Intelligent fault diagnosis method for lightweight mechanical equipment based on Kolmogorov-Arnold network

Abstract

A lightweight intelligent fault diagnosis method for mechanical equipment based on a Kolmogorov‑Arnold network. The method first obtains raw vibration monitoring data of mechanical equipment under different fault types and constructs an equipment vibration monitoring fault dataset. Then, a fault diagnosis model based on a KAN is constructed, and a learnable nonlinear function is used to replace the fixed linear weights in the convolution. The output features are then converted into one-dimensional feature vectors through a KAN convolution operation. The one-dimensional feature vectors are then input into a KANLinear layer to achieve mapping of high-dimensional features to different fault types. The training loss is then calculated, and the total loss is used as the optimization target in the training phase. The model parameters are updated using a gradient descent method. The fault diagnosis model is iteratively optimized. Finally, the fault diagnosis model is tested using a test set. The present invention can significantly reduce the number of model parameters and computational costs while maintaining high diagnostic accuracy, thereby improving overall computational efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical equipment fault diagnosis, and in particular to a lightweight intelligent fault diagnosis method for mechanical equipment based on a Kolmogorov-Arnold network. Background Art

[0002] Mechanical fault diagnosis is a key technology in industrial equipment maintenance. It can effectively prevent equipment failures, reduce maintenance costs and improve equipment reliability. At present, the methods of mechanical fault diagnosis mainly include model-based methods, signal processing-based methods and data-driven machine learning methods. Traditional model-based diagnosis methods rely on the physical model of the equipment, but the mechanical system in reality is complex and changeable, and it is very difficult to establish an accurate model; while signal processing-based methods, such as time domain and frequency domain analysis, can effectively extract some fault features, but are difficult to handle complex nonlinear faults and require a lot of experience and prior knowledge ([1] Li Han, Xiao Deyun. A review of data-driven fault diagnosis methods [J]. Control and Decision, 2011, 26(01): 1-9+16.).

[0003] In this context, data-driven machine learning methods have gradually attracted widespread attention. Due to its outstanding performance in the field of image processing, convolutional neural networks (CNNs) have been introduced into the field of fault diagnosis in recent years. They are used to automatically extract features from vibration signals, reducing the dependence on prior knowledge ([2] Wu Dinghai, Ren Guoquan, Wang Huaiguang, et al. A review of mechanical fault diagnosis methods based on convolutional neural networks [J]. Journal of Mechanical Strength, 2020, 42(05): 1024-1032.). CNN effectively captures local features in the data through convolution operations and is suitable for processing high-dimensional data. However, CNN models often contain a large number of parameters, resulting in high computational costs, and still have certain limitations when processing complex nonlinear signals.

[0004] To solve this problem, the Kolmogorov-Arnold network (KAN) provides a new architecture ([3] Ziming Liu. Kan: Kolmogorov–arnold networks. arXiv preprint arXiv: 2404.19756, 2024.). According to the Kolmogorov-Arnold theorem, any multivariable continuous function can be represented by the superposition of a finite number of unary functions. Based on this theorem, KAN introduces learnable B-spline functions between network layers to replace linear activation functions, thereby realizing the design of adjustable nonlinear activation functions. This design reduces the number of model parameters while enhancing the nonlinear expression ability of the model, effectively improving the generalization ability of the fault diagnosis model.

[0005] In summary, current fault diagnosis technologies suffer from shortcomings such as insufficient extraction of complex nonlinear features, large number of model parameters, and high computational costs. Combining the advantages of convolutional neural networks and KANs, a hybrid network architecture is constructed that reduces the number of model parameters while maintaining high diagnostic accuracy. This has become an effective approach to solving complex signal processing problems in mechanical fault diagnosis. Summary of the Invention

[0006] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a lightweight intelligent fault diagnosis method for mechanical equipment based on the Kolmogorov-Arnold network. By combining the spatial feature extraction capability of convolutional neural networks (CNN) with the nonlinear representation advantages of KAN, an efficient and lightweight hybrid neural network model is constructed. It can not only maintain high diagnostic accuracy while significantly reducing the number of model parameters and reducing computational costs, thereby improving overall computational efficiency.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] A lightweight intelligent fault diagnosis method for mechanical equipment based on a Kolmogorov-Arnold network comprises the following steps:

[0009] Step 1: Obtain the original vibration monitoring data of mechanical equipment under different fault types;

[0010] Step 2: Construct equipment vibration monitoring fault data set O = {O1, O2, ... m ,…O n}, indicating that there are n different fault types, where O m =(T m ,E m ), that is, each fault type contains training set data T m And the test set data E m ,in Indicates that the training set contains N T Samples; Indicates that the training set contains N E The length of each sample is L.

