On Affine Symbolic Regression Trees for the Solution of Functional Problems M. D. Champneys, N. Dervilis, and K. Worden Abstract Symbolic regression has emerged from the more general method of Genetic Programming (GP) as a means of solving functional problems in physics and engineering, where a functional problem is interpreted here as a search problem in a function space. A good example of a functional problem in structural dynamics would be to find an exact solution of a nonlinear equation of motion. Symbolic regression is usually implemented in terms of a tree representation of the functions of interest; however, this is known to produce search spaces of high dimension and complexity. The aim of this chapter is to introduce a new representation—the affine symbolic regression tree. The search space size for the new representation is derived, and the results are compared to those for a standard regression tree. The results are illustrated by the search for an exact solution to several benchmark problems. Keywords Symbolic regression · Genetic programming · Solutions of differential equations · Search space analysis 1 Introduction Differential equations are among the most fundamental tools available for scientific analysis. No other mathematical object has the same ability to describe change in the physical systems. It is therefore no wonder that differential equations have been so extensively studied. Tireless investigation into the specification, existence and uniqueness of solutions to differential equations has borne some of the most eminently useful techniques in engineering. Exact solutions to linear differential equations have enabled modal analysis, an invaluable analysis tool for linear dynamic systems. However, as the materials, geometries, demands and capabilities of engineering structures become ever more complex, a linear description of the physics becomes less and less applicable. By far, the majority of nonlinear differential equations are without exact solution. This reality hamstrings the development of powerful analysis tools that might be viewed as nonlinear alternatives to modal analysis. In the place of pure mathematical solutions, powerful heuristic methods have been developed. Heuristic methods fall into a number of distinct categories based on the form of the yielded solution. Numerical methods provide approximations to the true solution at a finite number of points in the domain. Analytical approximate methods such as the shooting method [1], or harmonic balance [2], provide analytic approximations, often by the consideration of series expansions. A third class of approaches, referred to hereafter as pure heuristic methods, offer the possibility of both an approximate and an exact solution. Such approaches are often (but not always [3, 4]) based on a standard symbolic regression (SR), a specific application of the more general method of Genetic Programming (GP) [5]. A large number of authors have proposed SR for solving differential equations in the last three decades. Highlights include the earliest suggestions of the approach in [5], a reverse-Polish notation-based tree structure in [6], a hybrid analytical approach in [7], a grammar-based approach with a benchmarking set in [8], and a Cartesian Genetic Programming (CGP) approach in [9]. M. D. Champneys ( ) Industrial Doctorate Centre in Machining Science, Advanced Manufacturing Research Centre with Boeing, University of Sheffield, Sheffield, UK Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: mdchampneys1@sheffield.ac.uk N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK © The Society for Experimental Mechanics, Inc. 2022 G. Kerschen et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-77135-5_11 95
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