Connecting GANs, Mean-Field Games, and Optimal Transport

IF 1.9 4区 数学 Q1 MATHEMATICS, APPLIED
Haoyang Cao, Xin Guo, Mathieu Laurière
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引用次数: 0

Abstract

SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1255-1287, August 2024.
Abstract. Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing and have recently attracted growing interest in financial modeling. This paper analyzes GANs from the perspectives of mean-field games (MFGs) and optimal transport: GANs are interpreted as MFGs under the Pareto optimality criterion or mean-field controls; meanwhile, GANs are to minimize the optimal transport cost indexed by the generator from the known latent distribution to the unknown true distribution of data. In particular, we provide a universal approximation result, which shows that there exists an appropriate neural network architecture for GANs training to capture the mean-field solution. The derivation of this universal approximation result leads to an explicit construction of the deep neural network for the transport mapping. The MFGs perspective of GANs leads to a GAN-based computational method (MFGANs) to solve MFGs: one neural network for the backward Hamilton–Jacobi–Bellman equation and one neural network for the forward Fokker–Planck equation, with the two neural networks trained in an adversarial way. Numerical experiments demonstrate superior performance of this proposed algorithm, especially in higher dimensional cases, when compared with existing neural network approaches.
连接 GAN、均场博弈和最优传输
SIAM 应用数学杂志》,第 84 卷第 4 期,第 1255-1287 页,2024 年 8 月。 摘要生成对抗网络(GANs)在图像生成和处理方面取得了巨大成功,最近在金融建模方面也引起了越来越多的兴趣。本文从均值场博弈(MFGs)和最优传输的角度分析了生成式对抗网络:GANs 被解释为帕累托最优准则或均值场控制下的 MFGs;同时,GANs 要最小化由生成器索引的从已知潜在分布到未知真实数据分布的最优传输成本。我们特别提供了一个通用近似结果,表明存在一个合适的神经网络架构用于 GANs 训练,以捕捉均值场解决方案。通过推导这一普遍近似结果,我们可以明确构建用于传输映射的深度神经网络。从 GANs 的 MFGs 视角出发,提出了一种基于 GANs 的计算方法(MFGANs)来求解 MFGs:一个神经网络求解后向汉密尔顿-雅各比-贝尔曼方程,一个神经网络求解前向福克-普朗克方程,两个神经网络以对抗方式进行训练。数值实验证明,与现有的神经网络方法相比,所提出的算法性能优越,尤其是在高维情况下。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
3.60
自引率
0.00%
发文量
79
审稿时长
12 months
期刊介绍: SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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