Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier-Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.