Multilayer perceptron integrated with Kolmogorov–Arnold networks for predicting the water level in the Gezhouba Sanjiang downstream approach channel

Inland waterway navigation is recognized as a more environmentally sustainable mode of transport compared to road or rail (Mihic et al. 2011). It produces fewer carbon emissions per ton-mile of cargo, thereby significantly reducing the overall carbon footprint relative to trucking and railway systems (Liu et al. 2015). Inland waterway transport is also highly cost-effective, especially for bulk commodities such as coal, grain, and construction materials (Trivedi et al. 2021). It facilitates the movement of substantial cargo volumes over extended distances with lower fuel consumption and reduced operational costs relative to road and rail transport (Irannezhad et al. 2018). Inland river navigation depends on a network of navigable waterways, encompassing natural rivers, lakes, man-made canals, and ship locks (Durajczyk & Drop 2021). The development and maintenance of these infrastructures are crucial for ensuring safe and efficient navigation. The ship lock is an essential engineering structure designed to raise and lower vessels between stretches of water of different levels on rivers and canals (Negi et al. 2024). It enables ships to navigate through varying terrains, including regions with significant elevation changes, such as those with dams or varying river gradients, thereby ensuring a continuous and navigable inland waterway (Zhang et al. 2019). The water levels in the DAC often dictate the maximum permissible draught of vessels (Wan et al. 2020). Therefore, accurate water level prediction is essential for guiding the proper loading of vessels, thereby enhancing navigation safety and capacity of ship locks.

With the advancement of artificial intelligence (AI) and the increasingly interdisciplinary nature of research, time series modeling and water level prediction methods have evolved from traditional physics-based and conceptual models to AI-based models, such as support vector regression (SVR), random forest (RF), fuzzy logic, heuristic algorithm (HA), and multilayer perceptron (MLP) (Behzad et al. 2010; Gurbuz et al. 2024). Over the past decades, these models have been successfully applied in time series modeling and water level forecasting. Based on the Water Cycle Optimization Algorithm (WCA) and Moth-Flame Optimization Algorithm (MFO), Adnan et al. (2021) developed a novel hybrid adaptive neuro-fuzzy inference system (ANFIS–WCAMFO) for monthly evapotranspiration. The results suggested this hybrid HA is accurate for monthly ET₀ prediction in a data-limited tropical humid region. Adnan et al. (2020a, b) employed a least-square SVR combined with a gravitational search algorithm (LSSVR–GSA) and the dynamic evolving neural-fuzzy inference system (DENFIS) to model ET₀ using limited data. According to their findings, the temperature or extraterrestrial radiation-based LSSVR–GSA models outperformed the DENFIS and M5 regression tree (M5RT) in estimating monthly ET₀. Adnan et al. (2020a, b) validated the feasibility of group method of data handling neural network (GMDHNN), multivariate adaptive regression spline (MARS), and M5RT for estimating monthly ET₀, and compared these models with empirical formulations. Their findings indicated that GMDHNN generally achieved the highest accuracy.

In recent years, MLP has emerged as a core element of DL algorithms, widely used by researchers to analyze and reveal patterns and characteristics in hydrological data (Pouyanfar et al. 2018; Ansarifard et al. 2024). Zhu et al. (2020) utilized feedforward neural networks and recurrent neural networks to predict monthly water levels in multiple temperate lakes, achieving robust performance. Similarly, Barzegar et al. (2021) coupled the boundary-corrected maximal overlap discrete wavelet transform with a CNN–LSTM model, demonstrating that their hybrid model surpassed SVR and RF models in multiscale lake water level forecasting. Furthermore, Sun et al. (2022) established backpropagation artificial neural network (BPANN), LSTM, and autoregressive integrated moving average (ARIMA) models to predict groundwater levels across five zones with distinct hydrogeological properties. Their comparative analysis revealed that the BPANN and LSTM models outperformed the ARIMA model across all five zones (Sun et al. 2022). Adnan et al. (2024) established advanced deep learning (DL) hybrid models to accurately forecast monthly and peak streamflow in the Gilgit River Basin, and compared them with traditional single models. Their findings demonstrated that CNN–BiGRU and CNN–BiLSTM significantly outperformed LSTM and GRU.

