A machine learning framework for solving high-dimensional mean field game and mean field control problems
A machine learning framework for solving high-dimensional mean field game and mean field control problems
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DOI：
10.1073/pnas.1922204117
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发表时间：
2020-04-28
影响因子：
11.1
通讯作者：
Fung, Samy Wu
中科院分区：
文献类型：
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作者：
Ruthotto, Lars;Osher, Stanley J.;Fung, Samy Wu
平均场博弈（MFG）和平均场控制（MFC）是多智能体模型的关键类别，用于有效分析大量相互作用的智能体。它们的应用领域涵盖经济学、金融学、博弈论、工业工程、人群运动等主题。在本文中，我们提供了一个灵活的机器学习框架，用于潜在的MFG和MFC模型的数值求解。用于解决此类问题的最先进的数值方法利用导致维数灾难的空间离散化。我们通过结合拉格朗日和欧拉观点并利用机器学习的最新进展来近似解决高维问题。更确切地说，我们的工作与拉格朗日制定的问题，并执行基本的汉密尔顿-雅可比-贝尔曼（HJB）方程，是从欧拉制定。最后，MFG/MFC解决方案的定制神经网络参数化有助于我们避免任何空间离散化。我们的数值结果包括近似的解决方案的100维的最佳运输和人群运动的问题，在一个标准的工作站和验证使用欧拉求解器在两个维度上。这些结果打开了大门，备受期待的应用程序的MFG和MFC模型是无法达到与现有的数值方法。
Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.