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.
翻译:在本文中,我们展示了一种可扩展的深层次学习方法,以解决意见动态和最佳控制问题,在动态和成本功能中以中值字段术语组合。我们的方法依赖于汉密尔顿-Jacobi-Bellman部分差异方程式解决方案的概率代表。以非线性版本的Feynman-Kac lemma为基础,汉密尔顿-Jacobi-Bellman部分差异方程式的解决方案与前方-后方史实差异方程式的解决方案相联系。这些方程式可以使用一个新颖的深层神经网络以数字方式解决,其结构要适合问题。由此产生的算法是在两极化的共识实验中测试的。大规模(10K)代理器实验证实了我们的算法的可扩展性和可概括性。拟议框架为未来应用极端大规模问题打开了可能性。