Neural-Network methods for irregular boundaries with Robin boundary conditions

Partial differential equations (PDEs) with mixed boundary conditions (Robin) defined on boundaries with simple geometry have been successfully treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the exact satisfaction of the boundary conditions. The method has been successfully tested on two-dimensional and three-dimensional PDEs and has yielded accurate results.