Short-term wind speed prediction method based on BDWPT-KAN model

Abstract

The invention relates to the technical field of wind speed prediction, and provides a short-term wind speed prediction method based on a BDWPT-KAN model, and the method comprises the steps: 1, obtaining original wind speed data; 2, carrying out data preprocessing on the original wind speed data; 3, for the preprocessed wind speed data, removing high-frequency noise in the original data by adopting Bayesian wavelet packet threshold noise reduction to obtain the denoised wind speed data; 4, dividing a training set and a test set; step 5, performing Z-score standardization processing on the wind speed data after the noise reduction processing; step 6, constructing a KAN model for wind speed prediction; 7, evaluating and verifying the trained KAN model by using the test set; and 8, according to the evaluation index obtained through calculation, iterating the step 6 and the step 7 to obtain an optimized wind speed prediction model. According to the method, a high-confidence prediction result can be provided, and reliable support is provided for application scenes such as wind power dispatching and power grid operation control.

Description

Technical Field

[0001] This invention relates to the field of wind speed prediction technology, and in particular to a short-term wind speed prediction method based on the BDWPT-KAN model. Background Technology

[0002] Driven by the global energy crisis, wind energy, as a clean, renewable, and zero-emission energy source, has received widespread global attention. However, the development and grid connection of wind energy face significant challenges, primarily due to the highly nonlinear, random, and intermittent nature of wind speed signals. These characteristics complicate tasks such as stable operation of wind power generation, grid dispatching, and power prediction, leading to increased system operating costs and enhanced power volatility. Therefore, high-precision short-term wind speed prediction has become a key technology for improving the efficiency and safety of wind power grid connection.

[0003] Currently, wind speed forecasting methods are mainly divided into two categories: those based on physical models and those based on data. The former constructs dynamic wind field models and extrapolates wind speed based on numerical weather prediction and topographic factors, suitable for medium- and long-term forecasts. However, its prediction resolution is limited due to high computational costs and heavy reliance on geographical and atmospheric boundary conditions. The latter includes traditional statistical models and artificial intelligence models, relying on historical wind speed and meteorological data to extract time-series evolution patterns. In recent years, with the development of AI technology, deep learning architectures have shown significant advantages in handling the time-series dependence and nonlinear modeling of wind speed.

[0004] Furthermore, to further improve prediction accuracy, preprocessing strategies have been widely introduced, including wavelet decomposition, empirical mode decomposition, and singular spectrum analysis. Among these, denoising, as an important means of improving data quality, can effectively suppress high-frequency disturbances, reduce model input noise, and improve model robustness. However, traditional wavelet soft-threshold denoising suffers from problems such as manual parameter setting, threshold sensitivity, and easy loss of effective high-frequency information. Summary of the Invention

[0005] This invention primarily addresses the technical problems of traditional denoising methods when processing wind speed data, such as over-denoising or under-denoising, leading to the loss of effective feature information or noise residue. It also addresses the limitations of traditional neural network models, which generally use fixed activation functions and have limited ability to fit high-frequency nonlinear features in wind speed signals, making it difficult to fully explore the dynamic changes in wind speed sequences. The invention proposes a short-term wind speed prediction method based on the BDWPT-KAN model. The aim is to construct a high-performance wind speed prediction framework with good generalization capabilities, adapting to wind speed prediction tasks in different wind farm environments and at multiple time scales, providing high-confidence prediction results, and offering reliable support for applications such as wind power dispatch and grid operation control.

[0006] The application provides a short-term wind speed prediction method based on a BDWPT-KAN model, comprising the following processes:

[0007] Step 1, obtaining original wind speed data;

[0008] Step 2, performing data preprocessing on the original wind speed data;

[0009] Step 3, for the preprocessed wind speed data, adopting Bayesian wavelet packet threshold denoising to remove high-frequency noise in the original data, to obtain denoised wind speed data;

[0010] Step 4, dividing the denoised wind speed data into a training set and a test set;

[0011] Step 5, performing Z-score standardization processing on the denoised wind speed data;

[0012] Step 6, constructing a KAN model for wind speed prediction;

[0013] Step 7, using the test set to evaluate and verify the trained KAN model;

[0014] Step 8, according to the calculated evaluation index, manually optimizing the hyperparameters in the KAN model, and iterating steps 6 and 7 to obtain an optimized wind speed prediction model.

