IRFO-VMD and ITCN-A-based hourly runoff combined prediction method

Abstract

The invention discloses a hourly runoff combined prediction method based on IRFO-VMD and ITCN-A. The hourly runoff combined prediction method comprises the following steps: collecting runoff data; optimizing the parameters of the VMD by using IRFO, and decomposing the runoff time sequence into K intrinsic mode functions (IMFs) by using the optimized VMD; the method comprises the following steps: extracting dynamic characteristics of a runoff time sequence by adopting phase-space reconstruction PSR, and then dividing runoff data into a training set, a verification set and a test set; and training the ITCN-A model by using the training set and the verification set containing the runoff time sequence dynamic characteristics, and predicting the test set by using the trained model. And analyzing a prediction result of the model by adopting three evaluation indexes including a root mean square error (RMSE), a mean absolute error (MAE) and a mean absolute percentage error (MAPE). According to the method, a good prediction effect is achieved on historical data of different hydrological stations, and the stability of the prediction model and the accuracy of the prediction effect are remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of runoff prediction technology, specifically to a method for joint hourly runoff prediction based on IRFO-VMD and ITCN-A. Background Technology

[0002] With the increasing severity of global climate change and the frequent occurrence of extreme weather events, the uncertainty in predicting watershed hydrological processes has also increased. Water scarcity and floods have become key factors threatening human security and sustainable development. Runoff forecasting is of great significance for watershed management, flood warning, reservoir operation, and ecological protection. Accurate runoff forecasting provides an important basis for efficient water resource allocation. However, runoff processes are influenced by a variety of complex factors, and their inherent nonlinear and non-stationary characteristics pose challenges to the accuracy of forecasts. Therefore, how to construct accurate and stable runoff forecasting models has become a critical problem that urgently needs to be solved in the field of hydrology.

[0003] Currently, runoff prediction methods can be mainly divided into three categories: runoff prediction based on traditional statistical methods, runoff prediction based on machine learning, and runoff prediction based on deep learning. In traditional methods, statistical models such as autoregressive moving average (AR) and autoregressive integral moving average (IMA) models can effectively capture the linear dynamic characteristics of runoff sequences, thus achieving simple and efficient predictions. However, traditional statistical models struggle to effectively characterize the nonlinear and nonstationary characteristics of runoff sequences, and their prediction accuracy is often limited in complex watershed environments. Machine learning methods, leveraging the excellent nonlinear fitting capabilities of machine learning models, can accurately capture complex nonlinear trends in runoff sequences, thus showing promising application prospects in runoff prediction. Common machine learning models include Extreme Learning Machine (ELM), Kernel Extreme Learning Machine (KLM), Random Forest, and Extreme Gradient Boosting Tree (ESBR). Although machine learning methods have achieved certain results in runoff prediction, their inability to fully characterize the dynamic characteristics and long-term dependencies of runoff processes poses a challenge to the accuracy of machine learning in runoff prediction tasks.

[0004] With the rapid development of artificial intelligence technology, deep learning has been widely used in runoff prediction tasks due to its ability to efficiently model the temporal dependence and dynamic change law in runoff sequences using neural networks. For example, in the literature [1]: Xu Junyang, Luo Yuanlin, Liu Yuexin, et al. Medium and long-term runoff forecast based on CEEMDAN-IASO-TCN combined model [J]. Yangtze River, 2025, 56(04): 128-135. However, due to the difficulty of accurately capturing the non-stationarity in runoff sequences by a single neural network model, researchers have begun to introduce signal decomposition technology into runoff prediction. By decomposing complex runoff sequences into several subsequences with different time scales or feature patterns, the complexity of the data is reduced. In addition, the prediction model is quite sensitive to parameter settings in practical applications, which may affect its prediction accuracy. With the rapid development of swarm intelligence optimization algorithms and metaheuristic algorithms, researchers have begun to use optimization algorithms to adaptively optimize model parameters. These algorithms can quickly find the optimal parameter combination in high-dimensional search space, providing an effective solution for parameter tuning in runoff prediction. Rüppell's fox optimizer (RFO) is a heuristic optimization algorithm, the specific formula of which can be found in reference [2]: Braik, M., Al-Hiary, H., 2025. Rüppell's fox optimizer: A novel meta-heuristic approach for solving global optimization problems. Cluster Comput.28, 292. However, the standard RFO is prone to getting trapped in local optima in complex optimization tasks. Summary of the Invention

[0005] To address the aforementioned technical issues, this invention proposes a joint hourly runoff prediction method based on IRFO-VMD and ITCN-A, which achieves good prediction results on historical data from different hydrological stations, significantly improving the stability of the prediction model and the accuracy of the prediction results.

[0006] The technical solution adopted in this invention is as follows:

[0007] The joint hourly runoff prediction method based on IRFO-VMD and ITCN-A includes the following steps: Step 1: Collect runoff data; Step 2: Optimize the parameters of VMD using IRFO, and decompose the runoff time series into K intrinsic mode functions (IMFs) using the optimized VMD. Step 3: Use phase space reconstruction (PSR) to extract the dynamic features of runoff time series, and then divide the runoff data into training set, validation set and test set; Step 4: Train the ITCN-A model using the training set and validation set containing the dynamic features of runoff time series obtained in Step 3, and then use the trained model to predict the test set.

[0008] Step 5: Analyze the model's prediction results using three evaluation metrics: root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE).

