SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1455-1485, January 2021.
We propose two algorithms for the solution of the optimal control of ergodic McKean--Vlasov dynamics. Both algorithms are based on approximations of the theoretical solutions by neural networks, the latter being characterized by their architecture and a set of parameters. This allows the use of modern machine learning tools, and efficient implementations of stochastic gradient descent. The first algorithm is based on the idiosyncrasies of the ergodic optimal control problem. We provide a mathematical proof of the convergence of the approximation scheme, and we analyze rigorously the approximation by controlling the different sources of error. The second method is an adaptation of the deep Galerkin method to the system of partial differential equations issued from the optimality condition. We demonstrate the efficiency of these algorithms on several numerical examples, some of them being chosen to show that our algorithms succeed where existing ones failed. We also argue that both methods can easily be applied to problems in dimensions larger than what can be found in the existing literature. Finally, we illustrate the fact that, although the first algorithm is specifically designed for mean field control problems, the second one is more general and can also be applied to the partial differential equation systems arising in the theory of mean field games.

中文翻译：

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平均场控制和博弈数值解的机器学习算法的收敛性分析 I：遍历案例

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1455-1485 页，2021 年 1 月。
我们提出了两种算法来求解遍历 McKean--Vlasov 动力学的最优控制。这两种算法都基于神经网络对理论解的近似，后者的特点是它们的架构和一组参数。这允许使用现代机器学习工具，以及随机梯度下降的有效实现。第一种算法基于遍历最优控制问题的特性。我们提供了近似方案收敛的数学证明，并通过控制不同的误差源来严格分析近似。第二种方法是将深度伽辽金方法应用于从最优性条件发出的偏微分方程组。我们在几个数值例子上证明了这些算法的效率，其中一些被选择来表明我们的算法在现有算法失败的情况下成功。我们还认为，这两种方法都可以很容易地应用于比现有文献中发现的维度更大的问题。最后，我们说明了一个事实，虽然第一个算法是专门为平均场控制问题设计的，但第二个算法更通用，也可以应用于平均场博弈理论中出现的偏微分方程系统。