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

20 Pages Posted: 23 Jul 2020

See all articles by Nacira Agram

Nacira Agram

Linnaeus University - International Center for Mathematical Modelling in Physics and Cognitive

Azzeddine Bakdi

University of Oslo

Bernt Oksendal

University of Oslo - Department of Mathematics

Multiple version iconThere are 2 versions of this paper

Date Written: June 30, 2020

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

Suggested Citation

Agram, Nacira and Bakdi, Azzeddine and Oksendal, Bernt, Deep Learning and Stochastic Mean-Field Control for a Neural Network Model (June 30, 2020). Available at SSRN: https://ssrn.com/abstract=3639022 or http://dx.doi.org/10.2139/ssrn.3639022

Nacira Agram

Linnaeus University - International Center for Mathematical Modelling in Physics and Cognitive ( email )

Kalmar
Sweden

Azzeddine Bakdi

University of Oslo ( email )

Oslo
Norway

Bernt Oksendal (Contact Author)

University of Oslo - Department of Mathematics ( email )

P.O. Box 1053
Blindern, N-0162, Os
Norway
+47-2285 5913 (Phone)
+47-2285 4349 (Fax)

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