[0009] Step 2 includes: 2.1: An IRFO algorithm is proposed, which effectively improves global search efficiency and convergence speed by introducing a reverse learning mechanism and combining it with a simulated annealing selection strategy; In the reverse learning mechanism, the reverse individual is constructed as follows: (1); In equation (1), Represents an individual that learns in reverse; Indicates the lower bound of an individual; Indicates the upper bound of an individual; For the first i Individual; If the fitness of the reversed solution is better than that of the original individual, then the original individual is replaced by the reversed solution; otherwise, the original individual remains unchanged. Its mathematical expression is as follows: (2); In equation (2), express X op The fitness function value; express X i The fitness function value.

[0010] Introducing a back-learning probability that dynamically decays with the number of iterations Only if the random numbers follow a uniform distribution (0,1) r Less than the reverse learning probability When the reverse learning operation is performed, the reverse learning probability is defined as follows: (3); In equation (3), a and b These represent the termination probability and the initial probability, respectively. t This represents the current iteration number; T This represents the maximum number of iterations. 2.2: Simulated annealing selection strategy avoids the algorithm getting trapped in local optima by accepting inferior solutions with a certain probability, and guides the algorithm to convergence by gradually decreasing the temperature. Let the... i The current solution for each individual isM i fitness is f ( M i After iteration, a new candidate solution is generated. X i1 The corresponding fitness is f ( X i1 If the fitness difference between the two is defined as follows: (4); In equation (4), This represents the difference in fitness function between the candidate solution and the current solution.

[0011] The formula for updating individual positions using simulated annealing is as follows: (5); In equation (5), r These are uniformly distributed random numbers following the interval (0, 1). T e The temperature is the current temperature, and it decays with the number of iterations. The decayed temperature is calculated as follows: (6); In equation (6), This indicates the updated temperature. α (0,1) represents the temperature decay coefficient; T f This is the lowest temperature.

[0012] 2.3: VMD decomposes complex non-stationary signals into several intrinsic mode functions (IMFs), each with a finite bandwidth and a specific center frequency. Its mathematical expression is as follows: (7); In equation (7), u k Indicates the first k One modal function; oh k Indicates the first k The center frequency of each modal function; Represents the set of modal functions; This represents the set of center frequencies corresponding to the modal functions; K Represents the number of decomposed modes; This represents taking the partial derivative with respect to time; It is the Dirac function; Represents the imaginary unit; t Represents time; Indicates the first k One modal function; Represents the multiplication operator; Represents the L2 norm; This represents the original vibration signal sequence.

[0013] Introducing Lagrange multipliers l and secondary penalty factor α This yields the extended Lagrange function. L The specific formula is defined as follows: (8); In equation (8), Represent the Lagrange function; Represents the Lagrange multiplier that changes over time; This indicates the inner product.

[0014] The saddle points of the augmented Lagrangian function are solved by the alternating direction multiplier method, thereby optimizing each mode function. u k ( t and center frequency oh k The detailed description of the updated formula is as follows: (9); (10); (11); In the above formula, , , and Represent f ( t ), u i ( t ), l ( t ), The Fourier transform expression; Indicates the first k Frequency domain representation of the modal functions; Indicates the first n +Lagrange multiplier at +1 iteration; Indicates the first n Lagrange multipliers at the next iteration; Indicates the first n In the +1st iteration, the... k The center frequency of each modal function; Indicates frequency.

[0015] By iteratively updating until the stopping condition is met, we obtain... K Modal functions uk ( t ) and its corresponding center frequency oh k .

[0016] 2.4: To achieve adaptive optimization of key VMD parameters, the IRFO obtained above is selected as the VMD modality number K and penalty factor. α Optimization is performed, using the minimum envelope entropy as the fitness function. This decomposes the runoff sequence into K intrinsic mode functions (IMFs) of different frequencies. The formula for the minimum envelope entropy is as follows: (12); (13); (14); In the above formula, Let be the parameter components to be optimized; min{} represents the minimum envelope entropy; K The number of modes in the VMD decomposition; Indicates the number of samples; p i [ n [ ] represents the normalized envelope probability; n For discrete time points; e i [ n ] is the first i The envelope amplitude of each mode; H{} is the Hilbert transform.

[0017] In step 3, phase space reconstruction (PSR) is performed on each intrinsic mode function (IMF), and the dataset is divided into training, validation, and test sets. Specifically: 3.1: Phase Space Reconstruction (PSR) through appropriate selection of embedding dimensions d and time delay t This maps the original one-dimensional time series to a high-dimensional phase space. (15); In the above formula, X i Represents a sequence sample; i For index, where i =1,2,3,……, R , R The length of the sequence after embedding the time delay. d For the embedded dimension; L The length of the original sequence; Represents the reconstructed phase space vector; Indicates transpose; R Represents the number of sampling points; Indicates the first sampling point. Indicates the 1st+ t One sampling point; Indicates the 1st + ( d -1) t Each sampling point Indicates the first i One sampling point; Indicates the first i + t One sampling point; Indicates the first i +( d -1) t One sampling point; Indicates the first R -( d -1) t One sampling point; Indicates the first R -( d -2) t One sampling point; Indicates the first R One sampling point.

[0018] The dynamic features of runoff time series are extracted by phase space reconstruction (PSR). Specifically, the original runoff data is reconstructed by phase space reconstruction using equation (15) to obtain the reconstructed runoff data.

[0019] 3.2: The reconstructed data containing the dynamic features of runoff time series is divided into training set, validation set and test set according to an appropriate ratio to obtain the final dataset.