[0015] Further, step 2 comprises steps 201 to 202 as follows:

[0016] Step 201, performing data inspection on the original wind speed data to ensure that it does not contain non-numerical records;

[0017] Step 202, using an interpolation regression method to fill in missing values to ensure the integrity of the time series.

[0018] Further, step 3 comprises steps 301 to 303 as follows:

[0019] Step 301, using Discrete Wavelet Packet Transform (DWPT) to perform multi-scale time-frequency domain signal decomposition processing on the original wind speed data f(t i )(i=1,2,...,N); the wind speed signal can be decomposed as follows:

[0020]

[0021] Step 302, for the decomposed high-frequency wavelet coefficients s j (k) and low-frequency scale coefficients w j (k), adopting Bayesian hypothesis testing to identify and eliminate noise;

[0022] Let the kth coefficient in the jth decomposition be denoted as d jk , and replace the decomposed wavelet coefficient s j (k) and scale coefficient w j (k) with this symbol i If f(t jk is decomposed by a three-level discrete wavelet packet transform, eight decomposition coefficients d j are obtained, including four wavelet coefficients s j (k) and four scale coefficients w jk (k);

[0023] Each d jk is considered to have two terms: a noise-free decomposition coefficient and a noise term ε j satisfying the standard normal distribution, as shown in the following formula:

[0024]

[0025] where σ jk represents the noise variance of the jth decomposition coefficient; the noise term ε jk is removed from d by hypothesis testing to obtain the noise-reduced coefficient

[0026] Step 303, reconstruct the original sequence according to the noise-reduced coefficient to obtain the noise-reduced wind speed sequence data The specific reconstruction method is as follows:

[0027]

[0028] Further, the Z-score standardization processing formula is as follows:

[0029]

[0030] where X is the Z-score standardized training set data, which is the input of the KAN model for wind speed prediction; x is the original data, μ is the mean of the training set data, and σ is the standard deviation of the training set data.

[0031] Further, in step 6, the KAN model uses the Adam optimizer to update the parameters, and the loss function selects the mean square error; the Z-score standardized training set data is used as the input to iteratively train the KAN model, and the KAN network weight parameters are optimized to minimize the prediction error;

[0032] The activation function is represented as follows:

[0033] φ(x) = w b b(x) + w s spline(x) (5)

[0034] where w b and w s are trainable weight parameters for improving the expressive ability of the activation function, b(x) represents the SiLU initialization function, and spline(x) represents the spline function, which is defined as:

[0035]

[0036] where B i (x) represents the i-th B-spline basis function, and c i is the corresponding learnable coefficient;

[0037] The above single-layer activation structure is extended to a multi-layer network structure to form a deep KAN network, and the overall mapping process is represented as:

[0038]

[0039] where Φ l represents the activation function matrix of the l-th layer, and L is the total number of layers; the specific input-output relationship of the l-th layer can be represented as:

[0040]

[0041] where x l is the input vector of the l-th layer; x l+1 is the output vector of the l+1-th layer; n l+1 and n l are the number of neurons of the l+1-th layer and the l-th layer, respectively; is the learnable spline activation function corresponding to the j-th neuron of the l-th layer to the i-th neuron of the l+1-th layer.

[0042] The short-term wind speed prediction method based on the BDWPT-KAN model provided by the application fuses the Bayesian wavelet packet threshold denoising (BDWPT) and the KAN model to perform wind speed prediction, and significantly improves the accuracy and stability of wind speed prediction. The method effectively filters out high-frequency noise interference in the wind speed data through wavelet multi-scale decomposition and Bayesian dynamic threshold technology, while retaining the key time sequence change characteristics, and can efficiently suppress noise while retaining key information. The application introduces the KAN network structure with strong nonlinear fitting capability, uses the spline activation function on the edge which can be learned to accurately model the complex dynamic law in the wind speed sequence; can prevent the spectrum deviation phenomenon of the traditional neural network model, further improve the fitting capability of the complex nonlinear relationship in the wind speed sequence, so as to realize the construction of the high-precision and interpretable wind speed prediction model. Compared with the traditional neural network, KAN has stronger high-frequency modeling capability and better interpretability, and can capture the rapid change of wind speed while maintaining the stability of prediction error.

