Closed-Form Solutions to Differential Equations via Differential Evolution

Abstract

We focus on solving ordinary differential equations using the evolutionary algorithm known as differential evolution (DE). The main purpose is to obtain a closed-form solution to differential equations. To solve the problem at hand, three steps are proposed. First, the problem is stated as an optimization problem where the independent variables areelementaryfunctions. Second, as the domain of DE is real numbers, we propose a grammar that assigns numbers to functions. Third, to avoid truncation and subtractive cancellation errors, to increase the efficiency of the calculation of derivatives, the dual numbers are used to obtain derivatives of functions. Some examples validating the effectiveness and efficiency of our method are presented.

On Affine Symbolic Regression Trees for the Solution of Functional Problems

“…Another common representation scheme is a grammar-based structure [8,13]; these are employed by several authors alongside techniques such as modular arithmetic to encode expressions as vectors of integers. The search space of grammar-based representations can conveniently be enumerated by considering the number of unique tree structures.…”