[0020] In step 4, the ITCN-A model is trained using the training set and validation set containing the dynamic features of runoff time series obtained in step 3, and the trained model is then used to predict the test set to obtain the prediction results; the specific steps are as follows: 4.1: Construct a multi-scale TCN module to extract multi-layer runoff temporal dependency features through parallel dilated convolution with different convolution kernels, thereby enhancing the expressive power of multi-scale features; TCN uses one-dimensional convolution as its basic operational unit to extract local temporal features of a sequence along the time dimension; the mathematical expression is as follows: (16); in: T t Represents the output of a one-dimensional convolution. W i These are the parameters of the convolution kernel. s t-i For the first t - i The input sequence at each time step, k-1 represents the kernel size. i Represents an index. b For bias parameters, t For time indexing, s The input sequence is .

[0021] To preserve temporal causality, TCN introduces a causal convolution mechanism into one-dimensional convolution, ensuring that the output at the current time step depends only on the current and previous inputs; the mathematical expression is as follows: (17); The parameters are consistent with those in equation (16).

[0022] TCN employs a residual connection structure. Each residual block consists of two convolutional layers and a nonlinear activation function; and through an identity mapping, the input features are directly added to the convolutional output to achieve cross-layer information transfer; the mathematical expression is as follows: (18); (19); Wherein: Conv1 and Conv2 are two convolutional layers; f 1 and f 2 are nonlinear activation functions; CC( s ) is the residual block output; s This represents the input sequence, as described above. J This represents the final output of the residual connection.

[0023] 4.2: An adaptive nonlinear projection layer is designed to expand and re-compress the runoff characteristic space through nonlinear mapping between upper and lower projection modules, thereby capturing higher-order nonlinear relationships; the mathematical expression is as follows: (20); (twenty one); in: and f These are nonlinear activation functions; W up and W down These are the learnable weights for the upper and lower projection layers, respectively; b up and b down These are the offset parameters for the upper and lower projection layers, respectively; h up and h down These are the outputs of the upper and lower projection layers, respectively. s The input sequence is .

[0024] The fusion ratio of the nonlinear projection is adaptively adjusted by learnable gating parameters, as shown in the following mathematical expression: (twenty two); in: g For gated functions, h down This is the output of the downward projection; s The input sequence; For gated fusion output; This is for element-wise multiplication.

[0025] The adjusted nonlinear projection is residually fused with the information from the multi-scale TCN module, and the mathematical expression is as follows: (twenty three); in: This is the output after fusion.

[0026] 4.3: To enhance ITCN-A's ability to represent key sequence information, a lightweight attention mechanism is introduced. Attention scores are generated along the time dimension through linear mapping, and a softmax function is used for weight normalization, thereby weighted aggregation of runoff features based on the importance of each time step. Details are as follows: Let the input feature sequence be , Let represent a three-dimensional feature vector with dimensions B, T, and C, where B , T , C These represent batch size, time step, and feature dimension, respectively.

[0027] Attention scores in the time dimension can be obtained through linear mapping. S The calculation formula is as follows: (twenty four); In equation (24), and All are learnable parameters; This is the output after the residual connection mentioned above; S Score for attention; The score vector in the time dimension is normalized using the softmax function to obtain the attention weights. a The specific expression is as follows: (25); in: t In terms of time dimension, a t represent t Attention weight at any moment S trepresent t Attention score at any moment S i represent i Attention score at any moment i Represents an index. This represents an exponential function.

[0028] Subsequently, the features at each time step are weighted and summed to achieve feature aggregation along the time dimension. The calculation form is as follows: (26); In equation (26), and Represent t Attention weights and feature sequences at each time step; T For time steps; t This is the current time step; O This is the result of weighted aggregation.

[0029] 4.4: The weighted aggregation results are used to obtain the runoff prediction results for each IMF component through linear layer mapping. The mathematical expression is as follows: (27); in: W z Represents learnable parameters; b z Represents the paranoia vector; Y z This represents the runoff prediction results for each IMF component.

[0030] In step 4, the ITCN-A model adopts a multi-layered, deeply integrated overall architecture design: First, construct a multi-scale TCN module based on equations (16) to (19); Secondly, a series adaptive nonlinear projection layer, as shown in equations (20) to (21), expands and re-compresses the feature space through the nonlinear mapping of the upper and lower projection modules to capture higher-order nonlinear relationships. Meanwhile, learnable gating parameters, as shown in Equation (22), are introduced to adaptively adjust the fusion ratio of the nonlinear projection; By performing residual fusion of the regulated nonlinear projection with information from the multi-scale TCN module, the flexibility and stability of the network are further improved.

[0031] Subsequently, the attention layer is connected in series, as shown in equations (24) to (26). The feature weighting mechanism assigns differentiated weights to different scales and channels. This highlights the contribution of key time steps and suppresses redundant information; Finally, as shown in Equation (27), the cascaded attention layer maps the fused features to the output space using a linear layer to obtain the prediction results of each IMF component, and then accumulates the prediction results of each component to generate the final runoff prediction result.

[0032] In step 5, the specific calculation formulas for the three evaluation indicators—Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE)—are as follows: (28); (29); (30); In the above formula: n Indicates the number of samples. and They represent the first i The actual and predicted values ​​of each sample.

[0033] This invention provides a joint hourly runoff prediction method based on IRFO-VMD and ITCN-A, with the following technical advantages: 1) This invention uses the IRFO-VMD method to decompose the photovoltaic power generation of different weather types into several subsequence components, thereby reducing nonlinearity and nonstationarity.

[0034] 2) This invention uses the ITCN-A model to predict each subsequence component separately, and accumulates the component prediction results to obtain the final runoff prediction result, thereby effectively improving the accuracy of runoff prediction.

[0035] 3) To enhance the diversity of initial solutions and improve the algorithm's ability to escape local optima, this invention proposes an IRFO mechanism. By introducing a reverse learning mechanism and combining it with a simulated annealing selection strategy, the global search efficiency and convergence speed are effectively improved.

