Integrated RBF networks for periodic extensions for solving boundary value problems

In this paper, the radial basis function networks (RBFNs) are trained to extend a non-periodic function to become a periodic one in a rectangular domain, from which Cartesian-grid-based partial-differential-equation (PDE) solvers can be applied. The networks are constructed through integration instead of the usual differentiation. The presence of the integration constants results in a higher dimension space in the hidden layer, which enhances the quality of approximation of the networks. In addition, the resulting integrated basis functions are applied to sinusoidal functions, which guarantees that the approximating function in the extended domain is naturally periodic. With the RBF width, amplitude and phase shift being adjustable parameters, the equations representing the networks are nonlinear and they can be solved by using the Levenberg–Marquardt method. Numerical experiments demonstrate that the iterative convergence is fast, the residual is low and the approximating function in the extension domain possesses a low level of fluctuation. The proposed networks are then utilised in conjunction with some high-order discretisation methods based on RBFs to enable the PDE in a non-rectangular domain to be solved in a rectangular one. The task of discretising a continuous spatial domain is very economical as a simple Cartesian grid can be used to represent the computational/extended domain. Linear and nonlinear problems are considered and the IRBF results are compared with the analytic solutions and numerical results produced by the finite difference and finite element methods. Numerical experiments demonstrate that the high-order discretisation methods are still able to produce their fast rates of convergence.