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Abstract-Partial differential equations (PDEs) with boundary conditions (Dirichlet or Neumann) 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.Index Terms-Boundary value problems, engineering problems, irregular boundaries, neural networks, partial differential equations (PDEs), penalty method, multilayer perceptron, radial basis function (RBF) networks.

Gaussian mixture models (GMMs) constitute a well-known type of probabilistic neural networks. One of their many successful applications is in image segmentation, where spatially constrained mixture models have been trained using the expectation-maximization (EM) framework. In this letter, we elaborate on this method and propose a new methodology for the M-step of the EM algorithm that is based on a novel constrained optimization formulation. Numerical experiments using simulated images illustrate the superior performance of our method in terms of the attained maximum value of the objective function and segmentation accuracy compared to previous implementations of this approach.