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Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-01-04 , DOI: 10.1137/22m151861x Jiequn Han 1 , Ruimeng Hu 2 , Jihao Long 3
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-01-04 , DOI: 10.1137/22m151861x Jiequn Han 1 , Ruimeng Hu 2 , Jihao Long 3
Affiliation
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 1-24, February 2024.
Abstract. One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus are inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [J. Han, R. Hu, and J. Long, Stochastic Process. Appl., 164 (2023), pp. 242–287]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker–Smale model, the cost of which depends on the full distribution of the forward process.
中文翻译：
学习具有一般分布依赖性的高维 McKean-Vlasov 前向-后向随机微分方程
SIAM 数值分析杂志，第 62 卷，第 1 期，第 1-24 页，2024 年 2 月。
摘要。平均场控制和平均场博弈的核心问题之一是求解相应的McKean-Vlasov前向-后向随机微分方程（MV-FBSDE）。大多数现有方法都是针对平均场相互作用仅取决于期望或其他矩的特殊情况而设计的，因此不足以解决平均场相互作用具有完全分布依赖性的问题。在本文中，我们提出了一种新颖的深度学习方法，用于计算具有平均场相互作用的一般形式的 MV-FBSDE。具体来说，在虚拟游戏的基础上，我们将问题重新转化为使用显式系数函数重复求解标准 FBSDE。这些系数函数用于近似具有完全分布依赖性的 MV-FBSDE 模型系数，并通过使用从最后迭代的 FBSDE 解决方案模拟的训练数据解决另一个监督学习问题来更新。我们使用深度神经网络来求解标准 BSDE，并使用近似系数函数来求解高维 MV-FBSDE。在对学习函数的正确假设下，我们通过使用先前在[J. Han、R. Hu 和 J. Long，随机过程。应用，164 (2023)，第 242–287 页]。证明的定理显示了该方法在高维情况下的优势。我们展示了高维 MV-FBSDE 问题的数值性能，包括著名的 Cucker-Smale 模型的平均场博弈示例，其成本取决于前向过程的完全分布。
更新日期：2024-01-05
Abstract. One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus are inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [J. Han, R. Hu, and J. Long, Stochastic Process. Appl., 164 (2023), pp. 242–287]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker–Smale model, the cost of which depends on the full distribution of the forward process.
中文翻译：
学习具有一般分布依赖性的高维 McKean-Vlasov 前向-后向随机微分方程
SIAM 数值分析杂志，第 62 卷，第 1 期，第 1-24 页，2024 年 2 月。
摘要。平均场控制和平均场博弈的核心问题之一是求解相应的McKean-Vlasov前向-后向随机微分方程（MV-FBSDE）。大多数现有方法都是针对平均场相互作用仅取决于期望或其他矩的特殊情况而设计的，因此不足以解决平均场相互作用具有完全分布依赖性的问题。在本文中，我们提出了一种新颖的深度学习方法，用于计算具有平均场相互作用的一般形式的 MV-FBSDE。具体来说，在虚拟游戏的基础上，我们将问题重新转化为使用显式系数函数重复求解标准 FBSDE。这些系数函数用于近似具有完全分布依赖性的 MV-FBSDE 模型系数，并通过使用从最后迭代的 FBSDE 解决方案模拟的训练数据解决另一个监督学习问题来更新。我们使用深度神经网络来求解标准 BSDE，并使用近似系数函数来求解高维 MV-FBSDE。在对学习函数的正确假设下，我们通过使用先前在[J. Han、R. Hu 和 J. Long，随机过程。应用，164 (2023)，第 242–287 页]。证明的定理显示了该方法在高维情况下的优势。我们展示了高维 MV-FBSDE 问题的数值性能，包括著名的 Cucker-Smale 模型的平均场博弈示例，其成本取决于前向过程的完全分布。
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