This section reports the results of all utilized models for predicting suspended sediment load daily. Table 3 lists the input combinations and names of the corresponding models. In the table, Q_t−1 and SSL_t−1 refer to the streamflow and SSL values of one previous day and vice versa. In the first four combinations, only streamflow data were considered inputs and then the lagged SSL data were added to the Q-based inputs obtained from the first four. Data covering the period from 01 January 2007 to 30 June 2011 (75% of the whole data) were used for the training stage, and data from 01 July 2011 to 31 December 2012 (25% of the whole data) were used for the testing stage. One hundred iterations were applied for all ANFIS-based models. Before applying ANFIS-based methods, MARS method was applied to the input combinations provided in Table 3 to decide the best input scenario. The rationale behind selecting these different input combinations was to systematically assess the impact of lagged streamflow and SSL data on model performance. We progressively added lagged values to capture temporal dependencies and identify the optimal balance between model complexity and predictive accuracy. This step was crucial in understanding how past observations influence SSL predictions and determining whether increasing the number of inputs would improve model performance or lead to redundancy and overfitting.

The MARS method has a simpler structure and is much easier to apply compared to ANFIS-based methods. Therefore, we evaluated whether the MARS method could effectively determine the best input combination for daily SSL prediction before applying more complex models. Table 4 summarizes the training and test outcomes of the MARS models for seven different input scenarios. The results indicate a clear trend: the MARS model’s accuracy improves in both the training and test stages as the number of input variables increases from Scenario 1 to Scenario 7, except for Scenario 4, which exhibits slightly worse performance than Scenario 3. This decline in accuracy could be attributed to the redundancy or noise introduced by additional input variables that do not contribute significantly to the model’s predictive capability. This phenomenon, often called the “curse of dimensionality,” suggests that including too many variables without proper feature selection can negatively impact model performance by increasing variance without substantial gains in accuracy.

The best predictive accuracy was obtained from the MARS model using the full set of input variables, achieving the lowest RMSE (387.6 kg/s), MAE (121.9 kg/s), and MAPE (82.69), along with the highest NSE (0.756) and R² (0.746) in the test stage. These results highlight the potential of MARS as a preliminary feature selection tool capable of identifying optimal input combinations before applying more sophisticated machine learning models. However, since MARS is a piecewise regression-based approach, it may not fully capture the nonlinear and dynamic relationships inherent in suspended sediment transport processes. Therefore, to further enhance predictive performance, we proceed with applying standard ANFIS and three hybrid ANFIS-based methods (ANFIS-PSO, ANFIS-GWO, and ANFIS-GBO) to the same input combinations, ensuring a more comprehensive evaluation of model effectiveness in daily SSL prediction.

Training and testing outcomes of the standard ANFIS models are given in Table 5 for SSL prediction. As observed from the table, increasing input number improves the prediction accuracy of the method; RMSE, MAE, MAPE, NSE and R² range from 388.1 kg/s, 134.1 kg/s, 76.36, 0.466 and 0.467 to 305.6 kg/s, 104 kg/s, 61.86, 0.709 and 0.711, respectively. The last input combination offered the best accuracy in predicting SSL for the training and testing stages. Considering previous SSL data as inputs improves the RMSE, MAPE, MAE, NSE and R² by 8%, 17.7%, 15.4, 8.7% and 8.7% in the test stage, respectively (compare the input cases 4 and 7). These improvements highlight the importance of capturing temporal dependencies in sediment load prediction. The historical SSL data likely contribute crucial contextual information about sediment behavior, particularly during peak events and extreme conditions.

Tables 6–8 illustrate the training and testing results of the hybrid ANFIS method, including ANFIS-PSO, ANFIS-GWO and ANFIS-GBO in predicting daily SSL. It is apparent from the tables that the accuracy of the models is improved by importing more inputs in all models and for both the training and testing stages. The RMSE, MAE, MAPE, NSE and R² range from 376.3 kg/s, 124.7 kg/s, 71.43, 0.502 and 0.505 to 270.2 kg/s, 71.4 kg/s, 47.31, 0.743 and 0.747 for ANFIS-PSO while the corresponding ranges are 366.6–228.3 kg/s, 115.8–63.8 kg/s, 67.21–43.72, 0.531–0.766 and 0.537–0.769 for ANFIS-GWO and 355.6–207.2 kg/s, 111.8–47.5 kg/s, 64.25–36.65, 0.549–0.816 and 0.554–0.822 for ANFIS-GBO in the test stage, respectively.

