Autoencoder-based analytic continuation method for strongly correlated quantum systems

Solving ill-posed problems is central to a variety of scientific investigations. We focus here on the analytic continuation of imaginary-time data obtained from quantum Monte Carlo (QMC) simulations to the real frequency axis, which involves the numerical inversion of a Laplace transform, a well-known ill-posed problem. We propose an unsupervised autoencoder-type neural network to address this problem, and we show that our encoder-decoder approach can extract high-quality real frequency spectral functions from imaginary-time Green's functions G(tau). With a deeply tunable architecture we demonstrate, for artificial test data with noise added, that the autoencoder neural network can locate sharp features of spectral functions, which may be lost using maximum entropy (MaxEnt) methods currently in use. We demonstrate the strength of the autoencoder approach by applying it to QMC results for a single-band Hubbard model as a function of density, and we show that it is more robust against noise in the input G(tau) compared to MaxEnt. The proposed method is general and can also be applied to other ill-posed inverse problems.