Solving equations by hybrid evolutionary computation techniques

Abstract

Evolutionary computation techniques have mostly been used to solve various optimisation and learning problems. This paper describes a novel application of evolutionary computation techniques to equation solving. Several combinations of evolutionary computation techniques and classical numerical methods are proposed to solve linear and partial differential equations. The hybrid algorithms have been compared with the well-known classical numerical methods. The experimental results show that the proposed hybrid algorithms outperform the classical numerical methods significantly in terms of effectiveness and efficiency. Index TermsLinear equations, Partial differential equations, Adaptation, Hybrid algorithms. I. INTRODUCTIONThere has been a huge increase in the number of papers and successful applications of evolutionary computation techniques in a wide range of areas in recent years. Almost all these applications can be classified as evolutionary optimisation (either numerical or combinatorial) or evolutionary learning (supervised, reinforcement or unsupervised). This paper presents a very different and novel application of evolutionary computation techniques in equation solving, i.e., solving linear and partial differential equations by simulated evolution.One of the best-known numerical methods for solving linear equations is the successive over relaxation (SOR) method [1]. However, it is often very difficult to estimate the optimal relaxation factor, which is a key parameter of the SOR method. This paper proposes a hybrid algorithm combining the SOR method with evolutionary computation techniques. The hybrid algorithm does not require a user to guess or estimate the optimal relaxation factor. The algorithm "evolves" it.Unlike most other hybrid algorithms where an evolutionary algorithm is used a wrapper around another algorithm (often a classical algorithm), the hybrid algorithm proposed in this paper integrates the SOR method with evolutionary computation techniques, such as crossover and mutation. It makes better use of a population by employing different equation solving strategies for different individuals in the population. Then these individuals can exchange information through crossover. Experimental results show that the hybrid algorithm can solve equations within a small fraction of time needed by the classical SOR method for a number of problems we have tested.Built on the work of solving linear equations, this paper also proposes two hybrid algorithms for solving partial differential equations, i.e., the Dirichlet problem [2]. Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.Fogel and Atmar [3] used linear equation solving as test problems for comparing crossover and inversion operators and Gaussian mutation in an evolutionary algorithm. A linear system of the form