A hybrid genetic programming approach for the analytical solution of differential equations

Abstract

This paper presents a novel addition to the current genetic programming techniques for solving differential equations. Rather than using numerical approximation of derivatives during fitness evaluation, automatically computed analytical derivatives of the candidate solutions are employed. Because analytical derivatives are used, symbolic constants can be incorporated in the solution. This permits the development of a single solution for a range of material properties, boundary conditions or other design parameters. Additionally, for the special case of linear differential equations, a modified Gram–Schmidt algorithm is used to reduce the set of general solutions located by genetic programming to a basis set.

Keywords:

• Genetic programming
• Evolutionary algorithm
• Differential equation
• Engineering design
• Human-in-the-loop
• Virtual engineering

Acknowledgements

This work was funded in part by a grant from the U.S. Department of Energy, Office of Industrial Technology and the U.S. Department of Energy, National Energy Technology Laboratory.

Notes

In 1998, Kenneth “Mark” Bryden completed his Ph.D. in Mechanical Engineering from the University of Wisconsin-Madison. Prior to earning his Ph.D., he worked for 14 years in a wide range of engineering positions at Westinghouse Electric Corporation. He is currently an associate professor of Mechanical Engineering, the program chair for the Complex Adaptive Systems, and a research associate of the Virtual Reality Applications Center at Iowa State University. His primary research interests are in the development of engineering tools for complex systems within a virtual engineering environment. Virtual engineering requires the integration of virtual reality, high performance computing, and new computational algorithms to solve complex, tightly coupled engineering and decision analysis problems. Dr Bryden is the recipient of several awards and fellowships, including the Power Engineering Education Fellowship.

Professor Ashlock holds a Chair in Bioinformatics in the Mathematics and Statistics Department of the University of Guelph in Guelph, Ontario, Canada. Professor Ashlock works in bioinformatics, computational biology, and development of biologically inspired algorithms that are applied to engineering, optimization, genomics, and theoretical biology. His publications cover a broad range of topics including game theory, thermal systems, genome assembly, virtual robotics, and pure mathematics. Dr. Ashlock has served as a founding member for interdisciplinary graduate programs in bioinformatics, complex adaptive systems, and human-computer interface. Dr Ashlock received his Ph.D. in Pure Mathematics from the California Institute of Technology in 1990 and currently resides in southern Ontario.

Steve Kirstukas (Ph.D., M.S., Mechanical Engineering, University of Minnesota; B.S., Civil Engineering, Catholic University of America) has worked in biomechanics and software engineering and has developed software for use in the health sciences, in the worldwide manufacturing community, and in academic research. He uses gradient-based optimization and genetic programming methods as engineering decision-making tools in the virtual environment, a place where communication barriers disappear and technical and non-technical people can reach some understanding. He has a special preference for the non-virtual environment of Mt. Desert Island.