[0036] 4) This invention verifies the superiority of a time-by-time runoff joint prediction method based on IRFO-VMD and ITCN-A by comparing and analyzing experimental results at three different hydrological stations. It achieves good prediction results on historical data from different hydrological stations, significantly improving the stability of the prediction model and the accuracy of the prediction results. Attached Figure Description

[0037] The present invention will be further described below with reference to the accompanying drawings and examples; Figure 1(a) is a schematic diagram of the TCN structure in the ITCN-A model proposed in this invention; Figure 1(b) is a schematic diagram of the multi-scale TCN structure in the ITCN-A model proposed in this invention.

[0038] Figure 2 This is a schematic diagram of the adaptive nonlinear mapping structure in the ITCN-A model proposed in this invention.

[0039] Figure 3 This is a flowchart of the present invention.

[0040] Figure 4(a) is a diagram of the decomposition results of the VMD method in this invention (Cuntan). Figure 4(b) is a decomposition result diagram of the VMD method in this invention (Wulong); Figure 4(c) is a diagram of the decomposition results of the VMD method in this invention (by Zhu Tuo).

[0041] Figure 5(a) is a comparison chart of the predicted value and the actual value using the present invention (Cuntan); Figure 5(b) is a comparison chart of the predicted value and the actual value using the present invention (Wulong); Figure 5(c) is a comparison chart of the predicted value and the actual value using the present invention (Zhu Tuo).

[0042] Figure 6 This is a comparison chart of the predicted and actual values ​​from the ablation experiment using the present invention.

[0043] Figure 7 This is a comparison chart of the predicted values ​​and actual values ​​obtained using the multi-step prediction method of this invention. Detailed Implementation

[0044] The present invention will now be described in further detail with reference to the accompanying drawings: This invention proposes a joint hourly runoff prediction method based on IRFO-VMD and ITCN-A, comprising: acquiring historical runoff data from hydrological stations; optimizing VMD parameters using IRFO; and decomposing the runoff time series into K inner IMFs using the optimized VMD. Subsequently, phase space reconstruction is used to extract dynamic features of the time series, and the data is then divided into training, validation, and test sets. The reconstructed data is then input into the ITCN-A model for prediction. First, multi-scale TCN is used for feature extraction, and then an adaptive nonlinear projection layer is used to enhance nonlinear modeling capabilities. Subsequently, an attention mechanism is used to weight and aggregate the feature sequences, highlighting key temporal features. Finally, a linear layer is used to output the runoff prediction results, and then the prediction results from all models are summarized to obtain the final runoff prediction. This invention has high prediction accuracy. The prediction process is as follows: Figure 3 As shown, the specific steps include the following: Step a: Collect runoff data and decompose it using VMD. Simultaneously, optimize the VMD parameters using the IRFO method to obtain the optimal decomposition result, ultimately obtaining K IMFs. The specific process is as follows: Step a1: RFO (Reverse Forward Search) is a heuristic optimization algorithm that achieves a balance between global search and local exploitation by simulating the foraging behavior of the Lüper fox. RFO effectively achieves global optimization through three stages: perception-driven prey search, cognitively regulated exploration-exploitation balance mechanism, and dynamic multi-source cooperative foraging strategy. However, standard RFO is prone to getting trapped in local optima in complex optimization tasks. To enhance the diversity of initial solutions and improve the algorithm's ability to escape local optima, an IRFO (Inverse Forward Search) is proposed. By introducing a reverse learning mechanism and combining it with a simulated annealing selection strategy, it effectively improves the global search efficiency and convergence speed. Reverse learning is a probabilistic heuristic search strategy that aims to accelerate convergence and improve the algorithm's ability to escape local optima by constructing a set of reverse individuals. Specifically, the reverse individuals are constructed as follows: ; in X i For the first i Individual.

[0045] If the fitness of the reversed solution is better than that of the original individual, then the original individual is replaced by the reversed solution; otherwise, the original individual remains unchanged. Its mathematical expression is as follows: ; To further improve the convergence efficiency of reverse learning, a reverse learning probability that dynamically decays with the number of iterations is introduced. p ( t Only if the random numbers follow a uniform distribution (0,1) r Less than the reverse learning probability p ( t When this occurs, a reverse learning operation is performed. The reverse learning probability is defined as follows: ; in: a and b These represent the termination probability and the initial probability, respectively, with values ​​of 0.05 and 0.4. t This represents the current iteration number. T This represents the maximum number of iterations.

[0046] Step a2: The simulated annealing selection strategy avoids the algorithm getting trapped in local optima by accepting inferior solutions with a certain probability, and guides the algorithm to convergence by gradually decreasing the temperature. Let the first step be... i The current solution for each individual is M i fitness is f ( M i After iteration, a new candidate solution is generated. X i1 The corresponding fitness is f( X i1 If the fitness difference between the two is defined as follows: ; The formula for updating individual positions using simulated annealing is as follows: ; in: r These are random numbers that follow a uniform distribution within the interval (0, 1). T e The temperature is the current temperature, and it decays with the number of iterations. The calculation is as follows: ; in: α (0,1) is the temperature decay coefficient. T f This is the lowest temperature.

[0047] Step a3: VMD is an adaptive signal decomposition method that aims to decompose complex non-stationary signals into several IMFs, each with a finite bandwidth and a specific center frequency. Its mathematical expression can be represented as: ; In the formula, f ( t ) represents the original vibration signal sequence. K Represents the number of decomposed modes. u k ( t ) indicates the first k One modal function, oh k Indicates the first k The center frequency of each modal function t Represents time, s ( t ) is the Dirac function.