The MLP models are based on the universal approximation theorem, while Liu et al. (2025) proposed the KAN, which, although inspired by the Kolmogorov–Arnold representation theorem, does not strictly conform to it. KAN models are distinguished by their parameter efficiency and superior capability to manage complex special functions and high-dimensional data. By applying sparsification regularization, the complexity of KAN can be reduced, thereby enhancing their generalization performance. Moreover, combining KAN with traditional MLP models may enhance performance in nonlinear regression tasks compared to using MLP alone.

Many scholars have pointed out the potential limitations of MLP models in handling non-stationary processes, noting that noise in hydrologic time series often degrades the prediction accuracy (Ebrahimi & Rajaee 2017; Freire et al. 2019; Bhardwaj et al. 2020; Hao et al. 2022). To address these challenges, various signal decomposition methods have been applied in data preprocessing, including time-varying filter empirical mode decomposition (TVFEMD) (Bokde et al. 2018; Eriksen & Rehman 2023). The TVFEMD employs a time-varying filter to mitigate the mode mixing problem inherent in empirical mode decomposition (EMD). To adapt to non-stationary signals, the cut-off frequency of this filter varies over time. Through the successive application of TVF, multiple local higher frequency (LHF) components and one local lower frequency (LLF) component can be extracted. The properties of these LHF components are nearly identical to the intrinsic mode functions (IMFs) obtained through EMD. Furthermore, TVFEMD increases the dimensionality of input features, and CNN is particularly effective in extracting features from these decomposed components (Jamei et al. 2023).

There is a strong and direct correlation between the water levels in the DAC and the dam discharge, and historical water levels also influence the current levels, due to the periodic emptying of the ship lock. Moreover, the significance of different feature values at various time points is not uniform, and these differences can be captured using a weight matrix derived from attention mechanisms. Numerous studies have demonstrated that attention mechanisms can enhance the predictive capabilities of MLP (Ding et al. 2020; Chang et al. 2022; Noor et al. 2022; Tao et al. 2022). Guo et al. (2023) proposed a novel external attention (EA) mechanism, which only has linear complexity and may yield superior results compared to the self-attention (SA) mechanism.

The major challenge in this study is the effect of water conservancy hub operation on water level prediction, which is seldom considered in previous studies (Yang et al. 2020; Tu et al. 2021). The hydropower station outputs are routinely regulated to satisfy peak shaving demands, which results in outflow and water level in the DAC changing frequently. Moreover, due to seasonal influences, the frequency components of the water level series vary over time, exhibiting pronounced non-stationary behavior. Consequently, an effective prediction model must capture the different fluctuation characteristics represented by multiple frequency components and demonstrate robust generalization capabilities. To overcome these difficulties and ensure forecasting accuracy, this study aims to develop an innovative hybrid model for predicting water levels in the DAC of the multi-line ship lock. Hybrid models have been extensively applied in many fields. In the field of computer networks, Afrifa et al. (2023a, b) developed a stacking ensemble model incorporating RF, decision tree (DT), and generalized linear model (GLM) to detect botnets in computer network traffic. However, choosing appropriate hyperparameters for RF remains a significant challenge in practice, often making it difficult to avoid overfitting. The DT is similarly vulnerable to overfitting, particularly when it becomes excessively deep with numerous branches. Overfitting can lead to poor generalization on unseen data, and reduce the predictive performance of models. Moreover, GLM may struggle to capture highly intricate nonlinear relationships between input and output features and require the correct choice of a link function to ensure valid results. To make meaningful and novel improvements in the proposed hybrid models, the MLPs serve as the backbone, with KAN integrated to simplify model structure, mitigate overfitting and enhance the generalization ability. The TVFEMD and EA were employed as data preprocessing techniques. TVFEMD extracted information across multiple frequency components, enabling the hybrid models to capture intricate nonlinear relationships between inputs and outputs. Meanwhile, the EA distinguished the effects of various features at different time steps on the prediction results. Additionally, the hybrid models incorporated CNN to extract features from the input matrix. The remainder of this paper is organized as follows: Section 2 introduces the study area and data. Section 3 details the hybrid AI model, KAN, TVFEMD, and EA, the development of the hybrid models, and the model evaluation criteria. Section 4 presents the experimental results of the models. Section 5 discusses the evaluation and comparison of the proposed models, along with a critical analysis. Finally, Section 6 provides concluding remarks and suggests directions for future research.

