Analytically continuing the von Neumann entropy from R\'enyi entropies is a
challenging task in quantum field theory. While the n-th R\'enyi entropy can
be computed using the replica method in the path integral representation of
quantum field theory, the analytic continuation can only be achieved for some
simple systems on a case-by-case basis. In this work, we propose a general
framework to tackle this problem using classical and quantum neural networks
with supervised learning. We begin by studying several examples with known von
Neumann entropy, where the input data is generated by representing TrρAn with a generating function. We adopt KerasTuner to determine the
optimal network architecture and hyperparameters with limited data. In
addition, we frame a similar problem in terms of quantum machine learning
models, where the expressivity of the quantum models for the entanglement
entropy as a partial Fourier series is established. Our proposed methods can
accurately predict the von Neumann and R\'enyi entropies numerically,
highlighting the potential of deep learning techniques for solving problems in
quantum information theory.Comment: 57 pages, 25 figure