[0043] Through the robust preprocessing mechanism and the flexible modeling framework, the application can still realize accurate prediction of short-term wind speed when facing complex and variable weather conditions, has good generalization ability and engineering practical value, and provides reliable technical support for safe operation and intelligent scheduling of wind farms. BRIEF DESCRIPTION OF DRAWINGS

[0044] Figure 1 is the implementation flowchart of the short-term wind speed prediction method based on the BDWPT-KAN model provided by the application;

[0045] Figure 2 is a process schematic diagram of the KAN model for wind speed prediction;

[0046] Figure 3 is a one-step, two-step and three-step wind speed point prediction result diagram based on the KAN model. DETAILED DESCRIPTION

[0047] In order to make the technical problems solved by the application, the technical solutions adopted and the technical effects reached more clear, the application will be further described in detail below in combination with the drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the application, not to limit the application. In addition, it should be noted that, in order to facilitate description, only the parts related to the application are shown in the drawings, not all contents.

[0048] As shown in Figure 1 , the short-term wind speed prediction method based on the BDWPT-KAN model provided by the embodiment of the application includes the following processes:

[0049] Step 1, obtaining original wind speed data.

[0050] Step 2, data preprocessing of the original wind speed data. Step 2 includes the following steps 201 to step 202:

[0051] Step 201, data checking of the original wind speed data to ensure that there is no non-numeric record in it.

[0052] Step 202, filling in the missing values by using the interpolation regression method to ensure the integrity of the time series.

[0053] Step 3, for the preprocessed wind speed data, Bayesian wavelet packet threshold denoising is used to remove high-frequency noise in the original data to obtain denoised wind speed data.

[0054] After the original data is filled with missing values in the time series data by interpolation regression, Bayesian wavelet packet threshold denoising (Bayesian Discrete Wavelet Packet Transform, BDWPT) is used. Bayesian wavelet packet threshold denoising is mainly composed of three parts: discrete wavelet packet transform, Bayesian adaptive threshold denoising and signal reconstruction. Step 3 includes the following steps 301 to step 303:

[0055] Step 301, using discrete wavelet packet transform (Discrete Wavelet Packet Transform, DWPT) to perform multi-scale time-frequency domain signal decomposition processing on the original wind speed data f(t i )(i = 1, 2,..., N). The wind speed signal can be decomposed as follows:

[0056]

[0057] Step 302, for the decomposed high-frequency wavelet coefficients s j (k) and low-frequency scale coefficients w j (k), noise identification and elimination are performed by using Bayesian hypothesis testing.

[0058] Let the kth coefficient in the jth decomposition be d jk , and use this symbol to replace the decomposed wavelet coefficients s j (k) and scale coefficients w j (k). If f(t i ) is decomposed by three levels of discrete wavelet packet transform, i.e. j = 3, eight decomposition coefficients d jk can be obtained, including four wavelet coefficients s j (k) and four scale coefficients w j (k).

[0059] For each d jkIt is considered to have two items: noise-free decomposition coefficients and noise items ε satisfying the standard normal distribution jk , which is expressed as follows:

[0060]

[0061] where σ j represents the noise variance of the jth decomposition coefficient. d jk is tested by hypothesis testing. jk , to obtain the noise-reduced coefficient

[0062] Step 303, according to the noise-reduced coefficient , the original sequence is reconstructed to obtain the noise-reduced wind speed sequence data The specific reconstruction method is as follows:

[0063]

[0064] Step 4, divide the training set and test set for the noise-reduced wind speed data.

[0065] The training set and test set are divided according to the ratio of 80% and 20%.

[0066] Step 5, the wind speed data after noise reduction is processed by Z-score standardization.

[0067] First, calculate the mean and standard deviation of the training set based on the training set data, and then use the mean and standard deviation of the training set data to standardize the training set and test set data respectively. This method can map all data to a distribution range with a mean of 0 and a standard deviation of 1, effectively eliminating the dimensional differences between different features and improving the stability and convergence efficiency of model training. The Z-score standardization formula is as follows:

[0068]

[0069] where X is the Z-score standardized training set data, which is the input of the KAN model for wind speed prediction; x is the original data, μ is the mean of the training set data, and σ is the standard deviation of the training set data.

[0070] Step 6, construct a KAN model for wind speed prediction.