AI refers to the simulation of human intelligence in machines that are programmed to think, learn, and solve problems like humans. It involves the development of algorithms and systems that can process data, recognize patterns, make decisions, and even improve over time through experience. In recent years, as a data-driven approach, AI models have achieved significant success in water level prediction according to the literature survey. Various AI models have been developed for water level forecasting. In addition to MLP, these include the Bayesian vine copula (BVC), ANFIS, support vector machine (SVM), extreme gradient boosting (XGB), RF, and M5 pruned (M5P).

Yadav & Eliza (2017) utilized a hybrid wavelet–SVM model to predict the water level fluctuation in Loktak Lake. Based on prediction results, the RMSD was 0.1547 and the R² was 0.9657. Li et al. (2016) compared the performance of RF and SVM in forecasting lake water levels, as exemplified by the Poyang Lake case study. Their results indicated that RF outperformed SVM in daily forecasting in terms of RMSD and R². Nguyen et al. (2021) developed hybrid models combining a genetic algorithm (GA) with XGB for hourly water level prediction. Their comparison showed that GA–XGBoost models outperformed RF in multistep-ahead predictions, with relative errors ranging from 2.18 to 9.21%, compared to 3.76 to 10.41% for RF. Nhu et al. (2020) predicted the daily water level of Zrebar Lake in Iran using M5P, RF and reduced error pruning tree. Their findings showed M5P performed best among the evaluated methods by comparing RMSD, MAE and R². Pham et al. (2022) applied the ANFIS coupled with the whale optimization algorithm to predict monthly water levels at Titicaca Lake. For the optimal scenario, MAE was 0.06 m, RMSD was 0.08 m, and R² was 0.96. Liu et al. (2021) proposed a hybrid BVC model for daily and monthly water level prediction. Their findings demonstrated that the hybrid BVC approach produced more reliable predictions than ANFIS, as evaluated using RMSD, R², and the Nash–Sutcliffe efficiency coefficient.

By reviewing 117 relevant papers, Afrifa et al. (2022) concluded that machine learning has been more popular than mathematical model approaches in recent years. Based on their findings, machine learning has lowered computational complexity and shortened model training times. Furthermore, they observed that researchers frequently use the SVM, RF and ANN in their studies. However, little emphasis is placed on other algorithms, such as ensemble and hybrid models. Afrifa et al. (2023a, b) developed a hybrid model (HM) combining Bayesian RF (BRF), Bayesian SVM (BSVM), and Bayesian ANN (BANN) to accurately forecast farmland groundwater levels. In the forecasting results of the HM, the R² ranged from 0.9051 to 0.9679, RMSD ranged from 0.0653 to 0.0727 and MAE ranged from 0.0121 to 0.0541. Their research provides valuable insights into agricultural water management.

Over the past decades, MLP has served as the foundation for numerous DL models, demonstrating strong performance in nonlinear regression, classification, and clustering tasks (Zhang et al. 2018). However, MLPs exhibit certain limitations. First, despite the development of approaches such as ResNet, the leaky-ReLU activation function, and batch normalization can ameliorate the gradient vanishing and exploding problems, these issues cannot be entirely solved (Yilmaz & Poli 2022). Second, MLPs generally adopt fully connected layers. Although dropout layers could be used to simplify the network architecture of MLPs, insufficient weight parameters in the linear transformation of neurons could impair the representational capacity of neural networks, leading to lower parameter efficiency. Third, while MLPs could approximate any function based on the universal approximation theorem, they struggle to capture long-term dependencies in input sequences, and various enhanced MLPs have not fully resolved this issue. Given the deficiencies mentioned above, KAN could be a promising alternative to MLP. Although KAN also utilizes fully connected layers, it differs significantly from MLP in its network structure:

