DWNN: Deep Wavelet Neural Network for Solving Partial Differential Equations

Abstract

In this paper, we propose a deep wavelet neural network (DWNN) model to approximate the natural phenomena that are described by some classical PDEs. Concretely, we introduce wavelets to deep architecture to obtain a fine feature description and extraction. That is, we constructs a wavelet expansion layer based on a family of vanishing momentum wavelets. Second, the Gaussian error function is considered as the activation function owing to its fast convergence rate and zero-centered output. Third, we design the cost function by considering the residual of governing equation, the initial/boundary conditions and an adjustable residual term of observations. The last term is added to deal with the shock wave problems and interface problems, which is conducive to rectify the model. Finally, a variety of numerical experiments are carried out to demonstrate the effectiveness of the proposed approach. The numerical results validate that our proposed method is more accurate than the state-of-the-art approach.

Keywords

• partial differential equations
• wavelet transforms
• deep neural network
• numerical solution