This study aims to evaluate the learning ability and performance of five meta-heuristic optimization algorithms in training forward and recurrent fuzzy-based machine learning models, such as adaptive neuro-fuzzy inference system (ANFIS) and RANFIS (recurrent ANFIS), to predict hydraulic jump characteristics, i.e., downstream flow depth (h₂) and jump length (L_j). To meet this end, the firefly algorithm (FA), particle swarm algorithm (PSO), whale optimization algorithm (WOA), genetic algorithm (GA), and moth-flame optimization algorithm (MFO) are embedded with the fuzzy-based models, which represent the main contribution of this study. The analysis of the results of predicting hydraulic jump characteristics shows that the embedded ANFIS and RANFIS models are more accurate than the empirical relations proposed by the previous researchers. Comparing the performance of the embedded RANFISs and ANFISs methods in predicting L_j represents the superiority of the RANFIS models to the ANFISs. The results of the sensitivity analysis show that among the input independent parameters, flow discharge (Q) is the most important factor in predicting downstream flow depth in weak, oscillating, and steady hydraulic jumps (1.7 < Froude number < 9), while the upstream flow depth (h₁) is more important than the other input parameters in strong hydraulic jumps (Froude number > 9).

这项研究旨在评估五种元启发式优化算法在训练正向和循环基于模糊的机器学习模型（例如自适应神经模糊推理系统（ANFIS）和RANFIS（循环ANFIS））中的预测预测液压的能力和性能。跳跃特性，即下游水深（h ₂）和跳跃长度（L _j）。为此，萤火虫算法（FA），粒子群算法（PSO），鲸鱼优化算法（WOA），遗传算法（GA）和蛾-火焰优化算法（MFO）都嵌入了基于模糊的模型，这代表了这项研究的主要贡献。对水力跳跃特性预测结果的分析表明，嵌入的ANFIS和RANFIS模型比以前的研究人员提出的经验关系更为精确。比较嵌入式RANFIS和ANFIS的方法在预测L _j方面的性能，代表了RANFIS模型相对于ANFIS的优越性。灵敏度分析的结果表明，在输入的独立参数中，流量流量（Q）是预测弱，振荡和稳定水力跳跃（1.7 <弗洛德数<9）中的下游水深的最重要因素，而在强水力跳跃中，上游水深（h ₁）比其他输入参数更重要（密码编号> 9）。