Emergent Quantumness in Neural Networks
This work provides a theoretical link between quantum mechanics and neural network dynamics, which could be significant for researchers exploring the fundamental principles underlying machine learning and potentially for theoretical physics.
This paper demonstrates that the Schrödinger equation can be derived from a grand canonical ensemble of neural networks, where the quantum phase is identified with the free energy of hidden variables. This derivation suggests that quantum mechanics offers a correct statistical description of the dynamics of such neural networks at learning equilibrium.
It was recently shown that the Madelung equations, that is, a hydrodynamic form of the Schrödinger equation, can be derived from a canonical ensemble of neural networks where the quantum phase was identified with the free energy of hidden variables. We consider instead a grand canonical ensemble of neural networks, by allowing an exchange of neurons with an auxiliary subsystem, to show that the free energy must also be multivalued. By imposing the multivaluedness condition on the free energy we derive the Schrödinger equation with "Planck's constant" determined by the chemical potential of hidden variables. This shows that quantum mechanics provides a correct statistical description of the dynamics of the grand canonical ensemble of neural networks at the learning equilibrium. We also discuss implications of the results for machine learning, fundamental physics and, in a more speculative way, evolutionary biology.