• (1) KAN selects learnable B-spline curves as activation functions, which are arranged on edges and consist of B-spline basis functions, whereas the activation functions in MLP models are fixed and placed on neurons;

• (2) In KAN neurons, input signals are simply added without any nonlinear transformation, whereas in MLP neurons, the results of linear transformation are fed to nonlinear activation functions;

• (3) KAN abandons linear weight matrices, instead merging weight parameters into the activation functions.

Guo et al. (2023) proposed TVFEMD to address the mode mixing problem, alleviate poor adaptability of parameters in EMD, and enhance robustness at low sampling rates. Previous studies have demonstrated that TVFEMD could improve the predictive performance of MLP models (Zhu et al. 2023). The TVFEMD utilizes Hilbert spectrum local narrowband signals instead of IMFs. The determination of local narrowband signals is based on instantaneous bandwidth, where the rate of amplitude change is significantly slower than that of phase change. The TVFEMD employs time-varying filtering to filter out low-frequency components (LFCs), with the filtered local high-frequency signals considered as local narrowband signals. The TVFEMD could decompose original time series into several LHF components and one LLF component. The data processing steps based on TVFEMD are as follows:

• (1) Estimating the local cut-off frequency by instantaneous amplitude and frequency information of the signal;

• (2) Designing an adaptive time-varying filter based on the local cut-off frequency;

• (3) Applying the time-varying filter to remove the LFCs from the original signal, with the remaining high-frequency component considered as the local narrowband signal.

The SA proposed by Vaswani et al. (2017) has become a broadly adopted technique in time series forecasting over the past few years. Xiong et al. (2022) utilized SA to dynamically assign weights to physical attribute data, effectively addressing the model's inability to differentiate the varying importance of input features. Niu et al. (2022) introduced three attention mechanism modules into the hidden state of BiGRU through mapping weight and a learning parameter matrix to enhance the influence of key information. In comparison to SA, the EA propounded by Guo et al. (2023) is more efficient, and exhibits superior generalization capability.

As mentioned in Section 2, the original dataset consisted of Miaozui water level and Gezhouba dam outflow, recorded at 6-hour intervals between January 1, 2022, and May 18, 2024. After max–min normalization, for hybrid models without TVFEMD, the normalized dataset was transformed into a supervised learning format through time delay. The first two samples of the dataset in the supervised learning format are shown in Table 2. For hybrid models with TVFEMD, the normalized Miaozui water level time series was decomposed into 11 LHF components, denoted as LHF1(L), LHF2(L), LHF3(L), … , LHF10(L) and LHF11(L) in descending order of frequency, along with one LLF component, denoted as LLF(L). Similarly, normalized Gezhouba dam outflow time series was decomposed into 11 LLF components, denoted as LHF1(Q), LHF2(Q), LHF3(Q), … , LHF10(Q) and LHF11(Q) in descending order of frequency, and one LLF component, denoted as LLF(Q). These components formed a new decomposed dataset, and then it was transformed into the supervised learning format through time delay. The first two samples of the decomposed dataset in the supervised learning format are presented in Table 3. Additionally, to verify the impact of variations in outflow on water levels, the prediction results using input features with and without outflow data were compared.

The dataset comprises 3,443 samples, of which 80% served as the training set and the remaining 20% served as the test set. Model parameters were determined and optimized on the training set, and the validation of these models was implemented on the test set. The key hyperparameters for EA, MLP, and KAN are presented in Table 4. To optimize the performance of the model, multiple sets of hyperparameters were experimented. To validate the robustness of the proposed hybrid models, they were compared with advanced models, including BRF, GA–XGBoost and SVM optimized by the seagull optimization algorithm (SOA–SVM).