[0048] By introducing Lagrange multipliers l and secondary penalty factor α This yields the extended Lagrange function. L The specific formula is defined as follows: ; The saddle points of the augmented Lagrangian function are solved by the alternating direction multiplier method, thereby optimizing each mode function. u k ( t and center frequency oh k The detailed description of the updated formula is as follows: ; ; ; In the formula, n Indicates the number of iterations. c For noise tolerance parameters, , , and Represent f ( t ), u i ( t ), l ( t )and The Fourier transform expression is obtained. Through iterative updates until a stopping condition is met, the following is obtained: K Modal functions u k ( t ) and its corresponding center frequency oh k .

[0049] Step a4: To achieve adaptive optimization of key VMD parameters, select the obtained IRFO for the number of VMD modes K and the penalty factor. α Optimization is performed, using the minimum envelope entropy as the fitness function. This decomposes the runoff sequence into K IMFs of different frequencies. The formula for the minimum envelope entropy is as follows: ; ; ; in: u i [ n [The first character after VMD decomposition] i One modality, f For the parameter components to be optimized, K Let be the number of modes in the VMD decomposition. n Let H{} be the index of the discrete time points, and H{} be the Hilbert transform. e i [ n ] is the first i The envelope amplitude of each mode, p i [ n ] represents the normalized envelope probability, and min{} represents the minimum envelope entropy.

[0050] Step b: Apply PSR to each IMF and divide it into training, validation, and test sets. The specific process is as follows: Step b1: PSR is an important method widely used in time series analysis. Its core idea is to appropriately select the embedding dimension. d and time delay t This maps the original one-dimensional time series to a high-dimensional phase space. Specifically, the phase space matrix is ​​constructed as follows: ; in: X i Represents a sequence sample; i For index, where i =1,2,3,……, R , R The length of the sequence after embedding the time delay. d For the embedded dimension; L The length of the original sequence.

[0051] Step b2: Divide the reconstructed data into training, validation and test sets according to an appropriate ratio to obtain the final dataset.

[0052] Step c: Input the reconstructed IMFs into the ITCN-A prediction model for modeling and training to obtain prediction results. The specific steps are as follows: Step c1: Construct a multi-scale TCN module to extract multi-layer temporal dependency features through parallel dilated convolutions with different kernels, thereby enhancing the expressive power of multi-scale features. Its structure is shown in Figure 1(a) and Figure 1(b): The TCN uses one-dimensional convolutions as its basic operational units to extract local temporal features of sequences along the time dimension. To maintain temporal causality, the TCN introduces a causal convolution mechanism in the one-dimensional convolution, ensuring that the output at the current moment depends only on the current and previous inputs, thus preventing future information leakage. To further enhance the modeling ability for long-term dependencies, dilated convolutions, by setting an interval stride in the convolution kernel, exponentially expand the receptive field, capturing long-term correlation information even in shallower networks. Furthermore, to alleviate the gradient decay problem in deep network training, the TCN employs a residual connection structure. Each residual block consists of two convolutional layers and a non-linear activation function, and the input features are directly added to the convolutional output through an identity mapping, achieving cross-layer information transfer and thus improving the training stability and convergence performance of the model. The TCN structure is shown in Figure 1.

[0053] Step c2: Design an adaptive nonlinear projection layer. This layer expands and re-compresses the feature space through nonlinear mapping between upper and lower projection modules to capture higher-order nonlinear relationships. The fusion ratio of the nonlinear projection is adaptively adjusted using learnable gating parameters. The adjusted nonlinear projection is then residually fused with multi-scale TCN information to further enhance the network's flexibility and stability. Its structure is as follows: Figure 2 As shown.

[0054] Step c3: To enhance the model's ability to represent key information in the sequence, a lightweight attention mechanism is introduced. Attention scores are generated along the time dimension through linear mapping, and a softmax function is used for weight normalization, thereby weighted aggregation of features based on the importance of each time step. Let the input feature sequence be... ,in B , T , C These represent batch size, time step, and feature dimension, respectively. The attention score in the time dimension can be obtained through linear mapping. S The calculation formula is as follows: ; in: and All are learnable parameters; This is the output after the residual connection mentioned above; S A score is given for attention.

[0055] The score vector along the time dimension is normalized using the softmax function to obtain the attention weights. a Subsequently, the features at each time step are weighted and summed to achieve feature aggregation along the time dimension. The calculation is as follows: ; in: and Represent t Attention weights and feature sequences at each time step; T For time steps; t This is the current time step; O This is the result of weighted aggregation.

[0056] Step c4: The ITCN-A model adopts a multi-layer deep fusion architecture. First, a multi-scale TCN module is constructed. Second, an adaptive nonlinear projection layer is connected in series. Through the nonlinear mapping of the upper and lower projection modules, the feature space is expanded and recompressed to capture higher-order nonlinear relationships. The fusion ratio of the nonlinear projection is adaptively adjusted through learnable gating parameters. The adjusted nonlinear projection is then residually fused with the multi-scale TCN information to further improve the network's flexibility and stability. Subsequently, an attention layer is connected in series. Through a feature weighting mechanism, differentiated weights are assigned to different scales and channels, thereby highlighting the contribution of key time steps and suppressing redundant information, enabling the model to more accurately focus on abrupt changes and major change patterns in the runoff process. Finally, a fully connected layer is connected in series to map the fused features to the output space, generating the final runoff prediction result.

[0057] Step d: The prediction results of each subsequence component are summed to obtain the final runoff prediction result.

