A neurocomputing approach to solving partial differential equations.

摘要

Differential equations in science and engineering are tools to describe the actual behavior of physical systems. Physical situations involving more than one variable can often be expressed using equations involving partial derivatives (PDEs). While there already exists many analytical and numerical techniques for solving PDEs, recent advances in an alternative problem solving approach, artificial neural networks, suggest that this methodology may have potential in this area. Neural nets are of interest to researchers in many areas of science. They are a powerful tool for modeling problems for which explicit solutions are not known or can not be easily obtained. Since PDEs can lead to challenging numerical problems, it is advantageous to develop methods for applying neural nets to solving partial differential equations.;This dissertation introduces a new approach to approximate solutions of differential equations using the recently developed Hopfield neural networks. The Hopfield neural network is designed to solve constrained optimization problems; it is a recurrent net where the weights are fixed to represent the constraints and the quantity to be optimized. Our approach consists of two new techniques developed to solve both boundary value problem (BVP) and PDE, by combining the two standard numerical methods, finite-differences and finite-elements, with the Hopfield neural net. The new approaches are denoted the Hopfield-finite-difference (HFD) and Hopfield-finite-element (HFE) methods.;The use of the HFD and HFE methods are illustrated for several simple problems. Sensitivity tests on the parameters involved in these methods demonstrate the robustness of these methods. Moreover, several forms of the Hopfield neural net are explored, namely, parallel computation, sequential mode, and random order of updates. The stability characteristics of the Hopfield nets developed in the research are summarized. The developed methods are then used to approximate the solutions of a variety of problems, and the results are compared with those obtained by numerical methods.;Many examples of basic PDEs and BVPs are successfully solved using the proposed approaches, demonstrating the effectiveness of these methods. The problems discussed in the research are a brief sample of the applications in which HFD and HFE can be used; they suggest the breadth of the neural net's applicability in approximating solutions to partial differential equations.