Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions

A method for solving boundary value problems (BVPs) is introduced using artificial neural networks (ANNs) for irregular domain boundaries with mixed Dirichlet/Neumann boundary conditions (BCs). The approximate ANN solution automatically satisfies BCs at all stages of training, including before training commences. This method is simpler than other ANN methods for solving BVPs due to its unconstrained nature and because automatic satisfaction of Dirichlet BCs provides a good starting approximate solution for significant portions of the domain. Automatic satisfaction of BCs is accomplished by the introduction of an innovative length factor. Several examples of BVP solution are presented for both linear and nonlinear differential equations in two and three dimensions. Error norms in the approximate solution on the order of 10^-4 to 10^-5 are reported for all example problems.