The water level in the downstream approach channel (DAC) of the multi-line ship lock exhibits intricately nonlinear fluctuations. This research integrated Kolmogorov–Arnold networks (KANs), convolutional neural networks (CNNs), external attention (EA), and time-varying filter empirical mode decomposition (TVFEMD) with long short-term memory (LSTM) or gate recurrent unit (GRU) to enhance prediction performance. Compared to the GRU, mean absolute error (MAE) of TVFEMD–EA–CNN–GRU–KAN decreased by 46% to 0.131 m, root mean square deviation (RMSD) by 46% to 0.153 m, mean absolute percentage error (MAPE) by 45 to 0.322%, combined accuracy (CA) index by 49% to 0.103, and coefficient of determination (R2) increased by 7% to 0.971. Compared to LSTM, MAE of TVFEMD–EA–CNN–LSTM–KAN decreased by 52% to 0.140 m, RMSD by 51% to 0.164 m, MAPE by 52 to 0.345%, CA by 55% to 0.111, and R2 increased by 11% to 0.968. A novel contribution was considering the influence of outflow changes caused by hydraulic project regulations in water level prediction, which was rarely addressed in existing studies. By collecting outflow data as one of the input features, the prediction accuracy of hybrid models was enhanced substantially. For TVFEMD–EA–CNN–GRU–KAN and TVFEMD–EA–CNN–LSTM–KAN, including outflow among the input features decreases CA by 20 and 22%, respectively.

  • Novel integration of KAN, CNN, EA, TVFEMD, and LSTM or GRU for water level prediction.

  • Focus on the influence of hydraulic project regulations on the water level.

  • Accurate DAC water level prediction as anticipated outcomes.

  • Practical applications for authorities of ship locks to establish dynamic draught limit standards.

  • Replicable methodology with potential for other hydraulic projects with hydropower generation and navigation.

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.

The Gezhouba project is located in Yichang City, Hubei Province, China, approximately 38 km downstream of the Three Gorges project. Its geographical coordinates are 30.738°N and 111.275°E. Often referred to as ‘the first dam on the Yangtze River’, the Gezhouba project is the earliest large-scale water conservancy project constructed on the mainstream of the Yangtze River (Liu et al. 2013). The geographical location and an aerial view of the Gezhouba project are illustrated in Figure 1. The Gezhouba project comprises three ship locks: No. 1 on the right bank and Nos 2 and 3 on the left bank. Figure 2 illustrates the trends in cargo volume and ship throughput through the Gezhouba ship locks from 2012 to 2023. The annual number of ships passing through the Gezhouba ship locks exceeded 40,000 vessels from 2012 to 2023 and exceeded 49,000 in 2023. During this period, the cargo volume exhibited an upward trend, achieving an average annual growth rate of 5.87%. In 2023, the total cargo volume of these three ship locks exceeded 170 million tons. The flood discharge and power generation dam sections are situated at the right of the No. 2 ship lock. The installed electricity generation capacity is 2.735 GW, with a maximum flood discharge of 110,000 m3/s. Ship lock Nos 2 and 3 share the DAC known as Sanjiang, where the chamber emptying processes occur. The Sanjiang DAC extends 4 km in length, and the minimum width is 120 m. According to Sanjiang DAC terrain data, the minimum water depth is found at the ship entrance, known as Miaozui. The location of Miaozui is shown in Figure 1.
Figure 1

Geographical location and an aerial view of the Gezhouba project (right is downstream in the aerial view). The blue line traces the mainstream of the Yangtze River. An orange inverted triangle marks the location of the Three Gorges Project. A red five-pointed star highlights the location of the Gezhouba Project. Province names are displayed in bold dark green, and provincial capital names are labeled in light green. The location of the Miaozui and ship lock Nos 2 and 3 are indicated in the aerial view.

Figure 1

Geographical location and an aerial view of the Gezhouba project (right is downstream in the aerial view). The blue line traces the mainstream of the Yangtze River. An orange inverted triangle marks the location of the Three Gorges Project. A red five-pointed star highlights the location of the Gezhouba Project. Province names are displayed in bold dark green, and provincial capital names are labeled in light green. The location of the Miaozui and ship lock Nos 2 and 3 are indicated in the aerial view.

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Figure 2

Trends in cargo volume and throughput through the Gezhouba ship locks.

Figure 2

Trends in cargo volume and throughput through the Gezhouba ship locks.

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The advantages of adopting this case study include practical value and challenges of accurate prediction. On the one hand, during dry seasons, the Miaozui water level may limit the draughts of ships passing through ship lock Nos 2 and 3. To take advantage of water depth fully and ensure navigation safety, the navigation authority dynamically controls the draughts of ships entering the Sanjiang DAC based on Miaozui water level prediction. Moreover, in recent years, vessels passing through the Gezhouba ship locks have exhibited a pronounced trend toward increased size. Figure 3 depicts the percentage of ships with rated deadweight tonnage less than 1,500 tonnes and greater than 4,000 tonnes passing through the Gezhouba ship locks. The percentage of ships with rated deadweight tonnage less than 1,500 tonnes exhibited a downward trend, while those exceeding 4,000 tonnes displayed an upward trend. Vessels with larger rated deadweight tonnage generally have deeper draughts, necessitating precise water level predictions at Miaozui to avoid stranding and ensure navigation efficiency. Therefore, accurate forecasting of the water level at Miaozui is meaningful and practical.
Figure 3

Percentage of ships with rated deadweight tonnage of <1,500 tonnes and >4,000 tonnes.

Figure 3

Percentage of ships with rated deadweight tonnage of <1,500 tonnes and >4,000 tonnes.

