Symbolic regression based genetic approximations of the Colebrook equation for flow friction

TL;DR

This paper develops explicit, low-cost approximations of the Colebrook equation using genetic programming, achieving high accuracy with minimal computational complexity for hydraulic flow friction calculations.

Contribution

It introduces a novel AI-driven symbolic regression approach to derive simplified, accurate explicit formulas for the Colebrook equation, reducing computational effort in hydraulic modeling.

Findings

Best approximation error up to 0.13%

Two logarithmic forms used in the best case

Single logarithm approximation with Pade approximation

Abstract

Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, and the relative roughness of inner pipe surface, with the output unknown parameter; the flow friction factor. In this paper, a few explicit approximations to the Colebrook equation are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers. The fact that the used genetic programming tool does not know the structure of the Colebrook equation which is based on computationally expensive logarithmic law, is used to obtain better structure of the approximations which is less demanding for calculation but also enough accurate. All generated approximations are with low computational cost because they contain a limited number of logarithmic forms used although for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor in best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Pade approximation, practically the same error is obtained also using only one logarithm.