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Deep learning method for solving mean field game theory problems

2022-06-27 00:01:00 【Shengsi mindspire】

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Mean field games (MFG, Mean field game theory ) and Mean field control (MFC, Mean field cybernetics ） It can simulate the game between a large number of objects , Explore in a competitive environment , How the object chooses the optimal decision . For example, in the stock market, there are a large number of investors who trade stocks according to the behavior of other users , A school of fish swimming in the sea , Spectators watching the football match at the World Cup . They are in Physics 、 Economics, data science and other disciplines play a central role . although MFG Our mathematical theory is quite mature , However, the development of numerical methods has not kept up with the growth of problem scale and massive data sets . because MFG There is usually no explicit solution , An effective numerical algorithm is very important . Most existing numerical methods use grids , Therefore, it is easily limited by the disaster of dimensionality .

In recent years , Combined with machine learning method MFG and MFC Problem solving has received a lot of attention from the academic community . Especially for solving problems with complex structures , High dimensional problems . This article will introduce the solution based on deep learning MFG and MFC Three kinds of methods .

Problem definition

MFG and MFC The model includes the following parameters ：

Here, the optimal control and state probability distribution estimated by the neural network are consistent with the theoretical values ：

1. Optimal control is a linear function of the state ;

2. The probability distribution of the state will go to 0 Gradually move .

Summary and prospect

This paper mainly introduces three methods of using deep learning to solve MFG and MFC Methods for related issues . The first method uses neural network to directly fit the control function , The second way is through Deep BSDE solve FBSDE, The last method uses neural network to solve partial differential equations , And then solve HJB Equation and FKP A system of equations consisting of equations . Based on the above method , The academic community has made various attempts on many high-dimensional and complex problems , Some preliminary results have been achieved . But because neural networks are usually composed of multiple items , So it is a highly nonconvex optimization problem . In the process of training , The items in the loss function may compete with each other , The training process may not be robust and stable enough , Convergence to the global minimum cannot be guaranteed . To solve this problem , It is necessary to develop more robust network structure and training algorithm .

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reference ：

[1] en.wikipedia.org/wiki/Fokker%25E2%2580%2593Planck_equation

[2] en.wikipedia.org/wiki/Hamilton%25E2%2580%2593Jacobi%25E2%2580%2593Bellman_equation

[3] papers.ssrn.com/sol3/papers.cfm%3Fabstract_id%3D2557457

[4] arxiv.org/abs/1811.08782

[5] DeepBSDE pnas.org/content/115/34/8505

[6] Optimal transport and crowd motion pnas.org/content/117/17/9183