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In this paper, we present a scalable deep learning approach to solve opiniondynamics stochastic optimal control problems with mean field term coupling inthe dynamics and cost function. Our approach relies on the probabilisticrepresentation of the solution of the Hamilton-Jacobi-Bellman partialdifferential equation. Grounded on the nonlinear version of the Feynman-Kaclemma, the solutions of the Hamilton-Jacobi-Bellman partial differentialequation are linked to the solution of Forward-Backward Stochastic DifferentialEquations. These equations can be solved numerically using a novel deep neuralnetwork with architecture tailored to the problem in consideration. Theresulting algorithm is tested on a polarized opinion consensus experiment. Thelarge-scale (10K) agents experiment validates the scalability andgeneralizability of our algorithm. The proposed framework opens up thepossibility for future applications on extremely large-scale problems.

Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control