It is understood from the ranges that the hybrid methods considerably improve the prediction accuracy of the standard ANFIS; the RMSE and MAE of the best ANFIS are improved by 11.6% and 31.3%, 25.3% and 38.7%, 32.2% and 54.3% applying ANFIS-PSO, ANFIS-GWO and ANFIS-GBO in the test stage, respectively. Among the hybrid approaches, the ANFIS-GBO acted as the best model with the lowest RMSE (207.2 kg/s), MAE (47.5 kg/s), MAPE (36.65) and the highest NSE (0.816) and R² (0.822) in daily SSL prediction.

Notably, the consistent improvement across all hybrid models highlights the effectiveness of metaheuristic optimization in refining ANFIS parameters and enhancing model learning. The significant reduction in RMSE and MAE further confirms that these optimization techniques effectively mitigate the limitations of standard ANFIS, such as slow convergence and sensitivity to local minima. It should be noted that the model’s performance is improved by taking into account the previous SSL data as inputs; improvements in testing RMSE are 6.3%, 14.5% and 20.1% for ANFIS-PSO, ANFIS-GWO and ANFIS-GBO (compare the inputs scenarios 4 and 7), respectively. This result suggests that incorporating historical SSL values allows the models to better capture temporal dependencies and sediment transport patterns, particularly in high-fluctuation conditions.

A statistical comparison of the models reveals that while ANFIS provides a robust baseline, including metaheuristic optimization, it significantly enhances its predictive capabilities. Among the hybrid methods, ANFIS-GBO demonstrated superior performance, likely due to its efficient gradient-based search mechanism, which optimally tunes ANFIS parameters while avoiding local minima. The improvement in RMSE and MAE of ANFIS-GBO over standard ANFIS was 32.2% and 54.3%, respectively, highlighting the effectiveness of the optimization approach. These findings emphasize the potential of ANFIS-GBO as a reliable tool for SSL prediction, particularly in complex hydrological systems where conventional models struggle to generalize. Future research should explore additional input features and optimization techniques to refine model accuracy.

Table 9 compares the hybrid ANFIS methods for catching peak sediment loads during the testing stage. The threshold value is 2000 kg/s and there are 18 peaks in the data. The total absolute relative errors clearly show that the ANFIS-GBO offered the best accuracy by providing the lowest error (1010.6).

Furthermore, the results confirm that metaheuristic optimization significantly enhances the ability of ANFIS to predict extreme sediment transport events, which are often challenging due to their irregularity and high variability. The improvements in peak estimation accuracy are notable, with reductions in error of 36.5%, 24.4%, and 22.7% when applying GBO, GWO, and PSO, respectively. This suggests that ANFIS-GBO effectively balances exploration and exploitation in its search space, leading to improved parameter tuning and better adaptation to peak sediment fluctuations.

Capturing peak sediment loads is crucial in hydrological modeling, as extreme sediment transport events can significantly impact river morphology, reservoir sedimentation, and flood risk management. The superior performance of ANFIS-GBO in this aspect highlights its potential as a robust tool for predicting high-impact sediment transport events with improved reliability. Future studies should investigate its applicability to different river basins and extreme hydrological conditions.

The bar plot in Fig. 4 presents the average RMSE, MAE, and MAPE for different ANFIS-based models in predicting suspended sediment load (SSL) during the test period. ANFIS-GBO outperforms all other models, showing the lowest RMSE, MAE, and MAPE among the four models. ANFIS, the standard model, has the highest errors, indicating that optimization significantly improves prediction performance. Hybrid models (ANFIS-PSO, ANFIS-GWO, and ANFIS-GBO) reduce errors, but ANFIS-GBO provides the most significant improvement. Error metrics follow a decreasing trend as optimization algorithms are applied. The findings emphasize the importance of metaheuristic algorithms in improving hydrological predictions.

