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 mean-field 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 mean-field 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 z^2 and the integral of u^2(t). This problem is too complicated to handle by mathematical methods alone. In the last section, we solve it using deep learning techniques.