Neural network representation of quantum systems

Needless to be exemplified by Boltzmann machine, Amari-Hopfield model and diffusion models, fundamental physics has provided a great influence on machine learning. Then a natural question arises -- to what extent do the fundamental physics and machine learning overlap with each other? For example, the notion of quantum is the central concept in microscopic physics. To what extent can quantum mechanics be formulated in terms of neural networks? Partial answers to this interesting question come from two developments at the intersection of machine learning and physics: (1) Gaussian processes and (2) stochastic neurodynamics, which we shall describe in order. Both of these have their roots in the research of random neural networks initiated by Amari [1, 3] and Rozonoer [2]. The random neural network is a fundamental tool to reveal the macroscopic properties of typical neural networks, as well as a key to control complicated learning dynamics of neural networks.