[0071] The KAN model uses the Adam optimizer to update parameters, and the loss function selects the mean square error (MSE). The Z-score standardized training set data is used as input to iteratively train the KAN model, and the KAN network weight parameters are optimized to minimize the prediction error.

[0072] KAN (Kolmogorov-Arnold) model is a new type of neural network structure based on Kolmogorov-Arnold representation theorem. The core idea is to realize the precise modeling of complex high-frequency changes by using a learnable parameterized spline function to perform nonlinear mapping on the input. This method is widely applicable to time series prediction tasks, and is particularly suitable for dealing with nonlinear, sudden and time-varying dynamic characteristics in wind speed signals.

[0073] Kolmogorov-Arnold representation theorem states that any continuous multivariate function f(x1, x2,..., x n ) can be represented as a combination of a finite number of univariate functions and addition operations:

[0074]

[0075] where Φ q and φ q,p are univariate functions; x p represents the input feature dimension.

[0076] KAN model is inspired by this and its innovation lies in deploying a learnable spline function on the network edge instead of a node activation function, which enhances the fitting ability of high-frequency and nonlinear characteristics. It can effectively solve the frequency bias phenomenon of traditional MLP fitting high-frequency parts.

[0077] In KAN (Kolmogorov-Arnold) model, the univariate functions Φ q and φ q,p in the original Kolmogorov representation theorem are replaced by learnable activation functions. To achieve learnability, a spline function is used to parameterize the activation function, and a B-spline function is used as the basis function of the activation function for learning. The activation function is represented as follows:

[0078] φ(x) = w b b(x) + w s spline(x) (5)

[0079] where w b and w s are trainable weight parameters used to improve the expression ability of the activation function, b(x) represents the SiLU initialization function, and spline(x) represents the spline function, which is defined as:

[0080]

[0081] where B i (x) represents the i-th B-spline basis function, c iare the corresponding learnable coefficients. This structure makes the activation function highly flexible, which can effectively fit the characteristics of nonlinear high-frequency signals.

[0082] The single-layer activation structure is extended to a multi-layer network structure to form a deep KAN network, and the overall mapping process is represented as:

[0083]

[0084] wherein l represents the activation function matrix of the lth layer, and L is the total number of layers. The specific input-output relationship of the lth layer can be represented as:

[0085]

[0086] x l is the input vector of the lth layer; x l+1 is the output vector of the l+1th layer; n l+1 and n l are the numbers of neurons of the l+1th layer and the lth layer, respectively; is the learnable spline activation function corresponding to the jth neuron of the lth layer to the ith neuron of the l+1th layer.

[0087] Each connection edge in the above structure corresponds to an independent learnable spline activation function, thereby enhancing the representation ability and generalization ability of the network in modeling high-dimensional nonlinear time series data. This structure of KAN is particularly suitable for wind speed prediction tasks with high-frequency disturbance and dynamic change characteristics, and can realize accurate modeling and prediction of wind speed values at future time points. The process of using the KAN model for wind speed prediction is shown in Figure 2 , wherein y t represents the wind speed value at time t, the wind speed at the past k time points is taken as the input, and then the wind speed prediction value at the future p time points is obtained through the multi-layer KAN network.

[0088] Step 7, using the test set, evaluating and verifying the trained KAN model.

[0089] Step 8, according to the calculated evaluation index, manually optimizing the hyperparameters in the KAN model, and iterating steps 6 and 7, finally obtaining an optimized wind speed prediction model.

[0090] The method proposed in the present application is verified by taking the actual operation data of a Norwegian offshore wind farm as an example:

[0091] The embodiment adopts the operation data of the SN2 station of the SCADA platform of a certain offshore wind farm in China at a distance of 150 m from the ground, with a time range from January 1, 2019 to June 30, 2019, and a total of 4344 sample points. First, the original wind speed data is preprocessed to eliminate outliers and interpolate missing values, thereby obtaining high-quality data for subsequent wind speed prediction model construction.

[0092] The preprocessed wind speed sequence is decomposed into 3-layer wavelet packet coefficients by discrete wavelet packet transform (DWPT). The noise threshold of each layer of coefficients is dynamically calculated by Bayesian hypothesis testing, and the coefficients below the threshold are set to zero, and only the high-frequency detail features are retained. Then, the denoised wind speed sequence is reconstructed by inverse wavelet packet transform.

