Assessment of Multi-Objective and Coevolutionary Genetic Programming for Predicting the Stokes Flow Around a Sphere

Abstract

Genetic Programming (GP) has been used in a variety of fields to learn the relationships between physical measurements of real-world problems. In this article, we combine different techniques from the area of evolutionary optimization and, particularly, GP to solve a fluid-dynamics problem: the Stokes flow around a rigid sphere. This serves as the starting point to explore the potential of applying different GP techniques to such complex physical problems. From the definition of the considered fluid-dynamics problem, six benchmark instances with different lengths and complexities are derived. We use single-and multi-objective GP methods and compare their performance using different objective functions. More precisely, we study how model complexity, correlation and the consistency with physical laws (i.e. physical units of measurement) can be included as different objective functions, and whether their inclusion is beneficial to the overall search process. In addition, we include the concept of Cooperative Coevolution, which maintains multiple independent populations of solutions, into our GP implementations and explore the capabilities and limitations of such coevolution-based optimization. We further propose a novel multi-phase approach, which alternates in the GP process between phases of traditional optimization and a mutation-only phase to reduce the model complexity. The results indicate that using multi-objective optimization is beneficial to the search process and can help finding numerically correct solutions to the problems, especially when including a transformed Spearman correlation as an additional optimization goal. Furthermore, the inclusion of the physical units of measurement also helps guide the GP toward numerically correct and physically meaningful equations. While the concept of coevolution did not lead to a superior performance in all cases, the precomputation of additional features, and the resulting reduction of the function set of the GP, leads to a drastic increase in performance and enables the algorithms to solve even the most complex of our benchmark instances. In the future, we aim to extend this research to more complex flows with multiple spheres and at higher Reynolds numbers, which involve a large number of input features.