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On the other hand, unlike many other case studies of water level prediction in navigation channels, the water level at Miaozui is influenced by routine hydropower station regulation and lock chamber emptying processes. The routine hydropower station regulation leads to changes in outflow, and the outflow from the Gezhouba dam significantly influences the water level at Miaozui, according to Figure 4. A single emptying process of ship lock No. 2 and the resulting variations in Miaozui water level are shown in Figure 5. Before the emptying process began, the Miaozui water level was already rising, and after it commenced, the rate of increase became faster. Obviously, lock chamber emptying processes also have a notable impact on the Miaozui water level. These factors certainly make precise water level prediction more challenging and place higher demands on prediction model performance. The results of this case study enable a more rigorous evaluation of the prediction models.
Figure 4

Outflow from the Gezhouba dam and Miaozui water level from January 1, 2022 to May 18, 2024.

Figure 4

Outflow from the Gezhouba dam and Miaozui water level from January 1, 2022 to May 18, 2024.

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Figure 5

A single emptying process of ship lock No. 2 and the resulting Miaozui water level variations. This emptying process began at 1:54 a.m. on 26 September 2023.

Figure 5

A single emptying process of ship lock No. 2 and the resulting Miaozui water level variations. This emptying process began at 1:54 a.m. on 26 September 2023.

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A water level measurement device was installed at Miaozui, which was used to collect Miaozui water level data, and the corresponding wireless data transmission module is shown in Figure 6. Historical Gezhouba dam outflow and Miaozui water level were selected as input features for the prediction models, and the Miaozui water level is the output feature. The outflow data of the Gezhouba dam was provided by the local navigation authority. The water level data at Miaozui and the outflow data of the Gezhouba dam were collected over the period from January 1, 2022, to May 18, 2024, at a sampling interval of 6 h. The first five and last samples of the original outflow and water level data are presented in Table 1.
Table 1

First five and last samples of the original outflow and water level data

TimeMiaozui water level (m)Outflow from the Gezhouba dam (m3/s)
January 1, 2022, 02:00 39.66 6,154 
January 1, 2022, 08:00 39.50 6,260 
January 1, 2022, 14:00 40.10 7,790 
January 1, 2022, 20:00 40.07 7,730 
January 2, 2022, 02:00 39.64 6,300 
… … … 
May 18, 2024, 08:00 43.02 14,700 
TimeMiaozui water level (m)Outflow from the Gezhouba dam (m3/s)
January 1, 2022, 02:00 39.66 6,154 
January 1, 2022, 08:00 39.50 6,260 
January 1, 2022, 14:00 40.10 7,790 
January 1, 2022, 20:00 40.07 7,730 
January 2, 2022, 02:00 39.64 6,300 
… … … 
May 18, 2024, 08:00 43.02 14,700 
Figure 6

Wireless water level data transmission module at Miaozui.

Figure 6

Wireless water level data transmission module at Miaozui.

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Hybrid AI model

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.

Kolmogorov–Arnold network

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.

During the training process, MLP could improve model performance by increasing the number of network layers and neurons per layer. However, a new MLP structure requires training from scratch. In contrast, KAN allows for a more efficient training process by initially using fewer parameters and training a simple network structure, with the option to increase the number of parameters through grid refinement while continuing training from the previous network. Moreover, in tasks involving the fitting of complex high-dimensional functions, the performance of KAN may continue to improve as the number of parameters increases, whereas MLP performance may plateau. This indicates KAN could maintain low error rates and exhibit superior generalization ability. This study integrated KAN with MLP to enhance prediction accuracy. The structure of a four-layer KAN is illustrated in Figure 7, with the amounts of neurons in each layer being 3, 3, 2 and 2, respectively.
Figure 7

The structure of a [3, 3, 2, 2] KAN.

Figure 7

The structure of a [3, 3, 2, 2] KAN.

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In Figure 7, each activation function ϕi,j,k is indexed by (l, i, j), where l denotes the layer index, i denotes the input neuron index, j denotes the output neuron index. The variable represents the output of activation function ϕi,j,k, while xl,i represents the ith neuron in the lth layer, and yi denotes ith neuron in output layer. The activation function (B-spline curves) ϕ(x) can be expressed by Equations (1)–(4):
(1)
(2)
(3)
(4)
where b(x) is referred to as the base function, with a default setting of SiLU(x). Although it can be configured to another function prior to training, it is not trainable. The parameter is a learnable coefficient for b(x), where nin denotes the number of input features, and , with a default value of . The parameter is a coefficient for spline(x), with a default value of 1. In Equation (2), G represents the grid size, k is the order of B-spline basis function Bi(x), and ci is a learnable coefficient. The grid refinement could enhance the ability of KAN to fit functions. As an example, we took the ϕ1,2,1 from Figure 7 to illustrate how to obtain a more fine-grained KAN, as shown in Figure 8.
Figure 8

Grid refinement process of KANs.

Figure 8

Grid refinement process of KANs.

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Time-varying filter empirical mode decomposition

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 determination of the local cut-off frequency is crucial in the TVFEMD. The computation and utilization of the local mean function in the EMD can be regarded as a low-pass filtering process. The TVFEMD adopts B-spline as a low-pass filter to screen out LFCs. Compared to traditional linear filters, the cut-off frequency of B-spline filter is easier to be constructed and can adaptively vary over time, making it more effective in handling nonlinear and non-stationary signals. The mathematical definition of the B-spline filter is as follows:
(5)
(6)
where [ · ] m and [ · ] m are the down-sampling and up-sampling operations by a factor m, respectively. The symbol * represents convolution operator, is a pre-filter, is a post-filter, βn denotes a B-spline function of order n, m is the step size of the knot sequence, t refers to time steps. The local cut-off frequency of B-spline filter depends on the input sequences, and the estimation process of it could be delineated as follows:
  • (1) By applying the Hilbert transform to the input sequence, the instantaneous frequency φ(t), phase ϕ(t) and amplitude A(t) could be obtained. Subsequently, the local maxima and minima, A({tmax}) and A({tmin}), are determined.
    (7)
  • (2) At these maxima and minima, the interpolation curves β1(t) and β2(t) can be constructed.

