Faculty Forum scheduled
Kevin McFall, Ph.D., assistant professor of engineering at Penn State Lehigh Valley, will be the featured speaker for the campus's first 2009-2010 Faculty Forum event. McFall will present his research, "Solution of Boundary Value Problems using an Artificial Intelligence technique," at 12:15 p.m. on Thursday, December 3, in room 207.
In summarizing his research, McFall writes:
Artificial intelligence (AI) is a broad field encompassing various algorithms inspired by biological systems. For example, genetic programming mirrors evolution where a population’s most successful individual solutions to a given problem are allowed to combine, producing offspring which are hopefully more successful at solving the problem. Fuzzy logic allows every day linguistic expressions posed in shades of gray to be manipulated with algebras normally reserved for binary logic values in black and white. The modeling powers of artificial neural networks (ANNs) derive from the dynamic and flexible electrical connections between neurons in the brain.
My area of research lies in developing AI-based numerical techniques to solve problems normally approached with more traditional methods. The main thrust of current work offers an alternative to the finite element method (FEM) used to solve boundary value problems (BVPs). Several concerns with the FEM such as complexity in dealing with nonlinear differential equations (DEs) and meshing difficulties are avoided when using ANNs to solve the BVP. An approximate solution to the BVP can be posed as a function of the ANN output, and gradient descent optimization is employed to modify the ANN so that the approximate solution more closely satisfies the DE. In such a manner the ANN is “trained” to satisfy the DE, thus producing a more accurate approximate solution.
Solving BVPs using ANNs is a relatively new concept, and my main contribution lies in developing an approximate solution automatically satisfying the boundary conditions (BCs) of the BVP. The approximate solution is designed in such a manner that it satisfies the BCs (whether Dirichlet or mixed) on boundaries of arbitrary shape, regardless of the ANN output. Such automatic BC satisfaction simplifies the learning process by removing the constraint of satisfying the BCs and DE simultaneously, and was made possible by development of a length factor based on mapping the domain shape onto a circle using thin plate splines. Besides enabling a simpler learning process, an approximate solution to the BVP defined in this way is guaranteed to converge to the exact analytical solution under certain circumstances which can be approximated during actual training of the ANN.
This seminar will present the basic outline of the work described here, focusing on its qualitative aspects rather than the technical details. The goal is to make the presentation more easily accessible to a broader audience.