[0058] Step g: Flow data from three hydrological stations in China—Cuntan, Wulong, and Zhutuo—were selected as experimental data. The data covered the period from May 1st to October 15th of a given year, comprising 4010 sampling points at hourly intervals. This data was used sequentially for model building, hyperparameter selection, and model validation. The training, validation, and test sets were set to 3000, 500, and 500 sets, respectively. The model's prediction results were evaluated using three metrics: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). The specific formulas for these metrics are as follows: ; ; ; In the formula: n Indicates the number of samples. and They represent the first i The actual and predicted values ​​of each sample.

[0059] To fully demonstrate the effectiveness of the proposed joint hourly runoff prediction method based on IRFO-VMD and ITCN-A, the proposed ITCN-A model was compared with five other models: LSTM, BiLSTM, BiGRU, TCN, and ICTN (which removes the adaptive nonlinear projection module from the proposed model). Seven models were also designed for ablation experiments as comparison models: TCN, M1 (multi-scale TCN), M2 (standard temporal convolutional network with attention mechanism), M3 (standard temporal convolutional network with adaptive nonlinear projection module), M4 (standard temporal convolutional network with both attention mechanism and nonlinear projection module), M5 (multi-scale temporal convolutional network with nonlinear projection module), and ITCN. Furthermore, the TCN and BiGRU models were selected as comparison models for multi-step prediction. To ensure experimental fairness, all models were decomposed using IRFO-VMD. The learning rate for both the proposed model and the comparison models at each site was set to 1×10⁻⁶. -4 The learning rate for the multi-step runoff prediction experiment was set to 5 × 10⁻⁶. -4 The prediction comparison results of the proposed model are shown in Figures 5(a) to 5(c), and the ablation experiment comparison results of the proposed model are shown in Figures 5(a) to 5(c). Figure 6 As shown, the multi-step prediction results of the proposed model are as follows: Figure 7As shown in Table 1, the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) were used to quantitatively evaluate the predictive performance of all experimental models.

[0060] From Figure 5(a) to Figure 5(c), Figure 6 , Figure 7 As can be seen, the runoff prediction curve of the ITCN-A model proposed in this invention is largely consistent with the actual curve, demonstrating high prediction accuracy and proving the superiority and effectiveness of this invention.

[0061]

[0062] Table 1 shows that the LSTM model performed poorly overall in the comparative experiments, with the lowest prediction accuracy. Compared to LSTM, BiLSTM improved across all metrics, with RMSE and MAE decreasing by 44.75% and 44.62% respectively at the Cuntan hydrological station. Furthermore, at the Wulong hydrological station, the BiGRU model outperformed BiLSTM, showing improvements across all metrics. Comparing the four initial models, TCN achieved the best prediction performance; its convolutional structure likely provides advantages in parallel feature extraction and gradient propagation, thus improving prediction accuracy. In contrast, the ITCN model, by introducing multi-scale convolution and attention mechanisms, demonstrated superior prediction performance compared to the basic models at all hydrological stations, indicating that the synergistic effect of multi-scale TCN and attention mechanisms enhances the model's comprehensive learning ability and generalization performance for runoff sequences at different time scales. Adding an adaptive nonlinear mapping module further improved the prediction accuracy of the proposed model. At the Cuntan hydrological station, the proposed model's RMSE, MAE, and MAPE decreased by 22.99%, 21.1%, and 20.0%, respectively, compared to ITCN. At the Wulong hydrological station, the corresponding indicators decreased by 22.1%, 22.6%, and 21.4%, respectively, indicating that the adaptive nonlinear mapping module can effectively enhance the model's ability to represent the nonlinear characteristics of runoff sequences. Compared to the aforementioned two stations, BiGRU's prediction results at the Zhutuo hydrological station were better than TCN. This may be related to the relatively stable runoff process at this station, indicating that BiGRU has better modeling ability in sequences with weak volatility, while TCN is more effective in capturing changing trends in sequences with strong volatility. In contrast, the proposed model still performs best in all indicators, indicating that the model has strong performance and reliability in runoff prediction.

[0063] In summary, the proposed model achieved the best performance across all evaluation indicators in runoff prediction tasks at different hydrological stations, demonstrating strong generalization ability and prediction stability, and fully verifying the synergistic advantages brought about by the integration of various modules.

[0064] In the ablation experiments, M2 and M3 performed poorly in the runoff prediction task, with all indicators lower than the standard TCN. This suggests that introducing an attention layer or adaptive nonlinear mapping module alone may introduce information redundancy, thereby reducing model performance. Based on the standard TCN, combining an attention layer and an adaptive nonlinear projection module, M4 outperformed TCN in all indicators, with RMSE reduced by 6.1% and MAE by 8.7%, showing that the synergistic effect of the two modules can effectively improve prediction accuracy and robustness. In contrast, the multi-scale TCN performed better in the prediction task, indicating that multi-scale convolution can enhance the model's ability to capture local trends. Introducing adaptive nonlinear projection on top of the multi-scale TCN further improved the performance of M5, demonstrating that nonlinear projection can synergistically optimize model performance with multi-scale convolution. Introducing an attention layer on top of M1 yielded the ITCN, which reduced RMSE by 17.9% and MAE by 22.8% compared to M1, but was still slightly lower than the proposed model. By further introducing adaptive nonlinear projection, the proposed model performs best on all evaluation metrics, demonstrating that the synergistic effect of multi-scale convolution, attention mechanism and adaptive nonlinear projection can effectively improve the prediction accuracy and robustness of the model.

[0065] In multi-step prediction experiments, the TCN model and the BiGRU model performed similarly across various metrics. The proposed model achieved optimal results across all evaluation metrics, demonstrating its superior predictive performance in multi-step prediction tasks.