Numerous evaluation metrics have been presented for assessing the performance of DL models. Afrifa et al. (2023a, b) utilized the models' area under the curve-receiver operating characteristics to evaluate the performance of models in detecting and classifying benignancy, malignancy, and normality in breast cancer cases. Afrifa et al. (2023a, b) applied MAE, RMSD, MAPE, median absolute error, and maximum error to assess the performance of models used for unemployment rate prediction. Adnan et al. (2024) used CA to compare the ability of grey wolf optimization–LSTM (GWO–LSTM), ANN, ANFIS, and GWO–SVM in the monthly streamflow prediction. Adnan et al. (2019) used CA to evaluate the performance of optimally pruned extreme learning machine method, ANFIS–particle swarm optimization, MARS and M5RT on forecasting the daily streamflow of Fujiang River.

The improvements presented in Table 5 indicated the degree to which TVFEMD, EA, CNN, and KAN enhanced the performance of LSTM and GRU models, as evidenced by the evaluation criteria. For LSTM-based hybrid models, using a single optimization method, EA provided the most significant boost, followed by KAN and CNN. When two optimization methods were employed, EA and KAN showed the greatest improvement, followed by TVFEMD and EA, while CNN and KAN ranked third. For GRU-based hybrid models, using one optimization method, the improvements from EA, CNN, and KAN were comparable and not significant. When two techniques were combined, EA–KAN and TVFEMD–EA demonstrated the greatest enhancement, far exceeding that of CNN–KAN. Among the LSTM-based models, TVFEMD–EA–CNN–LSTM–KAN outperformed all others, with TVFEMD–EA–LSTM–KAN following closely. Similarly, for GRU-based models, TVFEMD–EA–CNN–GRU–KAN ranked highest, followed by TVFEMD–EA–GRU–KAN. These results suggested that the cumulative application of optimization methods generally led to better model improvement, and these methods reinforced rather than conflicted with one another.

To illustrate the key hyperparameter tuning process, Table 6 presents the variation in model performance across different hyperparameter combinations, using the TVFEMD–EA–CNN–GRU–KAN as an example. Reducing the number of neurons per GRU layer had minimal impact on prediction performance. MAE increased by 4.58%, RMSD increased by 5.23%, MAPE increased by 6.25%, CA increased by 4.85%, and almost no change in R². However, simultaneously reducing the number of neurons per GRU layer and the number of GRU layers significantly degraded model performance. MAE increased by 58.78%, RMSD increased by 62.09%, MAPE increased by 59.38%, CA increased by 4.85%, and R² decreased by 5.25%. The other three sets of parameters had a moderate effect on model performance. For EA, reducing the hyperparameter S from 64 to 32, the MAE increased by 16.03%, RMSD increased by 13.73%, MAPE increased by 15.63%, CA increased by 13.59%, and R² decreased by 0.31%. For the hyperparameters in KAN, reducing the grid size (G) from 20 to 10 and the order of B-spline basis function (k) from 3 to 2 resulted in a 22.90% increase in MAE, a 20.92% increase in RMSD, a 25.00% increase in MAPE, a 20.39% increase in CA, and a 0.10% decrease in R². Keeping G and k constant, reducing both the number of neurons per KAN layer and the number of KAN layers resulted in a 29.01% increase in MAE, a 26.80% increase in RMSD, a 34.38% increase in MAPE, a 19.42% increase in CA and a 0.10% decrease in R². This suggested that the fitting ability of KAN was a critical determinant of hybrid model performance.

In Figure 14, a comparison of the prediction results between the LSTM–KAN and LSTM, as well as the GRU–KAN and GRU, indicated that KAN effectively enhanced the ability of MLP to fit high-frequency and high-amplitude fluctuations. This might be due to the superior capability of KAN in fitting complex high-dimensional functions. A comparison of the prediction results between the TVFEMD–EA–LSTM–KAN and LSTM–KAN, as well as the TVFEMD–EA–GRU–KAN and GRU–KAN, indicated that although TVFEMD increased the number of input features, the EA emphasized the most relevant information for prediction and ensured that hybrid models identified key features from the inputs. A comparison of the prediction results between the TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–LSTM–KAN, as well as the TVFEMD–EA–CNN–GRU–KAN and TVFEMD–EA–GRU–KAN, revealed that CNN enhanced the ability of models to extract effective features and fully utilized the information from the decomposed components, thereby further improving the fitting performance.