  • (3) The instantaneous amplitude a1(t) and a2(t) can be calculated using Equations (8) and (9).
    (8)
    (9)
  • (4) Calculate local cut-off frequency , and realign it.
    (10)
    (11)
    (12)
    (13)
    (14)

EA mechanism

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.

The calculation process of SA is illustrated in Figure 9 and Equations (15)–(19), where , , , ARN×N and represent key matrix, query matrix, value matrix, attention weight matrix and output matrix, respectively. Given an input matrix , where N is the feature dimension and Dx is the number of samples, SA linearly projects each input sequence xi to K, Q and V. The function S(Q, K) represents the attention scoring function. If we use scaled dot product attention scoring function, as shown in Equation (20), the output matrix H could be formulated as Equation (21). Additionally, an accepted pruned version of SA is illustrated in Figure 10 and Equations (22)–(23), which directly uses input matrix X to determine the attention weight matrix. However, the quadratic computational complexity of pruned SA may limit its applicability.
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
Figure 9

The calculation process of SA.

Figure 9

The calculation process of SA.

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Figure 10

The calculation process of pruned SA.

Figure 10

The calculation process of pruned SA.

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The SA uses a linear combination of the input matrix X to refine the input series. The query matrix can be viewed as a N-dimensional projection space constructed for the key matrix, resulting in a N × N attention weight matrix and leading to high computational effort. The EA uses two linear layers Mk and Mv, to replace the query matrix and key matrix, where Mk, MvRS×N are learnable and independent of the input matrix, and S is a predefined constant. The computational complexity of EA is O(NSDx). Mk and Mv serve as external memory units for the entire dataset. In contrast to SA, which only considers the correlation between input values within a single sample, EA can take the correlation between input values across different samples into account, potentially enhancing the capability and adaptability of the prediction models. The computation process of EA is illustrated in Figure 11 and Equations (24) and (25). The normalization method of EA is similar to that in SA and can be expressed as Equations (26)–(28).
(24)
(25)
(26)
(27)
(28)
Figure 11

The computation process of EA.

Figure 11

The computation process of EA.

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Model development and metrics

This study developed a series of coupling models leveraging TVFEMD, EA, CNN, GRU, KAN and MLP backbones (LSTM and GRU). The building process of TVFEMD–EA–CNN–GRU–KAN is depicted in Figure 12 as an example. The input features for these models were historical Miaozui water level and historical Gezhouba dam outflow, and the output was Miaozui water level. Figure 13 presents a schematic flowchart of the detailed data processing steps in the TVFEMD–EA– CNN–MLP–KAN hybrid model.
Figure 12

Architecture of the TVFEMD–EA–CNN–GRU–KAN coupling model.

Figure 12

Architecture of the TVFEMD–EA–CNN–GRU–KAN coupling model.

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Figure 13

Schematic flowchart illustrating the detailed data processing steps in the TVFEMD–EA–CNN–MLP–KAN hybrid model.

Figure 13

Schematic flowchart illustrating the detailed data processing steps in the TVFEMD–EA–CNN–MLP–KAN hybrid model.

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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.

Table 2

For hybrid models without TVFEMD, the first two samples of the dataset in the supervised learning format are shown (max–min normalization has finished)

No.Input
Output
TimeMiaozui water level (m)Outflow from Gezhouba dam (m3/s)TimeMiaozui water level (m)
2022/1/1
02:00 
0.052 0.045 2022/1/8
02:00 
0.038 
2022/1/1
08:00 
0.034 0.048 2022/1/8
08:00 
0.032 
2022/1/1
14:00 
0.099 0.103 2022/1/8
14:00 
0.067 
… … … 2022/1/8
20:00 
0.100 
2022/1/7
20:00 
0.053 0.064   
2022/1/1
08:00 
0.034 0.048 2022/1/8
08:00 
0.032 
2022/1/1
14:00 
0.099 0.103 2022/1/8
14:00 
0.067 
2022/1/1
20:00 
0.096 0.101 2022/1/8
20:00 
0.100 
… … … 2022/1/9
02:00 
0.045 
2022/1/8
02:00 
0.038 0.046   
… … … … … 
No.Input
Output
TimeMiaozui water level (m)Outflow from Gezhouba dam (m3/s)TimeMiaozui water level (m)
2022/1/1
02:00 
0.052 0.045 2022/1/8
02:00 
0.038 
2022/1/1
08:00 
0.034 0.048 2022/1/8
08:00 
0.032 
2022/1/1
14:00 
0.099 0.103 2022/1/8
14:00 
0.067 
… … … 2022/1/8
20:00 
0.100 
2022/1/7
20:00 
0.053 0.064   
2022/1/1
08:00 
0.034 0.048 2022/1/8
08:00 
0.032 
2022/1/1
14:00 
0.099 0.103 2022/1/8
14:00 
0.067 
2022/1/1
20:00 
0.096 0.101 2022/1/8
20:00 
0.100 
… … … 2022/1/9
02:00 
0.045 
2022/1/8
02:00 
0.038 0.046   
… … … … … 
Table 3

