Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs

In this paper, we propose two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (PDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully nonlinear PDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully nonlinear PDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multi-scale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. In numerical experiments, involving Allen-Cahn equations, Black-Scholes-Barentblatt equations, and Hamiltonian-Jacobi-Bellman equations, the first proposed method exhibits higher efficiency and accuracy than the existing method; the second proposed method can extend the dimensionality of the completely non-linear PDEs-2BSDEs system over $400$ dimensions, from which the numerical results illustrate the effectiveness of proposed methods.