Partial differential equations (PDEs) are used to describe a wide range of phenomena, such
as heat transfer, fluid dynamics, and quantum mechanics. By solving PDEs, we can ob-
tain insights into the behavior of the system and make predictions about its future evolution.
Conventional numerical methods for obtaining the approximate solutions of PDEs may re-
quire extensive computational resources and time, especially for complex PDEs and large-
scale problems. The recently developed physics-informed neural network (PINN) has shown
promise in a variety of scientific and engineering fields by incorporating physical laws into the
loss functions of the neural network (NN). In addition, the training of PINN does not require
ground truth data, but it has poor generalization ability to unseen domains. On the other hand,
a convolutional neural network (CNN) has fast inference and better generalization ability, but
it requires a large amount of training data.
Taking the advantages of PINN and CNN by using Legendre multiwavelets (LMWs) as
basis functions, we introduce a new method to approach the PDEs in this paper, namely
Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), in order to 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 pre-
computed training data, simultaneously providing fast and continuous solutions that gener-
alize 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, incompressible Navier-Stokes (N-
S) equation, and 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.
Computer Science and Mathematics, Computational Mathematics
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