[0011] Step 3: Construct a fault diagnosis model based on KAN, establish a KAN convolution layer, and train the training set data T obtained in step 2. m Perform feature extraction and use learnable nonlinear functions to replace fixed linear weights in convolution;

[0012] The convolution process is to convolve the window on the sample S LSlide up, the input data in the window first undergoes nonlinear transformation of B-spline function, and then extracts features through weighted combination of spline basis functions, and finally obtains the output result of KAN convolution; among them, the convolution kernel K M =[φ1 φ2 … φ M ] T Indicates that the convolution kernel size is M, S L =[a1 a2 … a L ] T Indicates that the length of a single sample is L; the convolution result at a certain position i of the sample is Each sample S L After one layer of KAN convolution, the output is O conv =[b1 b2 … b k … b N ] T , where b k =[c1 c2 … c l ], there are N convolution kernels in total, and the output length after convolution is l = L-N+1;

[0013] Step 4: The output feature O obtained by the KAN convolution operation in step 3 conv Flatten it and convert it into a one-dimensional feature vector O in =[d1 d2 … d FL ], where d FL =N×l;

[0014] Step 5: Create a KANLinear layer and transform the one-dimensional feature vector O obtained in step 4 in Input to the KANLinear layer to map high-dimensional features to different fault types; the output consists of the base linear layer output (BaseOutput) and the B-spline output (SplineOutput);

[0015] Step 6: Calculate the training loss, which includes cross entropy loss and regularization loss;

[0016] Step 7: Substitute the total loss L obtained in step 6 reg As the optimization target in the training phase, the model parameter η is updated using the gradient descent method:

[0017]

[0018] Where α represents the learning rate;

[0019] Step 8: Repeat steps 3 to 7 to iteratively optimize the fault diagnosis model until the maximum number of iterations is reached;

[0020] Step 9: Use the test set E divided in step 2 m ,The fault diagnosis model is tested by accuracy, precision, recall and F1-score indicators.

[0021] The definition of each element φ in the KAN convolution kernel in step 3 is:

[0022] φ=w1·spline(x)+w2·silu(x) (1)

[0023] Where x is the input data, spline(x) is the B-spline function, silu(x) is the SiLU activation function, w1 and w2 are the weights of the B-spline function and the SiLU activation function respectively; the calculation of the B-spline function depends on the input data x and the predefined grid grid, which is defined as follows:

[0024]

[0025] Among them, R represents the order of the B-spline function, grid k Represents the kth grid point in the grid; the grid is defined as follows:

[0026] grid=[{k·h+GLIM0|k∈{-R,...,D+R}}] (3)

[0027] Where GLIM is the specified grid range, GLIM1 is the upper limit, and GLIM0 is the lower limit; D is the grid size, and h is the step size, which is defined as:

[0028]

[0029] The output part in step 5 is composed of the base linear layer output (BaseOutput) and the B-spline output (SplineOutput), and the specific calculation method is as follows:

[0030] BaseOutput=W base ·SiLU(O in ) (5)

[0031] SplineOutput = W spline ·spline(O in ) (6)In the formula, The weight matrix corresponding to the base linear layer, The weight matrix corresponding to the B-spline; the final output is Output = BaseOutput + SplineOutput.

[0032] The specific calculation formula for the loss in step 6 is as follows:

[0033]

[0034] Where, L ce is the cross entropy loss, B is the number of data samples in the batch, y i,j represents the true label value of sample i under category c, p i,j represents the predicted probability that sample i belongs to category j; L reg is the total loss including cross entropy loss and regularization term, where λ is the regularization coefficient, μ1 and μ2 are the weights of μ2 regularization direction and entropy regularization term respectively, M represents the number of network layers, Φ m is the weight matrix of the mth layer.

[0035] The specific calculation formulas for the accuracy, precision, recall, and F1-score indicators in step 9 are as follows:

[0036]

[0037]

[0038] Where TP represents true positives, FP represents false positives, and FN represents false negatives.

