Numerical solution for high-dimensional partial differential equations based on deep learning with residual learning and data-driven learning

Abstract

Solving high-dimensional partial differential equations (PDEs) is a long-term computational challenge due to the fundamental obstacle known as the curse of dimensionality. This paper develops a novel method (DL4HPDE) based on residual neural network learning with data-driven learning elliptic PDEs on a box-shaped domain. However, to combine a strong mechanism with a weak mechanism, we reconstruct a trial solution to the equations in two parts: the first part satisfies the initial and boundary conditions, while the second part is the residual neural network algorithm, which is used to train the other part. In our proposed method, residual learning is adopted to make our model easier to optimize. Moreover, we propose a data-driven algorithm that can increase the training spatial points according to the regional error and improve the accuracy of the model. Finally, the numerical experiments show the efficiency of our proposed model.