Finding Approximate Analytic Solutions To Differential
Equations Using Genetic Programming
Technical Report, DSTO-TR-0838. Surveillance Systems Division, Defence Science and
Technology Organisation, Australia, Salisbury, SA, 5108, Austrlia, (February 1999)Abstract
The computational optimisation technique, genetic
programming, is applied to the analytic solution of
general differential equations. The approach generates
a mathematical expression that is an approximate or
exact solution to the particular equation under
consideration. The technique is applied to a number of
differential equations of increasing complexity in one
and two dimensions. Comparative results are given for
varying several parameters of the algorithm such as the
size of the calculation stack and the variety of
available mathematical operators. Several novel
approaches gave negative results. Angeline's module
acquisition (MA) and Koza's automatically defined
functions (ADF) are considered and the results of some
modifications are presented. One result of significant
theoretical interest is that the syntax-preserving
crossover used in Genetic Programming may be
generalised to allow the exchange of n-argument
functions without adverse effects.
The results show that Genetic Programming is an
effective technique that can give reasonable results,
given plenty of computing resources. The technique used
here can be applied to higher dimensions; although in
practice the algorithmic complexity may be too high.