For hybrid models with TVFEMD, the first two samples of the dataset in the supervised learning format are shown (max–min normalization has finished)

No.TimeInput
Output
Water level components (m)
Outflow components (m3/s)
TimeWater level (m)
LHF1 (L)LHF11 (L)LLF (L)LHF1 (Q)LHF11 (Q)LLF (Q)
2022/1/1
2:00 
0.015  −0.009 0.081 −0.013  0.002 0.081 2022/1/8
2:00 
0.038 
2022/1/1
8:00 
−0.014  −0.010 0.081 −0.015  0.001 0.081 2022/1/8
8:00 
0.032 
2022/1/1
14:00 
0.011  −0.010 0.081 0.029  0.001 0.081 2022/1/8
14:00 
0.067 
… … … … … … … … … 2022/1/8
20:00 
0.100 
2022/1/7
20:00 
−0.003  0.009 0.063 −0.006  0.002 0.072   
2022/1/1
8:00 
−0.014  −0.010 0.081 −0.015  0.001 0.081 2022/1/8
8:00 
0.032 
2022/1/1
14:00 
0.011  −0.010 0.081 0.029  0.001 0.081 2022/1/8
14:00 
0.067 
2022/1/1
20:00 
−0.005  −0.011 0.081 0.018  0.000 0.081 2022/1/8
20:00 
0.100 
… … … … … … … … … 2022/1/9
2:00 
0.045 
2022/1/8
2:00 
−0.005  0.012 0.057 −0.005 … 0.002 0.071   
… … … … … … … … … … … 
No.TimeInput
Output
Water level components (m)
Outflow components (m3/s)
TimeWater level (m)
LHF1 (L)LHF11 (L)LLF (L)LHF1 (Q)LHF11 (Q)LLF (Q)
2022/1/1
2:00 
0.015  −0.009 0.081 −0.013  0.002 0.081 2022/1/8
2:00 
0.038 
2022/1/1
8:00 
−0.014  −0.010 0.081 −0.015  0.001 0.081 2022/1/8
8:00 
0.032 
2022/1/1
14:00 
0.011  −0.010 0.081 0.029  0.001 0.081 2022/1/8
14:00 
0.067 
… … … … … … … … … 2022/1/8
20:00 
0.100 
2022/1/7
20:00 
−0.003  0.009 0.063 −0.006  0.002 0.072   
2022/1/1
8:00 
−0.014  −0.010 0.081 −0.015  0.001 0.081 2022/1/8
8:00 
0.032 
2022/1/1
14:00 
0.011  −0.010 0.081 0.029  0.001 0.081 2022/1/8
14:00 
0.067 
2022/1/1
20:00 
−0.005  −0.011 0.081 0.018  0.000 0.081 2022/1/8
20:00 
0.100 
… … … … … … … … … 2022/1/9
2:00 
0.045 
2022/1/8
2:00 
−0.005  0.012 0.057 −0.005 … 0.002 0.071   
… … … … … … … … … … … 

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).

Table 4

The hyperparameters in EA, MLP and KAN

Elements in the hybrid modelsHyperparameters
EA S: column sizes of Mk and row size of Mv 
MLP Network structure: number of neurons in each layer 
KAN G: grid size 
 k: order of B-spline basis function 
 Network structure: number of neurons in each layer 
Elements in the hybrid modelsHyperparameters
EA S: column sizes of Mk and row size of Mv 
MLP Network structure: number of neurons in each layer 
KAN G: grid size 
 k: order of B-spline basis function 
 Network structure: number of neurons in each layer 

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.

This study selected MSE as the loss function during the training processes of the hybrid models, as defined by Equation (29). The RMSD, MAE, MAPE, R2 and CA were employed as performance metrics, as defined by Equations (30)–(33), where N represented the length of the sequence, Yi and denoted the ith observed and predicted value, respectively, and denoted the means for observed and predicted values, respectively. RMSD, MAE, MAPE, and CA were utilized to evaluate the prediction error, with lower values indicating better model performance. Meanwhile, R2 was used to measure the capability of models to capture fluctuation trends, and higher values signified better model accuracy (with a maximum value of 1). Based on previous research, the R2 greater than 0.9 generally indicate satisfactory model performance (Afrifa et al. 2023a, 2023b; Adnan et al. 2019; 2024). Considering the ship safety and navigational efficiency, RMSD and MAE values less than 0.2 m indicate the prediction accuracy of models is acceptable. Considering that the Miaozui water level remains around 39 m even during the dry season (as illustrated in Figure 4), the MAPE must be less than 0.5% to demonstrate reliable predictive performance. In conjunction with the acceptable ranges for the above metrics, a CA value of less than 0.165 indicates good model performance. Furthermore, standard deviation (SD) and correlation coefficient (R) were calculated to draw the Taylor plots, as defined by Equations (34) and (35). R represents the correlation between observed and predicted values, with a maximum value of 1. Generally, a larger R indicates better prediction performance, and an R greater than 0.9 generally indicates a decent forecasting result.
(29)
(30)
(31)
(32)
(33)
(34)
(35)

