Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control

TL;DR

This paper introduces a scalable deep learning method for solving opinion dynamics stochastic control problems involving mean field interactions, using a neural network approach based on FBSDEs and PDE representations.

Contribution

The paper develops a novel deep neural network framework for efficiently solving large-scale opinion dynamics stochastic control problems with mean field coupling.

Findings

Successfully applied to a 10,000-agent opinion consensus experiment

Demonstrated scalability and generalizability of the approach

Enabled future applications to extremely large-scale problems

Abstract

In this paper, we present a scalable deep learning approach to solve opinion dynamics stochastic optimal control problems with mean field term coupling in the dynamics and cost function. Our approach relies on the probabilistic representation of the solution of the Hamilton-Jacobi-Bellman partial differential equation. Grounded on the nonlinear version of the Feynman-Kac lemma, the solutions of the Hamilton-Jacobi-Bellman partial differential equation are linked to the solution of Forward-Backward Stochastic Differential Equations. These equations can be solved numerically using a novel deep neural network with architecture tailored to the problem in consideration. The resulting algorithm is tested on a polarized opinion consensus experiment. The large-scale (10K) agents experiment validates the scalability and generalizability of our algorithm. The proposed framework opens up the possibility for future applications on extremely large-scale problems.