The head–discharge relationship of an overflow weir is a prerequisite for flow measurement. Conventionally, it is determined by regression methods. With machine learning techniques, data-driven modelling becomes an alternative. However, a standalone model may be inadequate to generate satisfactory results, particularly for a complex system. With the intention of improving the performance of standard gene expression programming (GEP), a hybrid evolutionary scheme is proposed, which is coupled with grey system theory and probabilistic technique. As a gene filter, grey relational analysis (GRA) eliminates noise and simulated annealing (SA) reduces overfitting by optimising the gene weights. The proposed GEP-based model was developed and validated using experimental data of a submerged pivot weir. Compared with standalone GEP, the GRA–GEP–SA model was found to generate more accurate results. Its coefficients of determination and correlation were improved by 3.6% and 1.7%, respectively. The root mean square error was lowered by 24.8%, which is significant. The number of datasets with an error of less than 10% and 20% was increased by 15% and 12%, respectively. The proposed approach outperforms classic genetic programming and shows a comparative error level with the empirical formula. The hybrid procedure also provides a reference for applications in other hydraulic issues.

中文翻译：

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用于出院预测的混合基因表达编程模型

溢流堰的水头-流量关系是流量测量的先决条件。传统上，它是通过回归方法确定的。借助机器学习技术，数据驱动建模成为一种替代方案。然而，独立模型可能不足以产生令人满意的结果，特别是对于复杂的系统。为了提高标准基因表达编程（GEP）的性能，提出了一种结合灰色系统理论和概率技术的混合进化方案。作为基因过滤器，灰色关联分析 (GRA) 可以消除噪声，模拟退火 (SA) 通过优化基因权重来减少过度拟合。所提出的基于 GEP 的模型是使用浸没枢轴堰的实验数据开发和验证的。与独立的 GEP 相比，GRA-GEP-SA 模型可以生成更准确的结果。其决定系数和相关系数分别提高了3.6%和1.7%。均方根误差降低了 24.8%，这是显着的。误差小于10%和20%的数据集数量分别增加了15%和12%。所提出的方法优于经典的遗传编程，并显示出与经验公式的比较误差水平。该混合程序也为其他液压问题的应用提供了参考。