  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.
