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<p>The Darcy-Weisbach friction factor is used to determine the head losses occurring due to friction in pressurised pipes. It is defined by the Colebrook-White Equation as an implicit function of the Reynolds number and relative roughness for which iterative methods are generally required. Explicitly derived approximations can be obtained very quickly but have parameter dependent errors, do not guarantee monotonicity, and give no indication of the level of uncertainty associated with these approximations. Most existing Machine Learning Surrogate models produce accurate representations but rarely constrain their predictions to enforce physically meaningful monotonic relationships, nor provide probabilistic estimates of their reliability. This paper develops a Monotonic Gradient Boosting (MGB) surrogate model that constrains its predictions based on physics-based monotonic relationships and provides both calibrated error bound approximations using quantile boosting and split conformal prediction methods as well as a transparent benchmarking protocol. On the fixed <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(24\times 7\)</EquationSource></InlineEquation> evaluation grid, rough-pipe subset (<InlineEquation ID="IEq2"><EquationSource Format="TEX">\(\varepsilon /D&gt;0\)</EquationSource></InlineEquation>, 144&#xa0;cases), the surrogate achieved MAPE&#xa0;=&#xa0;0.168% and maximum absolute error&#xa0;=&#xa0;<InlineEquation ID="IEq3"><EquationSource Format="TEX">\(1.46\times 10^{-3}\)</EquationSource></InlineEquation>; on the stratified held-out test set it achieved MAPE&#xa0;=&#xa0;0.25%. The empirical coverage for nominal 95% prediction intervals was 95.5%. There were no violations to monotonicity across all 1,829 Reynolds number and 1800 relative roughness changes. The explicit baseline models achieved an average absolute percentage error (MAPE) of 0.236% (Haaland), 0.546% (Swamee-Jain), and 0.012% (Serghides). While Serghides is still superior in terms of point accuracy than the others, the MGB provides both monotonicity constraints and calibration bounds on approximation errors to provide a reliability certificate that cannot be found using any closed form expressions.</p>

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Physics-constrained machine-learning surrogates for the colebrook friction factor: monotonic gradient boosting, uncertainty quantification, and open benchmarking

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  • 发表日期：

    2026年7月17日

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  • Tevfik Denizhan Muftuoglu

摘要

The Darcy-Weisbach friction factor is used to determine the head losses occurring due to friction in pressurised pipes. It is defined by the Colebrook-White Equation as an implicit function of the Reynolds number and relative roughness for which iterative methods are generally required. Explicitly derived approximations can be obtained very quickly but have parameter dependent errors, do not guarantee monotonicity, and give no indication of the level of uncertainty associated with these approximations. Most existing Machine Learning Surrogate models produce accurate representations but rarely constrain their predictions to enforce physically meaningful monotonic relationships, nor provide probabilistic estimates of their reliability. This paper develops a Monotonic Gradient Boosting (MGB) surrogate model that constrains its predictions based on physics-based monotonic relationships and provides both calibrated error bound approximations using quantile boosting and split conformal prediction methods as well as a transparent benchmarking protocol. On the fixed \(24\times 7\) evaluation grid, rough-pipe subset (\(\varepsilon /D>0\), 144 cases), the surrogate achieved MAPE = 0.168% and maximum absolute error = \(1.46\times 10^{-3}\); on the stratified held-out test set it achieved MAPE = 0.25%. The empirical coverage for nominal 95% prediction intervals was 95.5%. There were no violations to monotonicity across all 1,829 Reynolds number and 1800 relative roughness changes. The explicit baseline models achieved an average absolute percentage error (MAPE) of 0.236% (Haaland), 0.546% (Swamee-Jain), and 0.012% (Serghides). While Serghides is still superior in terms of point accuracy than the others, the MGB provides both monotonicity constraints and calibration bounds on approximation errors to provide a reliability certificate that cannot be found using any closed form expressions.