[0039] Compared with the prior art, the present invention has the following effective effects:

[0040] The present invention proposes a lightweight intelligent fault diagnosis method for mechanical equipment based on the Kolmogorov-Arnold network. During the convolution process, a learnable B-spline function is used to replace the fixed linear activation function in traditional convolution for nonlinear activation. The grid parameters are adaptively adjusted according to the distribution of input data, and fault features in vibration signals are extracted more efficiently, thereby improving the ability to capture complex nonlinear features. In addition, the KANLinear layer uses B-spline basis functions and a spline weight matrix to replace the linear weights in the traditional fully connected layer, thereby performing a more flexible nonlinear mapping between input features and output space. This not only enhances the flexibility of feature extraction, but also effectively reduces the number of parameters required to maintain high diagnostic accuracy, achieving faster convergence and better generalization. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 Flowchart of the present invention.

[0042] Figure 2 Schematic diagram of the model structure used in the embodiment of the present invention and the comparative method. Specific implementation methods

[0043] The present invention is described in further detail below with reference to the accompanying drawings and examples.

[0044] Reference Figure 1 A lightweight intelligent fault diagnosis method for mechanical equipment based on Kolmogorov-Arnold network includes the following steps:

[0045] Step 1: Obtain raw vibration monitoring data of mechanical equipment (such as rotating bearings) under different fault types;

[0046] Step 2: Construct equipment vibration monitoring fault data set O = {O1, O2, ... m ,…O n}, indicating that there are n different fault types, where O m =(T m ,E m ), that is, each fault type contains training set data T m And the test set data E m ,in Indicates that the training set contains N T Samples; Indicates that the training set contains N E The length of each sample is L.

[0047] Step 3: Construct a fault diagnosis model based on KAN, establish a KAN convolution layer, and train the training set data T obtained in step 2. m For feature extraction, a learnable nonlinear function is used to replace the fixed linear weights in ordinary convolution. Each element φ in the KAN convolution kernel is defined as:

[0048] φ=w1·spline(x)+w2·silu(x) (1)

[0049] Where x is the input data, spline(x) is the B-spline function, silu(x) is the SiLU activation function, w1 and w2 are the weights of the B-spline function and the SiLU activation function respectively; the calculation of the B-spline function depends on the input data x and the predefined grid grid, which is defined as follows:

[0050]

[0051] Among them, R represents the order of the B-spline function, grid k Represents the kth grid point in the grid; the grid is defined as follows:

[0052] grid=[{k·h+GLIM0|k∈{-R,…,D+R}}] (3)

[0053] Where GLIM is the specified grid range, GLIM1 is the upper limit, and GLIM0 is the lower limit; D is the grid size, and h is the step size, which is defined as:

[0054]

[0055] The convolution process is to convolve the window on the sample S L Slide up, the input data in the window first undergoes nonlinear transformation of B-spline function, and then extracts features through weighted combination of spline basis functions, and finally obtains the output result of KAN convolution; among them, the convolution kernel K M =[φ1 φ2 … φ M ] T Indicates that the convolution kernel size is M, S L =[a1 a2 … a L ] T Indicates that the length of a single sample is L; the convolution result at a certain position i of the sample is Each sample S L After one layer of KAN convolution, the output is O conv =[b1 b2 … b k … b N ] T , where b k =[c1 c2 … c l ], there are N convolution kernels in total, and the output length after convolution is l = L-N+1;

[0056] The KAN convolution kernel replaces fixed weights with a B-spline function defined on a preset grid, constructing a multi-scale grid-based feature capture mechanism. Compared to traditional fixed-step convolution, this mechanism is extremely flexible and can adaptively adjust the convolution operation according to the different ranges and scales of the input data, thereby more accurately capturing local feature changes and improving the model's generalization and expressiveness when processing complex data distributions.

[0057] Step 4: The output feature O obtained by the KAN convolution operation in step 3 conv Flatten it and convert it into a one-dimensional feature vector O in =[d1 d2 … d FL ], where d FL =N×l;

[0058] Step 5: Create a KANLinear layer and transform the one-dimensional feature vector O obtained in step 4 in Input to the KANLinear layer to map high-dimensional features to different fault types; the output part consists of the base linear layer output (BaseOutput) and the B-spline output (SplineOutput). The specific calculation method is as follows:

[0059] BaseOutput=W base ·SiLU(Oin ) (5)

[0060] SplineOutput = W spline ·spline(O in ) (6)In the formula, The weight matrix corresponding to the base linear layer, The weight matrix corresponding to the B-spline; the final output is Output = BaseOutput + SplineOutput;

[0061] The KANLinear layer achieves efficient feature mapping and extraction by combining the B-spline function with the base linear layer. Compared with the traditional fully connected layer, the KANLinear layer uses the weight matrix defined by the B-spline function on a preset grid, making the model more flexible and adaptable when processing data of different scales and distributions. This layer decomposes the input data through an adjustable multi-scale grid and combines it with learnable spline weights, so that it can capture local feature changes while maintaining the integrity of global features. The combination of the base linear layer and the spline function output further enhances the model's nonlinear expression capabilities.