Fig. 5 illustrates the scatterplots depicting the comparison between the observed and predicted suspended sediment loads during the test period. The scatter graphs indicate that the ANFIS-GBO model exhibits less scattered predictions compared to the other models. To assess the ANFIS-based models comprehensively, Fig. 6 presents a Taylor Diagram, which provides a statistical comparison of the applied models (ANFIS, ANFIS-PSO, ANFIS-GWO, and ANFIS-GBO) for suspended sediment load (SSL) prediction. The diagram evaluates model performance using three key statistical metrics: standard deviation, correlation coefficient (r), and Root Mean Square Error (RMSE). The standard deviation, represented along both axes, indicates the spread of predictions compared to the observed dataset (red dot). The curved green lines represent RMSE values, where models positioned closer to the observed point exhibit lower prediction errors. The blue radial lines indicate the correlation coefficient, showing the strength of agreement between the model predictions and actual observations. A model with a higher correlation, lower RMSE, and a standard deviation similar to the observed data is considered more accurate. As illustrated in Fig. 5, the ANFIS-GBO model (green point) demonstrates a higher correlation and lower RMSE compared to other models, indicating its superior performance in SSL prediction. Conversely, the ANFIS model (yellow point) exhibits relatively lower accuracy, highlighting the effectiveness of optimization techniques in improving predictive performance.

Fig. 7 presents a violin plot comparing the distribution of measured and predicted suspended sediment load (SSL) values across different models, including ANFIS, ANFIS-PSO, ANFIS-GWO, and ANFIS-GBO. The x-axis represents the different models and the observed (measured) values, while the y-axis indicates the SSL values in kg/s. The shape of each violin plot illustrates the probability density of SSL predictions, where wider sections indicate a higher frequency of values, and narrower sections indicate a lower frequency. The red boxplots within each violin highlight the interquartile range (IQR) and median values of SSL predictions. The measured data (first violin) provides a reference distribution, against which the models’ predictive accuracy is assessed. Among the predictive models, ANFIS-GBO exhibits a more consistent and concentrated distribution around the measured SSL values, suggesting improved prediction stability and reduced variance. The baseline ANFIS model shows a broader spread, indicating greater prediction variability. These results reinforce the effectiveness of metaheuristic optimization techniques in refining ANFIS-based models for SSL prediction, with ANFIS-GBO achieving the most reliable predictive performance.

The feasibility of a new hybrid ANFIS-GBO was investigated and compared with standard ANFIS and two hybrid ANFIS methods (ANFIS-PSO and ANFIS-GWO) in predicting daily SSL. The results showed that incorporating more input variables generally improved the prediction accuracy of all models. Adding SSL data from previous days as inputs had a particularly positive impact on the models’ performance. This indicates that historical SSL data contains valuable information for predicting future SSL values. The improvement in prediction accuracy, as measured by RMSE, MAE, and R², highlighted the importance of considering temporal dependencies in the SSL prediction task. On the other hand, adding streamflow data to the MARS model inputs resulted in a degradation of prediction accuracy, as evident from Table 4. This observation aligns with previous studies conducted by researchers such as Adnan et al. [49], Sharafati et al. [50] and Rahgoshay et al. [51]. These studies have consistently reported that an increase in the number of inputs does not necessarily lead to improved prediction accuracy and can have a negative impact on variance. Consequently, this may result in more complex models with poorer prediction performance. For instance, Adnan et al. [49] compared machine learning methods for predicting river flow. They found that including precipitation data as an input did not enhance the accuracy of the models.