Results

MAE, RMSD, MAPE, R2, and CA of the hybrid models discussed earlier are detailed in Table 5. The partial visual comparisons between the forecasted and observed Miaozui water levels are depicted in Figure 14. The RMSD and MAE of plain LSTM and GRU models were larger than 0.2 m, MAPE was larger than 0.5% and CA was larger than 0.165, which demonstrated MLPs were incapable of accurately predicting Miaozui water level. However, the integrations with TVFEMD, EA, CNN, and KAN significantly enhanced the performance of the base MLPs. As illustrated in Table 5 and Figure 14, TVFEMD–EA–LSTM–KAN, TVFEMD–EA–CNN–LSTM–KAN, TVFEMD–EA–GRU–KAN and TVFEMD–EA–CNN–GRU–KAN effectively captured the trend of Miaozui water level changes. Their predictive accuracy was sufficient to meet the requirements for formulating dynamic draught limit standards, and discrepancies between predicted and observed water levels will not impact the scheduling of vessels passing through Gezhouba ship locks. Notably, the TVFEMD–EA–CNN–GRU–KAN model outperformed all others, exhibiting the highest precision and underscoring excellent fitting ability.
Table 5

Average performance of models on the test set

ModelEvaluation criteria
Improvements on MLPs
MAE (m)RMSD (m)MAPER2CAMAE (m)RMSD (m)MAPER2CA
LSTM 0.292 0.335 0.714% 0.873 0.249 – – – – – 
LSTM–KAN 0.242 0.279 0.593% 0.895 0.207 0.050 (17%) 0.056 (17%) 0.121% (17%) 0.022 (3%) 0.042 (17%) 
CNN–LSTM 0.253 0.293 0.622% 0.877 0.221 0.039 (13%) 0.042 (13%) 0.092% (13%) 0.007 (1%) 0.029 (12%) 
EA–LSTM 0.208 0.248 0.510% 0.921 0.177 0.084 (29%) 0.087 (26%) 0.204% (29%) 0.048 (5%) 0.072 (29%) 
CNN–LSTM–KAN 0.221 0.267 0.544% 0.909 0.191 0.071 (24%) 0.068 (20%) 0.170% (24%) 0.036 (4%) 0.058 (23%) 
EA–LSTM–KAN 0.206 0.245 0.506% 0.923 0.174 0.086 (29%) 0.090 (27%) 0.208% (29%) 0.050 (6%) 0.075 (30%) 
TVFEMD–EA–LSTM 0.209 0.251 0.515% 0.920 0.178 0.083 (28%) 0.084 (25%) 0.199% (28%) 0.047 (5%) 0.071 (28%) 
TVFEMD–EA–LSTM–KAN 0.149 0.172 0.368% 0.974 0.114 0.143 (49%) 0.163 (49%) 0.346% (48%) 0.101 (12%) 0.135 (54%) 
TVFEMD–EA– CNN–LSTM–KAN (Inputs include outflow) 0.140 0.164 0.345% 0.968 0.111 0.152 (52%) 0.171 (51%) 0.369% (52%) 0.095 (11%) 0.138 (55%) 
TVFEMD–EA– CNN–LSTM–KAN (Inputs exclude outflow) 0.176 0.205 0.435% 0.950 0.142 0.116 (40%) 0.130 (39%) 0.279% (39%) 0.077 (9%) 0.107 (43%) 
GRU 0.241 0.283 0.589% 0.909 0.203 – – – – – 
GRU–KAN 0.234 0.276 0.572% 0.910 0.198 0.007 (3%) 0.007 (2%) 0.017% (3%) 0.001 (0%) 0.005 (2%) 
CNN–GRU 0.233 0.272 0.574% 0.910 0.196 0.008 (3%) 0.011 (4%) 0.015% (3%) 0.001 (0%) 0.007 (3%) 
EA–GRU 0.228 0.272 0.561% 0.914 0.193 0.013 (5%) 0.011 (4%) 0.028% (5%) 0.005 (1%) 0.010 (5%) 
CNN–GRU–KAN 0.221 0.256 0.541% 0.933 0.180 0.020 (8%) 0.027 (10%) 0.048% (8%) 0.024 (3%) 0.023 (11%) 
EA–GRU–KAN 0.187 0.211 0.459% 0.948 0.149 0.054 (22%) 0.072 (25%) 0.130% (22%) 0.040 (4%) 0.055 (27%) 
TVFEMD–EA–GRU 0.182 0.219 0.447% 0.952 0.148 0.059 (24%) 0.064 (23%) 0.142% (24%) 0.044 (5%) 0.055 (27%) 
TVFEMD–EA–GRU–KAN 0.160 0.184 0.396% 0.972 0.123 0.081 (34%) 0.099 (35%) 0.193% (33%) 0.063 (7%) 0.080 (39%) 
TVFEMD–EA–CNN–GRU–KAN (Inputs include outflow) 0.131 0.153 0.322% 0.971 0.103 0.110 (46%) 0.130 (46%) 0.267% (45%) 0.062 (7%) 0.100 (49%) 
TVFEMD–EA–CNN–GRU–KAN (Inputs exclude outflow) 0.165 0.191 0.406% 0.967 0.128 0.076 (32%) 0.092 (33%) 0.183% (31%) 0.060 (7%) 0.075 (37%) 
ModelEvaluation criteria
Improvements on MLPs
MAE (m)RMSD (m)MAPER2CAMAE (m)RMSD (m)MAPER2CA
LSTM 0.292 0.335 0.714% 0.873 0.249 – – – – – 
LSTM–KAN 0.242 0.279 0.593% 0.895 0.207 0.050 (17%) 0.056 (17%) 0.121% (17%) 0.022 (3%) 0.042 (17%) 
CNN–LSTM 0.253 0.293 0.622% 0.877 0.221 0.039 (13%) 0.042 (13%) 0.092% (13%) 0.007 (1%) 0.029 (12%) 
EA–LSTM 0.208 0.248 0.510% 0.921 0.177 0.084 (29%) 0.087 (26%) 0.204% (29%) 0.048 (5%) 0.072 (29%) 
CNN–LSTM–KAN 0.221 0.267 0.544% 0.909 0.191 0.071 (24%) 0.068 (20%) 0.170% (24%) 0.036 (4%) 0.058 (23%) 
EA–LSTM–KAN 0.206 0.245 0.506% 0.923 0.174 0.086 (29%) 0.090 (27%) 0.208% (29%) 0.050 (6%) 0.075 (30%) 
TVFEMD–EA–LSTM 0.209 0.251 0.515% 0.920 0.178 0.083 (28%) 0.084 (25%) 0.199% (28%) 0.047 (5%) 0.071 (28%) 
TVFEMD–EA–LSTM–KAN 0.149 0.172 0.368% 0.974 0.114 0.143 (49%) 0.163 (49%) 0.346% (48%) 0.101 (12%) 0.135 (54%) 
TVFEMD–EA– CNN–LSTM–KAN (Inputs include outflow) 0.140 0.164 0.345% 0.968 0.111 0.152 (52%) 0.171 (51%) 0.369% (52%) 0.095 (11%) 0.138 (55%) 
TVFEMD–EA– CNN–LSTM–KAN (Inputs exclude outflow) 0.176 0.205 0.435% 0.950 0.142 0.116 (40%) 0.130 (39%) 0.279% (39%) 0.077 (9%) 0.107 (43%) 
GRU 0.241 0.283 0.589% 0.909 0.203 – – – – – 
GRU–KAN 0.234 0.276 0.572% 0.910 0.198 0.007 (3%) 0.007 (2%) 0.017% (3%) 0.001 (0%) 0.005 (2%) 
CNN–GRU 0.233 0.272 0.574% 0.910 0.196 0.008 (3%) 0.011 (4%) 0.015% (3%) 0.001 (0%) 0.007 (3%) 
EA–GRU 0.228 0.272 0.561% 0.914 0.193 0.013 (5%) 0.011 (4%) 0.028% (5%) 0.005 (1%) 0.010 (5%) 
CNN–GRU–KAN 0.221 0.256 0.541% 0.933 0.180 0.020 (8%) 0.027 (10%) 0.048% (8%) 0.024 (3%) 0.023 (11%) 
EA–GRU–KAN 0.187 0.211 0.459% 0.948 0.149 0.054 (22%) 0.072 (25%) 0.130% (22%) 0.040 (4%) 0.055 (27%) 
TVFEMD–EA–GRU 0.182 0.219 0.447% 0.952 0.148 0.059 (24%) 0.064 (23%) 0.142% (24%) 0.044 (5%) 0.055 (27%) 
TVFEMD–EA–GRU–KAN 0.160 0.184 0.396% 0.972 0.123 0.081 (34%) 0.099 (35%) 0.193% (33%) 0.063 (7%) 0.080 (39%) 
TVFEMD–EA–CNN–GRU–KAN (Inputs include outflow) 0.131 0.153 0.322% 0.971 0.103 0.110 (46%) 0.130 (46%) 0.267% (45%) 0.062 (7%) 0.100 (49%) 
TVFEMD–EA–CNN–GRU–KAN (Inputs exclude outflow) 0.165 0.191 0.406% 0.967 0.128 0.076 (32%) 0.092 (33%) 0.183% (31%) 0.060 (7%) 0.075 (37%) 
Figure 14

