Convolution-Based Model-Solving Method for Three-Dimensional, Unsteady, Partial Differential Equations

Neural networks are increasingly used widely in the solution of partial differential equations (PDEs). This letter proposes 3D-PDE-Net to solve the three-dimensional PDE. We give a mathematical derivation of a three-dimensional convolution kernel that can approximate any order differential operator within the range of expressing ability and then conduct 3D-PDE-Net based on this theory. An optimum network is obtained by minimizing the normalized mean square error (NMSE) of training data, and L-BFGS is the optimized algorithm of second-order precision. Numerical experimental results show that 3D-PDE-Net can achieve the solution with good accuracy using few training samples, and it is of highly significant in solving linear and nonlinear unsteady PDEs.

Automated summary

This letter presents 3D-PDE-Net, a neural network designed for solving three-dimensional partial differential equations (PDEs). It introduces a mathematical derivation of a three-dimensional convolution kernel that can approximate any order differential operator, and utilizes this theory to construct 3D-PDE-Net. Experimental results demonstrate that 3D-PDE-Net can achieve accurate solutions with few training samples, and it is highly significant in solving linear and nonlinear unsteady PDEs.