Neural Network Method for Solving Partial Differential Equations

Authors:
Lucie P. Aarts;Peter Van Der Veer
Affiliations:
Delft University of Technology, Department of Civil Engineering, Faculty of Civil Engineering and Geosciences 2628 CN Delft, The Netherlands. E-mail: l.aarts@citg.tudelft.nl;Delft University of Technology, Department of Civil Engineering, Faculty of Civil Engineering and Geosciences 2628 CN Delft, The Netherlands
Venue:
Neural Processing Letters
Year:
2001

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Abstract

A method is presented to solve partial differential equations (pde's) and its boundary and/or initial conditions by using neural networks. It uses the fact that multiple input, single output, single hidden layer feedforward networks with a linear output layer with no bias are capable of arbitrarily well approximating arbitrary functions and its derivatives, which is proven by a number of authors and well known in literature. Knowledge about the pde and its boundary and/or initial conditions is incorporated into the structures and the training sets of several neural networks. In this way we obtain networks of which some are specifically structured. To find the solution of the pde and its boundary and/or initial conditions we have to train all obtained networks simultaneously. Therefore we use an evolutionary algorithm to train the networks. We demonstrate the working of our method by applying it to two problems.