Partial comparison between the forecasted and observed Miaozui water level on the test set.

Figure 14

Partial comparison between the forecasted and observed Miaozui water level on the test set.

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Moreover, it could be intuitively observed from Figures 15 to 18 that the RMSD and R of the TVFEMD–EA–LSTM–KAN, TVFEMD–EA–CNN–LSTM–KAN, TVFEMD–EA–GRU–KAN and TVFEMD–EA–CNN–GRU–KAN in the Taylor diagrams and scatter plots were significantly superior to other LSTM-based and GRU-based hybrid models, respectively. The enhancement on the predicted results when the input features included the outflow was also evident in Table 5. For TVFEMD–EA–CNN–LSTM–KAN, including outflow as an input feature reduced the MSE and RMSD by 20%, MAPE by 21%, CA by 22%, and increased R2 by 2%. Similarly, for TVFEMD–EA–CNN–GRU–KAN, including outflow data reduced the MSE by 21%, the RMSD by 20%, the MAPE by 21%, and the CA by 20%. These findings confirmed that variations in discharge flow significantly impacted Miaozui water levels. Including outflow data as an input feature markedly enhanced prediction accuracy.
Figure 15

Taylor diagram of hybrid models based on LSTMs.

Figure 15

Taylor diagram of hybrid models based on LSTMs.

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Figure 16

Taylor diagram of hybrid models based on GRUs.

Figure 16

Taylor diagram of hybrid models based on GRUs.

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Figure 17

Scatter plot of the LSTM, LSTM–KAN, CNN–LSTM–KAN, TVFEMD–EA–LSTM–KAN, and TVFEMD–EA–CNN–LSTM–KAN predicted values and observed values.

Figure 17

Scatter plot of the LSTM, LSTM–KAN, CNN–LSTM–KAN, TVFEMD–EA–LSTM–KAN, and TVFEMD–EA–CNN–LSTM–KAN predicted values and observed values.

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Figure 18

Scatter plot of the GRU, GRU–KAN, CNN–GRU–KAN, TVFEMD–EA–GRU–KAN, and TVFEMD–EA–CNN–GRU–KAN predicted values and observed values.

Figure 18

Scatter plot of the GRU, GRU–KAN, CNN–GRU–KAN, TVFEMD–EA–GRU–KAN, and TVFEMD–EA–CNN–GRU–KAN predicted values and observed values.

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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.