[0093] The processed data is divided into training set and test set in the ratio of 8:2, and then the mean and standard deviation of the training set data are calculated, and the two statistics are used to perform Z-score standardization processing on the training set and test set respectively, so as to convert them to the interval with mean value of 0 and standard deviation of 1, so as to eliminate the difference between different dimensions.

[0094] A Kolmogorov-Arnold network (KAN) prediction model is constructed using the training set. The KAN model introduces a learnable parameterized spline activation function on the network connection edge to enhance the fitting ability of the model to high-frequency nonlinear features. The connection between each layer of neurons corresponds to an independent spline function, which is a weighted combination of linear functions and B-spline basis functions, thereby realizing flexible nonlinear mapping of input features.

[0095] During the model training process, the mean square error (MSE) is used as the loss function to minimize the error between the predicted value and the true value. The network output is calculated by forward propagation, and the loss function is combined for back propagation to update the weight parameters and spline function coefficients of each layer. The optimization algorithm uses Adam to improve the convergence efficiency and training stability.

[0096] During the training process, to prevent model overfitting and improve generalization ability, only the training set data is used for parameter updating. The mean and standard deviation used for standardization processing are only derived from the training set to avoid information leakage of the test set.

[0097] After the model training is completed, the test set data is input into the trained KAN model to output the corresponding single-step or multi-step wind speed prediction results. By comparing with the true value, the prediction performance and robustness of the method in the short-term wind speed prediction task are evaluated.

[0098] Single-step and multi-step wind speed prediction using KAN model with manually adjusted parameters

[0099] Table 1 compares the performance indicators of the four wind speed prediction methods, covering the mean absolute error (MAE), mean square error (MSE), and root mean square error (RMSE). The results show that the BDWPT-KAN method performs best in all three indicators in the one-step, two-step, and three-step wind speed prediction tasks: the RMSE of single-step prediction (0.1258 m / s) is only about 14% of the original KAN (0.0359 m / s), the MAE (0.0917 m / s) is reduced by 44.66% compared to BDWPT-LSTM (0.1657 m / s) and by 34.17% compared to BDWPT-Informer (0.1393). This result not only verifies the effectiveness of the BDWPT method in wind speed data denoising, but also further demonstrates the synergistic optimization advantage brought by the combination of BDWPT adaptive denoising technology and KAN high-frequency feature modeling capability. Through this combined model, the fitting ability and prediction accuracy of non-stationary and strongly fluctuating wind speed signals are significantly improved, showing good generalization performance and engineering application prospects.

[0100] Figure 3 The one-step, two-step, and three-step wind speed point prediction results based on the KAN model are shown. The red curve represents the model prediction value, and the blue curve represents the actual observation value. It can be seen that the KAN model prediction curve is highly consistent with the true value in terms of overall trend and local fluctuation, and the error distribution remains stable on the time axis without systematic deviation. This indicates that the model has good short-term dynamic modeling capability and can effectively capture the time series characteristics of wind speed changes. The experimental results verify the high-precision prediction ability of the KAN model in handling complex nonlinear wind speed sequences, making it suitable for practical application scenarios such as wind power dispatching that require high real-time and accuracy.

[0101] Table 1 Performance indicators of four wind speed prediction methods

[0102]

[0103] In summary, the wind speed prediction method proposed in the present application, which combines Bayesian wavelet packet adaptive denoising and Kolmogorov-Arnold network (KAN) modeling, significantly improves the accuracy and stability of wind speed prediction. This method effectively filters out high-frequency noise interference in wind speed data through wavelet multi-scale decomposition and Bayesian dynamic threshold technology, while preserving key time series characteristics. The KAN network structure with strong nonlinear fitting capability is introduced, and its edge learnable spline activation function is used to accurately model the complex dynamic rules in wind speed sequences.

[0104] Compared with traditional neural networks, KAN has stronger high-frequency modeling capability and better interpretability, and can capture the rapid changes of wind speed while keeping the stability of prediction error. Through the robust preprocessing mechanism and the flexible modeling framework, this method can still achieve accurate prediction of short-term wind speed under complex and variable weather conditions, has good generalization ability and engineering practical value, and provides reliable technical support for the safe operation and intelligent scheduling of wind farms.

