Analytic Solutions to Differential Equations under Graph-Based Genetic Programming

Cartesian Genetic Programming (CGP) is applied to solving differential equations (DE). We illustrate that repeated elements in analytic solutions to DE can be exploited under GP. An analysis is carried out of the search space in tree and CGP frameworks, examining the complexity of different DE problems. Experimental results are provided against benchmark ordinary and partial differential equations. A system of ordinary differential equations (SODE) is solved using multiple outputs from a genome. We discuss best heuristics when generating DE solutions through evolutionary search.

[1]  Declan A. Diver Applications of genetic algorithms to the solution of ordinary differential equations , 1993 .

[2]  Lishan Kang,et al.  Evolutionary Modeling of Systems of Ordinary Differential Equations with Genetic Programming , 2000, Genetic Programming and Evolvable Machines.

[3]  James P. Braselton,et al.  Differential Equations With Maple V , 1994 .

[4]  John R. Koza,et al.  Genetic programming - on the programming of computers by means of natural selection , 1993, Complex adaptive systems.

[5]  Willy Klier,et al.  Comparison of Automatic and Symbolic Differentiation in Mathematical Modeling and Computer Simulation of Rigid-Body Systems , 2002 .

[6]  Julian Francis Miller,et al.  Cartesian genetic programming , 2000, GECCO '10.

[7]  H. Bateman Partial Differential Equations of Mathematical Physics , 1932 .

[8]  Maarten Keijzer,et al.  Scientific discovery using genetic programming , 2001 .

[9]  Hod Lipson,et al.  Comparison of tree and graph encodings as function of problem complexity , 2007, GECCO '07.

[10]  Daniel A. Ashlock,et al.  A hybrid genetic programming approach for the analytical solution of differential equations , 2005, Int. J. Gen. Syst..

[11]  Isaac E. Lagaris,et al.  Solving differential equations with genetic programming , 2006, Genetic Programming and Evolvable Machines.

[12]  James P. Braselton,et al.  Differential Equations with Maple V with Cdrom , 1999 .

[13]  E. T. WHITTAKER,et al.  Partial Differential Equations of Mathematical Physics , 1932, Nature.

[14]  Julian Francis Miller,et al.  Investigating the performance of module acquisition in cartesian genetic programming , 2005, GECCO '05.