Deep Learning Numerical Methods For High-dimention Nonlinear Stochastic Partial Differential Equations

In recent years,we encounter the difficult problem known as the ”curse of dimensionality”,when solving high dimensional nonlinear stochastic partial differential equations(SPDEs).Since deep learning is widely used in solving high-dimensional partial differential equations(PDEs),we will introduce a numerical method based on deep learning.The method can handle high-dimensional nonlinear SPDEs driven by Stratonovich noise and multiplicative noise.Firstly,the SPDEs are reformulated using stochastic differential equations(SDEs).Then,we use the deep neural networks to approximate the solutions of the SPDEs and the SDEs.In deep neural networks,we use the stochastic gradient descent-type(SGD)algorithm and adaptive moment estimation(Adam)algorithm to minimize the cost function.Finally,numerical results on examples including the high-dimensional nonlinear Black-Scholes equation,the reaction diffusion equation,and the European financial derivatives pricing equation driven by Stratonovich noise or multiplicative noise.The experimental results suggest that the proposed numerical experiment method can effectively solve high-dimensional nonlinear SPDEs.