Analytic Solutions to Differential Equations under Graph-Based Genetic Programming

Abstract

Abstract. 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.

On Affine Symbolic Regression Trees for the Solution of Functional Problems

“…Each point in the space E is a structurally (but not necessarily mathematically) unique representation of a mathematical expression. Points in the encoding space are mapped via a decoding function P into the space of representable expressions (Denoted as Ω following [9]).…”

“…An ongoing body of work investigates heuristics for optimal selection of these components. This paper will focus on the selection of the encoding space E. Seaton et al define a heuristic metric for comparison of encoding spaces termed unguided complexity in [9]. The method relies on estimating the ratio,…”

“…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]. Despite some progress, no single approach has yet broken through as a significant step forward.…”

“…A recent body of work has focussed on ways that the performance of SR algorithms applied to the solution of differential equations might be improved. This paper opens with a thorough mathematical description (based in part on the work of Seaton et al in [9]) of the search problem underpinning SR-based solution of differential equations. Of the elements defined in this description, the current study is focused on the encoding space.…”

Sniffer Technique for Numerical Solution of Korteweg-de Vries Equation Using Genetic Algorithm

“…Considerable work has been reported in the literature for obtaining numerical solution for ODE equations. Seaton et al [1] have used Cartesian Genetic Programming for solving differential equations. John Butcher [2] describes a variety of numerical methods for solving differential equations.…”