Deep Learning and Stochastic Mean-Field Control for a Neural Network Model

Abstract

We study a membrane voltage potential model by means of stochastic control of meanfield stochastic differential equations (SDEs) and by deep learning techniques. The mean-field stochastic control problem is a new type, involving the expected value of a combination of the state X(t) and the running control u(t) at time t. Moreover, the control is two-dimensional, involving both the initial value z of the state and the running control u(t). We prove a necessary condition for optimality of a control (u; z) for such a general stochastic meanfield control problem, and we also prove a verification theorem for such problems. The results are then applied to study a particular case of a neural network problem, where the system has a drift given by E[X(t)u(t)] and the problem is to arrive at a terminal state value X(T) which is close in terms of variance to a given terminal value F under minimal costs, measured by z2 and the integral of u2(t). This problem is too complicated to handle by mathematical methods alone. In the last section, we solve it using deep learning techniques.

Keywords: Deep learning, neural network, mean-field control

JEL Classification: C45, C61