inherent advantage of the present framework and may lead to substantial improvements in solving inverse problems.