Deep Learning Method-based Numerical Algorithm For Solving High-dimensional PDEs

There are lots of practical problems in this world can be described by partial differential equations(PDEs),such as the phenomenon that the virus scatter among people,the price of financial products and so on.From all of these,we can definitely say that it is important for a number of fields like physics,biology,and finance to solve the PDEs.The fact is that there have been many successful methods researched by some researchers among the world,including the finite element method,the finite difference method etc.However,it has been still a puzzle in academic circle that find an effective method for solving high-dimensional PDEs.And the main reason is that those conventional techniques tend to be suffered with "curse of dimension",generally speaking,the complexity of calculation will appear explosive growth with the increase of dimensions.In this paper,we propose a numerical method called deep C-N method based on deep BSDE method described by E W etc.in 2017.Comparing with deep BSDE method,there are two optimizations in our method,including the accuracy of discrete scheme and the complexity of information in neural network.At the end,we compare the numerical results between these two methods via two different numerical examples-Allen-Cahn equation and Loan price equation.The fact is that deep C-N method not only overcome the "curse of dimensionality",and also get more successful computing results in approaching the real value of equations and reducing the value of loss function.However,comparing with deep BSDE method,deep C-N method shows a poor performance in convergence speed due to this more complex discrete scheme.In addition,this method is extended to solve the reflected PDE,using the penalty method to approximate the reflected PDE to a general format PDE.Similarly,the Allen-Cahn equation with reflection and the stock loan pricing equation with reflection are used to verify the effect of the deep learning method in solving the reflected PDE.The results show that the deep learning method also has a good effect in solving the reflected PDE problem.