Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities

• Published in: Journal of the Franklin Institute 2013-03, Vol.350 (2), p.300-317
• Main Author: McFall, Kevin Stanley
• Format: Article
• Language: English
• ISSN: 0016-0032; 1879-2693
• DOI: 10.1016/j.jfranklin.2012.11.003

The length factor artificial neural network (ANN) method for solving coupled systems of partial differential equations (DEs) is unique among ANN methods in that the approximate solution exactly satisfies boundary conditions (BCs) on arbitrary geometries regardless of the ANN output. Besides removing the BC constraint from the optimization process, this property allows the method to accurately solve problems with discontinuous BCs despite the continuous nature of ANNs. An automated design parameter selection process is developed to choose a single ANN from an ensemble comprising numerous combinations of design parameters and random starting weights and biases. The selection is made completely independently of the human designer by comparing the magnitude and uniformity of each approximate solution's error in satisfying the DE(s). The automated selection process successfully chooses a solution with error on the same order of magnitude as the best solution in the ensemble. The resulting approximations provide low error solutions for the three different thermal-fluid science example problems explored, including the Navier–Stokes equations. ► Accurately solves discontinuous differential equations using neural networks. ► The approximate solution exactly satisfies boundary conditions even before training. ► Handles coupled systems of nonlinear partial differential equations. ► Automated design parameter selection which is independent of the designer's bias.