Table 6

Average performance of TVFEMD–EA–CNN–GRU–KAN on different hyperparameter combinations

RankSGRU network structureGkKAN network structureMAE (m)RMSD (m)MAPER2CA
64 [700, 700, 600, 600, 600] 20 [600, 1, 1, 3, 4] 0.131 0.153 0.322% 0.971 0.103 
64 [300, 300, 300, 300, 200] 20 [200, 1, 1, 3, 4] 0.137 0.161 0.337% 0.972 0.108 
32 [700, 700, 600, 600, 600] 20 [600, 1, 1, 3, 4] 0.152 0.174 0.371% 0.968 0.117 
64 [700, 700, 600, 600, 600] 10 [600, 1, 1, 3, 4] 0.161 0.185 0.398% 0.970 0.124 
64 [700, 700, 600, 600, 600] 20 [600, 4] 0.169 0.194 0.426% 0.970 0.123 
64 [200, 200, 200, 200] 20 [200, 1, 1, 3, 4] 0.208 0.248 0.510% 0.920 0.177 
RankSGRU network structureGkKAN network structureMAE (m)RMSD (m)MAPER2CA
64 [700, 700, 600, 600, 600] 20 [600, 1, 1, 3, 4] 0.131 0.153 0.322% 0.971 0.103 
64 [300, 300, 300, 300, 200] 20 [200, 1, 1, 3, 4] 0.137 0.161 0.337% 0.972 0.108 
32 [700, 700, 600, 600, 600] 20 [600, 1, 1, 3, 4] 0.152 0.174 0.371% 0.968 0.117 
64 [700, 700, 600, 600, 600] 10 [600, 1, 1, 3, 4] 0.161 0.185 0.398% 0.970 0.124 
64 [700, 700, 600, 600, 600] 20 [600, 4] 0.169 0.194 0.426% 0.970 0.123 
64 [200, 200, 200, 200] 20 [200, 1, 1, 3, 4] 0.208 0.248 0.510% 0.920 0.177 

A violin plot is a combination of a box plot and a kernel density plot. Violin plots focus less on outliers and more on data distribution, emphasizing the distribution profile and the most concentrated region of the data. As shown in Figures 19 and 20, violin plots were used to visually compare the distributions of predicted and observed values. In Figure 19, the distribution regions of observed and predicted values for five hybrid models were nearly parallel and close to the same value. However, in the six box plots, the upper and lower quartiles, boundaries, medians, box lengths, and skewness differed significantly. By combining the box plot and violin plot, it was evident that TVFEMD–EA–CNN–LSTM–KAN outperformed other hybrid models in data distribution and outliers. Similarly, as shown in the six violin plots of Figure 20, TVFEMD–EA–CNN–GRU–KAN outperformed other hybrid models.
Figure 19

Violin plot of prediction results for (1) LSTM, (2) LSTM–KAN, (3) CNN–LSTM–KAN, (4) TVFEMD–EA–LSTM–KAN, and (5) TVFEMD–EA–CNN–LSTM–KAN.

Figure 19

Violin plot of prediction results for (1) LSTM, (2) LSTM–KAN, (3) CNN–LSTM–KAN, (4) TVFEMD–EA–LSTM–KAN, and (5) TVFEMD–EA–CNN–LSTM–KAN.

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Figure 20

Violin plot of prediction results for (1) GRU, (2) GRU–KAN, (3) CNN–GRU–KAN, (4) TVFEMD–EA–GRU–KAN, and (5) TVFEMD–EA–CNN–GRU–KAN.

Figure 20

Violin plot of prediction results for (1) GRU, (2) GRU–KAN, (3) CNN–GRU–KAN, (4) TVFEMD–EA–GRU–KAN, and (5) TVFEMD–EA–CNN–GRU–KAN.

Close modal

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.

As mentioned previously, for hybrid models with TVFEMD, the normalized water level and outflow time series were decomposed into 11 LHF components and one LLF component. For water level prediction, TVFEMD could adaptively adjust the response frequency of the filter according to the characteristics of the input series (outflow and water level series), which enabled more effective extraction of their different frequency components while filtering noise during the process. Therefore, using the extracted frequency components as inputs to the hybrid models enabled them to better grasp the patterns of changes in the Miaozui water level, thereby improving the accuracy of predictions. The results of TVFEMD are shown in Figures 21 and 22. According to the model evaluation metrics in Table 5, compared to EA–LSTM–KAN, TVFEMD–EA–LSTM–KAN reduced MAE by 27.67%, RMSD by 29.80%, MAPE by 27.27%, CA by 5.53%, and increased R2 by 34.48%. Compared to EA–GRU–KAN, TVFEMD–EA–GRU–KAN reduced MAE by 14.44%, RMSD by 12.80%, MAPE by 13.73%, CA by 2.53%, and increased R2 by 17.45%. The Taylor diagrams in Figures 15 and 16 and scatter plots in Figures 17 and 18 further confirmed that hybrid models with TVFEMD significantly outperformed the other models.
Figure 21

TVFEMD result of the water level time series at Miaozui.

Figure 21

TVFEMD result of the water level time series at Miaozui.

Close modal
Figure 22

TVFEMD result of outflow time series from the Gezhouba dam.

Figure 22

TVFEMD result of outflow time series from the Gezhouba dam.

