Partial Differential Equation Solving Based On Deep Neural Network

Partial differential equations(PDEs)can describe many practical problems,such as the spread of viruses in the population,the prices of financial derivatives,etc.Therefore,solving PDEs is of great significance to many fields,including physics,biology,and finance.High-dimensional PDEs are usually difficult to solve,and traditional algorithms will encounter the phenomenon of "curse of dimensionality" when dealing with high-dimensional PDEs,that is,the amount of computation presents an exponential increase in dimension d.In this paper we are introduced the method and related theory of deep neural network to solve PDE,including the comparison of three commonly used important methods:physical information neural network method(PINN),deep neural network approximation of linear PDE solution combined with Feynman-Kac formula and application of Deep Backward Stochastic Differential Equation(BSDE)solver that approximates semilinear PDE solutions.Among them,the PINN method is relatively simple,requires little training data,and has a good effect on physical problems.The other two methods can be used for a class of high-dimensional PDEs derived from issues such as financial portfolio investment and extreme climate probability.In Chapter 4,we introduce the theory of deep neural network overcoming the curse of dimensionality,and point out that the neural network combined with multi-stage Picard iteration can effectively overcome the curse of dimensionality of a class of high-dimensional semilinear thermal partial differential equations.At this time,the number of parameters of the neural network is at most polynomially increased with respect to the dimension of the PDE and the reciprocal of the required precision.