Accurate forecasting of the Miaozui water level is essential for the navigation authority to establish dynamic draught limit standards in the Gezhouba Sanjiang DAC. To enhance prediction precision, this study built a series of hybrid models. The CNN and KAN were combined with LSTM or GRU to form the main model backbone, while TVFEMD and EA were employed to preprocess the water level and outflow input series. The contributions of TVFEMD, EA, CNN, and KAN in enhancing the performance of LSTM and GRU were analyzed. The prediction performance of proposed hybrid models was compared using multiple evaluation indices, including RMSD, MAE, MAPE, R² and CA. Taylor diagrams were used to visually compare the R and RMSD. Violin plots were employed to illustrate the distribution and outliers of predicted and observed values. Moreover, TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN were compared with new advanced models to validate their predictive accuracy and robustness. The main conclusions are as follows:

• (1) Due to routine hydropower output regulations and lock chamber emptying processes, the Miaozui water level exhibits complex fluctuations, and plain LSTM and GRU models are incapable for prediction. Furthermore, existing research on water level prediction seldom considers the effects of water conservancy hub operations.

• (2) The hydraulic project regulations could cause significant changes in outflow, thereby affecting the Miaozui water level. This study innovatively collected the Gezhouba outflow data as an input series and compared predictive performance with and without its inclusion. For TVFEMD–EA–CNN–LSTM–KAN, including outflow as an input feature reduced the MSE and RMSD by 20%, the MAPE by 21%, the CA by 22% and increased R² by 2%. Similarly, for TVFEMD–EA–CNN–GRU–KAN, including the outflow feature reduced the MSE by 21%, the RMSD by 20%, the MAPE by 21%, and the CA by 20%. The inclusion of the outflow in the input features notably enhanced prediction accuracy.

• (3) TVFEMD–EA–CNN–GRU–KAN outperformed all other hybrid models. It achieved a MAE of 0.131 m, a RMSE of 0.153 m, a MAPE of 0.322%, an R² of 0.971 and a CA of 0.103. In Figure 16, the R between predicted and observed values reflected its ability to accurately predict water level fluctuation trends. In Figure 18, the comparisons in violin plots indicated its forecasting results exhibited an apparently similar data distribution to the observed water level. The proposed hybrid models were compared with BRF, GA–XGBoost, and SOA–SVM to validate their robustness and accuracy. The evaluation criteria in Table 7 and the visual comparisons in Figures 23 and 24 declared that TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN achieved higher prediction accuracy and the distributions of their predicted results were more similar to the observed values.

• (4) Due to its superior ability to fit complex high-dimensional functions, KAN effectively enhanced the ability of MLPs to fit high-frequency and high-amplitude fluctuations. The EA ensured hybrid models focused on the most crucial information for accurate prediction. The integration of CNN effectively reduced the discrepancy between the loss function values on the training and test sets, contributing to a more robust model. Performance metric comparisons indicated that KAN, TVFEMD, EA, and CNN effectively enhanced the prediction capability of MLPs. The optimization methods reinforced rather than conflicted with others.

This research provided valuable support for the establishment of reasonable dynamic draught limit standards and could serve as a reference for other hydraulic projects with hydropower generation and navigation. However, the performance of hybrid models when applied to other projects required further exploration. And these models are only applicable to predict the water level at critical locations, but not the whole DAC. These limitations need further study.

We acknowledge the Three Gorges Navigation Authority for their consent, cooperation, and regulations regarding data confidentiality in deploying water level monitoring equipment. We also thank Professor Jianfeng An for his contributions to field equipment installation and figure preparation during revision.

This research was supported by the National Key R&D Program of China (Grant No. 2023YFC3206101) and the Specialized Research Funding of Nanjing Hydraulic Research Institute (Grant Nos Y123012 and Y122007). These funds facilitated the purchase of water level measurement and data transmission equipment, as well as travel expenses for site installation.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.