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To explore the changes of EA weights in both temporal and decomposed component dimensions, the EA visualization of the TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN are, respectively, presented in Figure 23(a) and 23(b). In Figure 23, the X-coordinate of each subplot represents the historical time, a total of 28 time steps, the Y-coordinate represents the input decomposed components, and the shade of color represents EA weights. Figure 23(a) and 23(b) exhibit several similar features. First, the LLF(Q) and LLF(L) exhibited the highest and second-highest attention weight values, respectively, in all time steps. This was attributed to the LLF components in TVFEMD results of the water level and outflow time series possessed the largest amplitudes among the decomposed components, as shown in Figures 21 and 22. Consequently, they exerted a greater impact on the prediction results. Furthermore, the fact that the LLF(Q) held the highest weight value confirmed the assertion in Section 2 that variations in the outflow from the Gezhouba dam significantly influenced the water level at Miaozui. Second, in terms of the temporal dimension, the time steps immediately preceding the prediction period consistently showed higher EA weights than other time steps. This suggested that the component values at the time steps closest to the prediction period exerted the strongest influence on the prediction results. Third, on the temporal dimension, the weights of each input component changed gently, indicating that the weights of adjacent time steps were close, with no abrupt changes.
Figure 23

EA weight visualization.

Figure 23

EA weight visualization.

Close modal
To delve deeper into the performance of these models, their training processes are visually compared in Figure 24. All models were trained for 10,000 epochs, and the trends of the loss function values for both the training and test sets indicated that all models successfully converged by the end of training, with no signs of overfitting or underfitting. A comparison of the training processes between the LSTM–KAN and CNN–LSTM–KAN, as well as the GRU–KAN and CNN–GRU–KAN, indicated that incorporating CNN effectively reduced the discrepancy between the loss function values on the training and test sets. This reduction in the gap suggested that the CNN-enhanced models achieved better generalization. Furthermore, during the training processes of TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN, the fluctuation of loss function values on training and test sets was lower compared to the TVFEMD–EA–LSTM–KAN and TVFEMD–EA–GRU–KAN. This reduced volatility suggested that the integration of CNN contributed to a more stable learning process, which ensured the model was reliable on unseen input data.
Figure 24

Training processes of hybrid models.

Figure 24

Training processes of hybrid models.

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To demonstrate the robustness of the proposed hybrid models in water level prediction, the BRF, GA–XGBoost, and SOA–SVM models were compared with them. The evaluation criteria are presented in Table 7. For CA, TVFEMD–EA–CNN–LSTM–KAN was 48.37% lower than SOA–SVM, 48.85% lower than BRF and 54.69% lower than GA–XGBoost. Likewise, the CA of TVFEMD–EA–CNN–GRU–KAN was 52.09% lower than SOA–SVM, 52.53% lower than BRF and 57.96% lower than GA–XGBoost. The TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN exhibited higher prediction accuracy than the other models. The visual comparisons are shown in Figures 2527. Based on the R and RMSD in Figures 25 and 26, TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN demonstrated greater robustness in capturing the trends of the Miaozui water level. According to the box plots and violin plots in Figure 27, the prediction results of TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN were closer to the observed values in terms of data distribution and outliers. In summary, TVFEMD–EA–CNN–LSTM–KAN and TVFEMD–EA–CNN–GRU–KAN exhibited higher prediction accuracy and superior fitting performance.
Table 7

Performance comparison of TVFEMD–EA–CNN–GRU–KAN, TVFEMD–EA–CNN–LSTM–KAN, BRF, GA–XGBoost and SOA–SVM

RankModelMAE (m)RMSD (m)MAPER2CA
TVFEMD–EA–CNN–GRU–KAN 0.131 0.153 0.322% 0.971 0.103 
TVFEMD–EA–CNN–LSTM–KAN 0.140 0.164 0.345% 0.968 0.111 
SOA–SVM 0.223 0.332 0.548% 0.903 0.215 
BRF 0.230 0.334 0.568% 0.906 0.217 
GA–XGBoost 0.246 0.373 0.604% 0.877 0.245 
RankModelMAE (m)RMSD (m)MAPER2CA
TVFEMD–EA–CNN–GRU–KAN 0.131 0.153 0.322% 0.971 0.103 
TVFEMD–EA–CNN–LSTM–KAN 0.140 0.164 0.345% 0.968 0.111 
SOA–SVM 0.223 0.332 0.548% 0.903 0.215 
BRF 0.230 0.334 0.568% 0.906 0.217 
GA–XGBoost 0.246 0.373 0.604% 0.877 0.245 
Figure 25

Visual performance comparison of TVFEMD–EA–CNN–GRU–KAN, TVFEMD–EA–CNN–LSTM–KAN, SOA–SVM, BRF, and GA–XGBoost.

Figure 25

Visual performance comparison of TVFEMD–EA–CNN–GRU–KAN, TVFEMD–EA–CNN–LSTM–KAN, SOA–SVM, BRF, and GA–XGBoost.

Close modal
Figure 26

Scatter plot of predicted values and observed values of the TVFEMD–EA–CNN–GRU–KAN, TVFEMD–EA–CNN–LSTM–KAN, SOA–SVM, BRF, and GA–XGBoost.

Figure 26

Scatter plot of predicted values and observed values of the TVFEMD–EA–CNN–GRU–KAN, TVFEMD–EA–CNN–LSTM–KAN, SOA–SVM, BRF, and GA–XGBoost.

Close modal
Figure 27

Violin plot of prediction results for (1) TVFEMD–EA–CNN–GRU–KAN, (2) TVFEMD–EA–CNN–LSTM–KAN, (3) SOA–SVM, (4) BRF, and (5) GA–XGBoost.

Figure 27

Violin plot of prediction results for (1) TVFEMD–EA–CNN–GRU–KAN, (2) TVFEMD–EA–CNN–LSTM–KAN, (3) SOA–SVM, (4) BRF, and (5) GA–XGBoost.

Close modal

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.

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