Claims

1. A joint hourly runoff prediction method based on IRFO-VMD and ITCN-A, characterized in that... Includes the following steps: Step 1: Collect runoff data; Step 2: Optimize the parameters of VMD using IRFO, and decompose the runoff time series into K intrinsic mode functions (IMFs) using the optimized VMD. Step 3: Use phase space reconstruction (PSR) to extract the dynamic features of runoff time series, and then divide the runoff data into training set, validation set and test set; Step 4: Train the ITCN-A model using the training set and validation set containing the dynamic features of runoff time series obtained in Step 3, and then use the trained model to predict the test set.

2. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 1, characterized in that: It also includes step 5: using three evaluation indicators, root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), to analyze the model's prediction results.

3. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 1, characterized in that: Step 2 includes: proposing an IRFO, which effectively improves global search efficiency and convergence speed by introducing a reverse learning mechanism and combining it with a simulated annealing selection strategy; In the reverse learning mechanism, the reverse individual is constructed as follows: （1）； In equation (1), Represents an individual that learns in reverse; Indicates the lower bound of an individual; Indicates the upper bound of an individual; For the first i Individual; If the fitness of the reversed solution is better than that of the original individual, then the original individual is replaced by the reversed solution; otherwise, the original individual is left unchanged. Its mathematical expression is as follows: （2）； In equation (2), express X op The fitness function value; express X i The fitness function value; Introducing a back-learning probability that dynamically decays with the number of iterations Only if the random numbers follow a uniform distribution (0,1) r Less than the reverse learning probability When the reverse learning operation is performed, the reverse learning probability is defined as follows: （3）； In equation (3), a and b These represent the termination probability and the initial probability, respectively. t This represents the current iteration number; T This represents the maximum number of iterations.

4. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 3, characterized in that: The simulated annealing selection strategy avoids the algorithm getting trapped in local optima by accepting inferior solutions with a certain probability, and guides the algorithm to convergence by gradually decreasing the temperature; let the th... i The current solution for each individual is M i fitness is f ( M i After iteration, a new candidate solution is generated. X i1 The corresponding fitness is f ( X i1 If the fitness difference between the two is defined as follows: （4）； In equation (4), This represents the difference in fitness function between the candidate solution and the current solution; The formula for updating individual positions using simulated annealing is as follows: （5）； In equation (5), r These are uniformly distributed random numbers following the interval (0, 1). T e The temperature is the current temperature, and it decays with the number of iterations. The decayed temperature is calculated as follows: （6）； In equation (6), This indicates the updated temperature. α (0,1) represents the temperature decay coefficient; T f This is the lowest temperature.

5. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 4, characterized in that: VMD decomposes complex non-stationary signals into several intrinsic mode functions (IMFs), each with a finite bandwidth and a specific center frequency. Its mathematical expression is as follows: （7）； In equation (7), u k Indicates the first k One modal function; ω k Indicates the first k The center frequency of each modal function; Represents the set of modal functions; This represents the set of center frequencies corresponding to the modal functions; K Represents the number of decomposed modes; This represents taking the partial derivative with respect to time; It is the Dirac function; Represents the imaginary unit; t Represents time; Indicates the first k One modal function; Represents the multiplication operator; Represents the L2 norm; This represents the original vibration signal sequence; Introducing Lagrange multipliers λ and secondary penalty factor α This yields the extended Lagrange function. L The specific formula is defined as follows: （8）； In equation (8), Represent the Lagrange function; Represents the Lagrange multiplier that changes over time; Indicates the inner product; The saddle points of the augmented Lagrangian function are solved by the alternating direction multiplier method, thereby optimizing each mode function. u k ( t and center frequency ω k The detailed description of the updated formula is as follows: （9）； （10）； （11）； In the above formula, , , and Represent f ( t ), u i ( t ), λ ( t ), The Fourier transform expression; Indicates the first k Frequency domain representation of the modal functions; Indicates the first n +Lagrange multiplier at +1 iteration; Indicates the first n Lagrange multipliers at the next iteration; Indicates the first n In the +1st iteration, the... k The center frequency of each modal function; Indicates frequency; By iteratively updating until the stopping condition is met, we obtain... K Modal functions u k ( t ) and its corresponding center frequency ω k .

6. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 5, characterized in that: To achieve adaptive optimization of key VMD parameters, the IRFO obtained above is selected to determine the VMD modality number K and penalty factor. α Optimization is performed, using the minimum envelope entropy as the fitness function; thus, the runoff sequence is decomposed into K intrinsic mode functions (IMFs) of different frequencies; the formula for the minimum envelope entropy is as follows: （12）； （13）； （14）； In the above formula, These are the parameter components to be optimized; min{} represents the minimum envelope entropy; K The number of modes in the VMD decomposition; Indicates the number of samples; p i [ n [ ] represents the normalized envelope probability; n For discrete time points; e i [ n ] is the first i The envelope amplitude of each mode; H{} is the Hilbert transform.

7. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 6, characterized in that: In step 3, phase space reconstruction (PSR) is performed on each intrinsic mode function (IMF), and the dataset is divided into training, validation, and test sets; specifically as follows: 3.1: Phase Space Reconstruction (PSR) through appropriate selection of embedding dimensions d and time delay τ This maps the original one-dimensional time series to a high-dimensional phase space. （15）； In the above formula, X i Represents a sequence sample; i For index, where i =1,2,3,……, R , R The length of the sequence after embedding the time delay. d For the embedded dimension; L The length of the original sequence; Represents the reconstructed phase space vector; Indicates transpose; R Represents the number of sampling points; Indicates the first sampling point. Indicates the 1st+ τ One sampling point; Indicates the 1st + ( d -1) τ Each sampling point Indicates the first i One sampling point; Indicates the first i + τ One sampling point; Indicates the first i +( d -1) τ One sampling point; Indicates the first R -( d -1) τ One sampling point; Indicates the first R -( d -2) τ One sampling point; Indicates the first R One sampling point; The dynamic features of runoff time series are extracted by phase space reconstruction (PSR). Specifically, the original runoff data is reconstructed by phase space reconstruction using equation (15) to obtain the reconstructed runoff data. 3.2: The reconstructed data containing the dynamic features of runoff time series is divided into training set, validation set and test set according to an appropriate ratio to obtain the final dataset.

8. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 7, characterized in that: In step 4, the ITCN-A model is trained using the training set and validation set containing the dynamic features of runoff time series obtained in step 3, and the trained model is then used to predict the test set to obtain the prediction results; the specific steps are as follows: 4.1: Construct a multi-scale TCN module to extract multi-layer runoff temporal dependency features through parallel dilated convolution with different convolution kernels, thereby enhancing the expressive power of multi-scale features; TCN uses one-dimensional convolution as its basic operational unit to extract local temporal features of a sequence along the time dimension; The mathematical expression is as follows: （16）； in: T t Represents the output of a one-dimensional convolution. W i These are the parameters of the convolution kernel. s t-i For the first t - i The input sequence at each time step, k -1 represents the kernel size. i Represents an index. b For bias parameters, t For time indexing, s The input sequence; To preserve temporal causality, TCN introduces a causal convolution mechanism into one-dimensional convolution, ensuring that the output at the current time step depends only on the current and previous inputs; the mathematical expression is as follows: （17）； TCN employs a residual connection structure; each residual block consists of two convolutional layers and a nonlinear activation function; and it achieves cross-layer information transfer by directly adding the input features to the convolutional output through an identity mapping; the mathematical expression is as follows: （18）； （19）； Wherein: Conv1 and Conv2 are two convolutional layers; f 1 and f 2 are nonlinear activation functions; CC( s ) is the residual block output; s Represents the input sequence; J This represents the final output of the residual connection; 4.2: An adaptive nonlinear projection layer is designed to expand and re-compress the runoff characteristic space through nonlinear mapping between upper and lower projection modules, thereby capturing higher-order nonlinear relationships; the mathematical expression is as follows: （20）； （21）； in: and φ These are nonlinear activation functions; W up and W down These are the learnable weights for the upper and lower projection layers, respectively; b up and b down These are the offset parameters for the upper and lower projection layers, respectively; h up and h down These are the outputs of the upper and lower projection layers, respectively. s The input sequence; The fusion ratio of the nonlinear projection is adaptively adjusted by learnable gating parameters, as shown in the following mathematical expression: （22）； in: g For gated functions, h down This is the output of the downward projection; s The input sequence; For gated fusion output; For element-wise multiplication; The adjusted nonlinear projection is residually fused with the information from the multi-scale TCN module, and the mathematical expression is as follows: （23）； in: The output after fusion; 4.3: To enhance ITCN-A's ability to represent key sequence information, a lightweight attention mechanism is introduced. Attention scores are generated along the time dimension through linear mapping, and a softmax function is used for weight normalization, thereby weighted aggregation of runoff features based on the importance of each time step; details are as follows: Let the input feature sequence be , Let represent a three-dimensional feature vector with dimensions B, T, and C, where B , T , C These represent batch size, time step, and feature dimension, respectively. Attention scores in the time dimension can be obtained through linear mapping. S The calculation formula is as follows: （24）； In equation (24), and All are learnable parameters; This is the output after the residual connection mentioned above; S Score for attention; The score vector in the time dimension is normalized using the softmax function to obtain the attention weights. a The specific expression is as follows: （25）； in: t In terms of time dimension, a t represent t Attention weight at any moment S t represent t Attention score at any moment S i represent i Attention score at any moment i Represents an index. Represents an exponential function; Subsequently, the features at each time step are weighted and summed to achieve feature aggregation along the time dimension. The calculation form is as follows: （26）； In equation (26), and Represent t Attention weights and feature sequences at each time step; T For time steps; t This is the current time step; O This is the result of weighted aggregation; 4.4: The weighted aggregation results are used to obtain the runoff prediction results for each IMF component through linear layer mapping. The mathematical expression is as follows: （27）； in: W z Represents learnable parameters; b z Represents the paranoia vector; Y z This represents the runoff prediction results for each IMF component.

9. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 8, characterized in that: In step 4, the ITCN-A model adopts a multi-layered, deeply integrated overall architecture design: First, construct a multi-scale TCN module based on equations (16) to (19); Secondly, a series adaptive nonlinear projection layer, as shown in equations (20) to (21), expands and re-compresses the feature space through the nonlinear mapping of the upper and lower projection modules to capture higher-order nonlinear relationships. Meanwhile, learnable gating parameters, as shown in Equation (22), are introduced to adaptively adjust the fusion ratio of the nonlinear projection; The network's flexibility and stability are further enhanced by residual fusion of the regulated nonlinear projection with information from the multi-scale TCN module. Subsequently, the attention layer is connected in series, as shown in equations (24) to (26). The feature weighting mechanism assigns differentiated weights to different scales and channels. This highlights the contribution of key time steps and suppresses redundant information; Finally, as shown in Equation (27), the cascaded attention layer maps the fused features to the output space using a linear layer to obtain the prediction results of each IMF component, and then accumulates the prediction results of each component to generate the final runoff prediction result.

10. The hourly runoff joint prediction method based on IRFO-VMD and ITCN-A according to claim 2, characterized in that: In step 5, the specific calculation formulas for the three evaluation indicators—Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE)—are as follows: （28）； （29）； （30）； In the above formula: n Indicates the number of samples. and They represent the first i The actual and predicted values ​​of each sample.

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