Analytic Solutions to Differential Equations under Graph-Based Genetic Programming

  • Tom Seaton
  • Gavin Brown
  • Julian F. Miller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6021)


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.


Genetic Programming Cartesian Genetic Programming Maximum Path Length Tree Genetic Programming Bloat Control 


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  1. 1.
    Sobolob, S.L.: Partial Differential Equations of Mathematical Physics. Pergamen Press, Oxford (1964)Google Scholar
  2. 2.
    Abell, M.L., Braselton, J.P.: Differential Equations with Maple V. Academic Press, London (2000)Google Scholar
  3. 3.
    Diver, D.A.: Applications of genetic algorithms to the solution of ordinary differential equations. J. Phys. A: Math. Gen. 26, 3503–3513 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Koza, J.R.: Genetic Programming, on the Programming of Computers By Means of Natural Selection. MIT Press, Cambridge (1992)MATHGoogle Scholar
  5. 5.
    Cao, H., Lishan, K., Chen, Y.: Evolutionary Modelling of Systems of Ordinary Differential Equations with Genetic Programming (2000)Google Scholar
  6. 6.
    Tsoulos, I., Lagaris, I.E.: Solving differential equations with genetic programming. Genet. Program. Evolvable Mach. 7, 33–54 (2006)CrossRefGoogle Scholar
  7. 7.
    Kirstukas, S.J., Bryden, K.M., Ashlock, D.A.: A hybrid genetic programming approach for the analytical solution of differential equations. International Journal of General Systems 34, 279–299 (2005)MATHCrossRefGoogle Scholar
  8. 8.
    Miller, J.F., Thomson, P.: Cartesian Genetic Programming. In: Poli, R., Banzhaf, W., Langdon, W.B., Miller, J., Nordin, P., Fogarty, T.C. (eds.) EuroGP 2000. LNCS, vol. 1802, pp. 121–132. Springer, Heidelberg (2000)Google Scholar
  9. 9.
    Walker, J., Miller, J.F.: Investigating the performance of module acquisition in cartesian genetic programming. In: Genetic and Evolutionary Computation Conference, pp.1649–1655, 25-06 (2005)Google Scholar
  10. 10.
    Durrbaum, A., Klier, W., Hahn, H.: Comparison of Automatic and Symbolic Differentiation in Mathematical Modeling and Computer Simulation of Rigid-Body Systems. Multibody System Dynamics 7, 331–355 (2002)CrossRefGoogle Scholar
  11. 11.
    Scmidt, M., Hod, L.: Comparison of Tree and Graph Encodings as Function of Problem Complexity. In: Genetic and Evolutionary Computation Conference, pp. 1674–1679 (2007)Google Scholar
  12. 12.
    Keijzer, M.: Scientific Discovery using Genetic Programming. PhD thesis, Department for Mathematical Modelling, Technical University of Denmark (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tom Seaton
    • 1
  • Gavin Brown
    • 2
  • Julian F. Miller
    • 1
  1. 1.Department of ElectronicsUniversity of York 
  2. 2.School of Computer ScienceUniversity of Manchester 

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