Genetic programming and automatic differentiation algorithms applied to the solution of ordinary and partial differential equations

Abstract

This paper investigates the potential of evolutionary algorithms, developed by the combination of genetic programming (GP) and automatic differentiation methods (AD), in determining analytic solutions to ordinary and partial differential equations (ODE and PDE). In turn, AD is a set of techniques based on the mechanical application of the chain rule to numerically evaluate the derivative of a function specified by a computer program. The AD method has a fundamental role in this work since it calculates the exact values of the derivatives of a function for a given set of input values while numerical differentiation methods introduce unacceptable round-off errors in the discretization process. With this purpose, and using the Matlab programming environment, we developed several algorithms (namely GPAD) and addressed problems of different kinds of differential equations. The results are promising, with exact solutions obtained for most of the addressed problems, which include equations where not even commercial systems could find a symbolic solution. These results empirically indicate that GPAD can be an efficient and robust methodology to find analytic solutions for ODE and PDE.