Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations

TL;DR

The paper introduces a physics-informed multiresolution wavelet neural network (PIMWNN) that efficiently solves various PDEs with higher accuracy and speed, overcoming limitations of traditional PINNs and handling complex boundary conditions.

Contribution

It proposes a novel PIMWNN approach combining wavelet neural networks with physics-informed training for PDEs, enhancing accuracy and computational efficiency.

Findings

PIMWNN outperforms PINNs in accuracy and speed.

The method effectively handles boundary conditions.

It is suitable for both stationary and time-dependent PDEs.

Abstract

In this paper, a physics-informed multiresolution wavelet neural network (PIMWNN) method is proposed for solving partial differential equations (PDEs). This method uses the multiresolution wavelet neural network (MWNN) to approximate unknown functions, then substituting the MWNN into PDEs and training the MWNN by least-squares algorithm. We apply the proposed method to various problems, including stationary/nonstationary advection, diffusion and advection-diffusion problems, and linear/nonlinear time-dependent problems. Numerical experiments show that the PIMWNN method can achieve higher accuracy and faster speed than Physics Informed Neural Networks (PINNs). Moreover, the PIMWNN method, being mesh-free, can handle different boundary conditions easily and solve the time-dependent problems efficiently. The proposed method is expected to solve the spectral bias problem in network training. These characteristics show the great potential of the PIMWNN method used in the field of numerical solving methods for PDEs.