Reconstructing spectral functions via automatic differentiation

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 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 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.