Proper approximation of friction capacity (FC) of driven piles is a noticeable issue in geotechnical engineering. Hence, the pivotal focus of the current research is on proposing reliable predictors for evaluating this parameter. Three artificial neural networks (ANNs) improved by firefly algorithm (FA), multi-tracker optimization algorithm (MTOA), and black hole algorithm (BHA) are used to estimate the FC when it is affected by four key factors, namely pile length, pile diameter, vertical effective stress, and undrained shear strength. Checking the optimization process of different population sizes revealed that the MTOA and BHA need the population twice as many as the FA does (400 vs. 200). The results showed that all three models can properly comprehend the association of FC to the mentioned parameters. According to the values of root mean square error (RMSE) as well as the coefficient of determination (R2) obtained in the prediction phase (6.9606 and 0.8827, 8.5411 and 0.8370, 6.0454 and 0.9194, respectively, for the FA-ANN, MTOA-ANN, and BHA-ANN), the BHA-ANN is more accurate than two other algorithms, however, the MTOA-ANN gained the largest accuracy of training. The suggested models proved the high efficiency of hybrid predictors and can be potential alternatives to experimental and traditional approaches.

中文翻译：

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使用三种复杂的搜索方案确定打入桩的摩擦能力

打入桩的摩擦能力 (FC) 的适当近似值是岩土工程中的一个值得注意的问题。因此，当前研究的重点是提出可靠的预测指标来评估该参数。使用萤火虫算法 (FA)、多跟踪器优化算法 (MTOA) 和黑洞算法 (BHA) 改进的三种人工神经网络 (ANN) 来估计受四个关键因素影响时的 FC，即桩长、桩径、竖向有效应力和不排水抗剪强度。检查不同种群规模的优化过程表明，MTOA 和 BHA 需要的种群数量是 FA 的两倍（400 对 200）。结果表明，所有三个模型都可以正确理解 FC 与上述参数的关联。根据预测阶段获得的均方根误差 (RMSE) 值和决定系数 (R2)（分别为 6.9606 和 0.8827、8.5411 和 0.8370、6.0454 和 0.9194，对于 FA-ANN，MTOA- ANN 和 BHA-ANN），BHA-ANN 比其他两种算法更准确，然而，MTOA-ANN 获得了最大的训练准确度。建议的模型证明了混合预测器的高效率，并且可以作为实验和传统方法的潜在替代方案。