[0105] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: the modification of the technical solutions recorded in the foregoing embodiments, or the equivalent replacement of part or all of the technical features, does not make the essence of the corresponding technical solution deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A short-term wind speed prediction method based on the BDWPT-KAN model, characterized in that, Includes the following processes: Step 1: Obtain raw wind speed data; Step 2: Perform data preprocessing on the raw wind speed data; Step 3: For the preprocessed wind speed data, Bayesian wavelet packet thresholding is used to remove high-frequency noise from the original data to obtain the denoised wind speed data. Step 4: Divide the denoised wind speed data into training and testing sets; Step 5: Perform Z-score standardization on the noise-reduced wind speed data; Step 6: Construct the KAN model for wind speed prediction; Step 7: Use the test set to evaluate and validate the trained KAN model; Step 8: Based on the calculated evaluation index, manually fine-tune the hyperparameters in the KAN model, and iterate through steps 6 and 7 to obtain the optimized wind speed prediction model.

2. The short-term wind speed prediction method based on the BDWPT-KAN model according to claim 1, characterized in that, Step 2 includes the following steps 201 to 202: Step 201: Perform a data check on the raw wind speed data to ensure that it does not contain any non-numerical records; Step 202: Use interpolation regression to fill in missing values ​​to ensure the integrity of the time series.

3. The short-term wind speed prediction method based on the BDWPT-KAN model according to claim 1, characterized in that, Step 3 includes the following steps 301 to 303: Step 301: Use Discrete Wavelet Packet Transform (DWPT) to process the raw wind speed data f(t). i (i = 1, 2, ..., N) undergoes multi-scale time-frequency domain signal decomposition processing; the wind speed signal can be decomposed into the following equation: Step 302, for the decomposed high-frequency wavelet coefficients s j (k) and low-frequency scaling coefficient w j (k) uses Bayesian hypothesis testing for noise identification and elimination; Let the k-th coefficient in the j-th decomposition be denoted as d. jk This symbol is used to replace the decomposed wavelet coefficients s. j (k) and scaling factor w j (k), if f(t) is subjected to discrete wavelet packet transform i The three-level decomposition yields eight decomposition coefficients d. jk This includes four wavelet coefficients s j (k) and the four scaling factors w j (k); For each of the above d jk It is generally believed to have two properties: noise-free decomposition coefficient. And the noise term ε that follows a standard normal distribution jk It can be represented by the following formula: Where σ j The noise variance of the coefficients in the j-th decomposition is represented; hypothesis testing is used to evaluate d. jk Remove noise term ε jk To obtain the noise-reduced coefficients Step 303, based on the noise reduction coefficients The original sequence is reconstructed to obtain the denoised wind speed sequence data. The specific reconstruction method is as follows:

4. The short-term wind speed prediction method based on the BDWPT-KAN model according to claim 1, characterized in that, The Z-score standardization formula is as follows: Where X is the Z-score standardized training set data, which serves as the input to the KAN model for wind speed prediction; x is the original data, μ is the mean of the training set data, and σ is the standard deviation of the training set data.

5. The short-term wind speed prediction method based on the BDWPT-KAN model according to claim 1, characterized in that, In step 6, the KAN model uses the Adam optimizer to update its parameters and the mean squared error is selected as the loss function. The KAN model is iteratively trained using the Z-score standardized training set data as input to optimize the KAN network weight parameters in order to minimize the prediction error. The activation function is expressed as follows: φ(x)=w b b(x)+w s spline(x) (5) Among them, w b and w s Here are the trainable weight parameters used to improve the expressive power of the activation function, b(x) represents the SiLU initialization function, and spline(x) represents the spline function, defined as follows: Among them, B i (x) represents the i-th B-spline basis function, c i These are the corresponding learnable coefficients; Extending the above single-layer activation structure to a multi-layer network structure, forming a deep KAN network, the overall mapping process is expressed as follows: Where, Φ l This represents the activation function matrix of the l-th layer, where L is the total number of layers; the specific input-output relationship of the l-th layer can be represented as: Where, x l x is the input vector of layer l; l+1 n is the output vector of layer l+1; l+1 and n l , which are the number of neurons in the (l+1)th layer and the lth layer, respectively; Let be the learnable spline activation function corresponding to the j-th neuron in layer l to the i-th neuron in layer (l+1).

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