Deep Learning and Stochastic Mean-Field Control for a Neural Network Model
22 Pages Posted: 19 Oct 2020
There are 2 versions of this paper
Deep Learning and Stochastic Mean-Field Control for a Neural Network Model
Deep Learning and Stochastic Mean-Field Control for a Neural Network Model
Date Written: August 31, 2020
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.
Keywords: Deep learning, neural network, mean-field, stochastic control
JEL Classification: C61, C63
Suggested Citation: Suggested Citation