Deep Backward and Galerkin Methods for the Finite State Master Equation
arxiv(2024)
摘要
This paper proposes and analyzes two neural network methods to solve the
master equation for finite-state mean field games (MFGs). Solving MFGs provides
approximate Nash equilibria for stochastic, differential games with finite but
large populations of agents. The master equation is a partial differential
equation (PDE) whose solution characterizes MFG equilibria for any possible
initial distribution. The first method we propose relies on backward induction
in a time component while the second method directly tackles the PDE without
discretizing time. For both approaches, we prove two types of results: there
exist neural networks that make the algorithms' loss functions arbitrarily
small, and conversely, if the losses are small, then the neural networks are
good approximations of the master equation's solution. We conclude the paper
with numerical experiments on benchmark problems from the literature up to
dimension 15, and a comparison with solutions computed by a classical method
for fixed initial distributions.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要