A deep learning algorithm for computing mean field control problems via forward-backward score dynamics

TL;DR

This paper introduces a deep learning method to solve mean field control problems involving Fokker-Planck and Hamilton-Jacobi-Bellman equations by using forward-backward score dynamics and neural network approximation.

Contribution

It presents a novel deterministic forward-backward characteristic formulation for mean field control, differing from traditional stochastic approaches, and demonstrates its effectiveness through numerical examples.

Findings

Effective neural network approximation of characteristic lines

Successful application to entropy potential energy control problems

Demonstrated accuracy in linear quadratic regulator and systemic risk scenarios

Abstract

We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the entropy, namely the score function, we first formulate the deterministic forward-backward characteristics for the mean field control system, which is different from the classical forward-backward stochastic differential equations (FBSDEs). We further apply the neural network approximation to fit the proposed deterministic characteristic lines. Numerical examples, including the control problem with entropy potential energy, the linear quadratic regulator, and the systemic risks, demonstrate the effectiveness of the proposed method.