Adding SSL data from previous days as inputs had a particularly positive impact on the models’ performance. This indicates that historical SSL data contains valuable information for predicting future SSL values. The improvement in prediction accuracy, as measured by RMSE, MAE, and R², highlighted the importance of considering temporal dependencies in the SSL prediction task. On the other hand, it was found from the models’ outcomes that only streamflow-based input scenarios also provided acceptable accuracy in SSL prediction. As Tao et al. [52] discussed, a crucial concern in modeling SSL is the common reliance on previous SSL values as model inputs. However, this approach poses practical challenges due to the difficulty in accurately measuring SSL data, particularly during extreme events. Alternatively, emphasizing water level data alone, rather than streamflow, holds greater significance in SSL modeling. This is especially pertinent for developing countries where essential variables may be unavailable or lacking at many monitoring stations. By prioritizing water level data, the limitations associated with SSL measurements can be mitigated, facilitating more feasible and effective SSL modeling approaches.

Comparing the different modeling approaches, the hybrid ANFIS methods (ANFIS-PSO, ANFIS-GWO, and ANFIS-GBO) consistently outperformed the standard ANFIS model in terms of prediction accuracy. Among the hybrid methods, ANFIS-GBO emerged as the best-performing model, achieving the lowest RMSE, MAE, MAPE, NSE and the highest R² in daily SSL prediction. This indicates that the combination of ANFIS with the GBO optimization algorithm led to superior performance in capturing the complex relationships between input variables and SSL. The considerable improvements in prediction accuracy obtained by the hybrid ANFIS methods highlighted the effectiveness of using nature-inspired optimization algorithms (PSO, GWO, and GBO) to enhance the learning and adaptation capabilities of the ANFIS model. These algorithms enabled the models to search the solution space more effectively and find better parameter configurations, leading to improved predictions.

While the evaluation metrics (RMSE, MAE, MAPE, NSE and R²) demonstrate the improved performance of ANFIS-GBO compared to other models, statistical validation is necessary to confirm the significance of these improvements. Table 10 presents the results of the paired t-test, comparing the performance of each model against ANFIS. The results indicate that all optimization-based ANFIS models significantly outperform standard ANFIS (p < 0.05). ANFIS-GWO exhibited the highest statistical significance (t = −7.120, p = 2.584E−12), confirming its superior predictive ability. The negative t-values suggest that the optimized models have lower mean prediction errors than the baseline ANFIS model. These results validate the effectiveness of using metaheuristic optimization techniques in SSL modeling, with ANFIS-GBO providing the most stable and accurate predictions.

Overall, the study demonstrated the effectiveness of ANFIS-based models, especially the hybrid approaches, in predicting daily suspended sediment load in the Bailong River Basin. Incorporating historical SSL data and utilizing nature-inspired optimization algorithms significantly improved the models’ performance. As shown in Table 8, ANFIS-GBO achieved the highest accuracy with the lowest RMSE and MAE due to its efficient optimization process and reduced computational cost. Compared to other models, ANFIS-GBO required fewer control parameters while maintaining a robust prediction capability. These findings have implications for water resource management and environmental planning, as accurate prediction of SSL can aid in assessing and mitigating the impacts of sediment transport on river systems and associated ecosystems.

This study evaluated the effectiveness of hybrid Adaptive Neuro-Fuzzy Inference System (ANFIS) models integrated with different optimization techniques—Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and the Gradient-Based Optimizer (GBO)—for predicting daily suspended sediment load (SSL). The findings demonstrated that hybrid models significantly improved predictive accuracy compared to the standard ANFIS model. The key conclusions of this study are summarized as follows: –

Among the tested models, the ANFIS-GBO method achieved the highest prediction accuracy, with the lowest RMSE (207.2 kg/s), MAE (47.5 kg/s), and MAPE (36.65) and the highest NSE (0.816) and R² (0.822), confirming its capability to effectively model the nonlinear and dynamic behavior of SSL.

However, despite the promising results, this study has some limitations. The reliance on historical SSL data may pose practical challenges due to the difficulty in measuring SSL accurately, particularly during extreme weather events. Additionally, the study was conducted on a single river basin, and further validation is required to assess the generalizability of the proposed approach to different hydrological conditions.

To further improve SSL prediction accuracy, future studies should: –

Incorporate additional environmental and climatic variables such as rainfall, temperature, and land-use characteristics.

Overall, this study contributes to advancing AI-driven hydrological modeling and provides a strong foundation for further research in sediment transport prediction.