Deep Learning Methods for Mean Field Control Problems with Delay

TL;DR

This paper introduces deep learning algorithms for solving complex mean field control problems with delay, including neural network parameterization of controls and solving McKean-Vlasov equations, supported by theoretical principles.

Contribution

It develops two novel deep learning algorithms for delayed mean field control problems and establishes a stochastic maximum principle with existence and uniqueness results.

Findings

Neural network-based control parameterization effectively solves delayed mean field problems.

The MV-FABSDE system can be numerically solved using the proposed algorithms.

Theoretical results provide a foundation for existence and uniqueness of solutions.

Abstract

We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly parameterize the optimal control using neural networks, the other is based on numerically solving the McKean-Vlasov forward anticipated backward stochastic differential equation (MV-FABSDE) system. In addition, we establish a necessary and sufficient stochastic maximum principle for this class of mean field control problems with delay based on the differential calculus on function of measures, as well as existence and uniqueness results for the associated MV-FABSDE system.