A Bessel-Class Radial Basis Function for Neural Networks in Solving Helmholtz and Laplace Equations

Helmholtz and Laplace equations are important in mechanics. A Bessel-class radial basis functions (RBFs) is introduced in neural networks to solve Laplace and Helmholtz equations. This class of RBFs is proved to be continuous and infinite positive definite. The presented Bessel-class RBF can degenerate to Gaussian in an infinite smooth case. The presented RBF satisfies Helmholtz equations in the domain, and does not need physics regularization. It can also be applied to Laplace equation by applying a small artificial parameter. Thus, the presented RBF can be used in RBF neural networks to solve Helmholtz and Laplace equations only by training data on the boundary. Several numerical examples including Helmholtz and Laplace equations have been carried out to show the effectiveness of this Bessel-class RBFs in 1-D, 2-D and 3-D domain, respectively.