[0062] Step 6: Calculate the training loss. The loss includes cross entropy loss and regularization loss. The specific calculation formula is as follows:

[0063]

[0064] Where, L ce is the cross entropy loss, B is the number of data samples in the batch, y i,j represents the true label value of sample i under category c, p i,j represents the predicted probability that sample i belongs to category j; L reg is the total loss including cross entropy loss and regularization term, where λ is the regularization coefficient, μ1 and μ2 are the weights of μ2 regularization direction and entropy regularization term respectively, M represents the number of network layers, Φ m is the weight matrix of the mth layer;

[0065] Step 7: Substitute the total loss L obtained in step 6 reg As the optimization target in the training phase, the model parameter η is updated using the gradient descent method:

[0066]

[0067] Where α represents the learning rate;

[0068] Step 8: Repeat steps 3 to 7 to iteratively optimize the fault diagnosis model until the maximum number of iterations is reached;

[0069] Step 9: Use the test set E divided in step 2 m ,The fault diagnosis model is tested by indicators such as accuracy, precision, recall, and F1-score. The specific calculation formula is as follows:

[0070]

[0071] Where TP represents true positives, FP represents false positives, and FN represents false negatives.

[0072] The effectiveness of the present invention is verified below by taking the fault diagnosis of Case Western Reserve's public bearing fault dataset as an example.

[0073] From Case Western Reserve's public bearing fault dataset, we selected normal bearing data under the operating conditions of 0 motor load and 1797 speed, as well as data for nine fault types. We divided each data type into a training set and a test set at a 1:1 ratio, each containing 50 training samples and 50 test samples respectively. The training set data was then fed into a fault diagnosis model for training. The fault diagnosis model consisted of two KAN convolutional layers and one KANLinear layer, with the dimensions of the KAN convolutional layers being 5 and 25, respectively. The model training parameters are shown in Table 1:

[0074] Table 1 Fault diagnosis model training parameters

[0075]

[0076] After the fault diagnosis model training is completed, the test set data is input into the fault diagnosis model to obtain the test results of the model. In order to fully verify the effectiveness of the method of the present invention, the test accuracy, precision, recall rate and F1 score are used to evaluate the performance of the model. At the same time, the method of the present invention is compared with the comparative method. Figure 2The network model used in comparison method 1 has the same number of parameters as the method of the present invention, but uses ordinary convolutional layers and fully connected layers; the network model used in comparison method 2 has the same number of parameters as the method of the present invention, but uses ordinary convolutional layers and KANLinear layers; the network model used in comparison method 3 has the same number of parameters as the method of the present invention, but uses KAN convolutional layers and ordinary fully connected layers; comparison method 4 uses ordinary convolutional layers and fully connected layers, but the number of parameters is much larger than that of the method of the present invention. As can be seen from Table 2, under the condition of the same number of parameters, the fault diagnosis model combining the KAN convolutional layer and the KANLinear layer significantly outperforms the comparison method in multiple performance indicators such as precision, recall rate, F1 score, and test accuracy, fully verifying the effectiveness and superiority of the method of the present invention in complex nonlinear signal feature extraction and mapping. Compared with models that only use traditional convolutional layers or fully connected layers, the introduction of KAN convolutional layers or KANLinear layers alone can also improve the diagnostic precision and accuracy of the model to a certain extent; while ensuring high diagnostic accuracy, the number of parameters required by traditional convolutional neural networks is more than that of the method of the present invention, which proves that the method of the present invention can maintain a high performance level while optimizing the model structure and reducing parameter consumption.

[0077] Table 2 Comparison of fault diagnosis performance of different methods

[0078]

Claims

1. A lightweight intelligent fault diagnosis method for mechanical equipment based on Kolmogorov-Arnold network, characterized in that: The following steps are involved: Step 1: Obtain the original vibration monitoring data of mechanical equipment under different fault types; Step 2: Construct equipment vibration monitoring fault data set O = {O1, O2, ... m ,…O n }, indicating that there are n different fault types, where O m =(T m ,E m ), that is, each fault type contains training set data T m And the test set data E m ,in Indicates that the training set contains N T Samples; Indicates that the training set contains N E The length of each sample is L. Step 3: Construct a fault diagnosis model based on KAN, establish a KAN convolution layer, and train the training set data T obtained in step 2. m Perform feature extraction and use learnable nonlinear functions to replace fixed linear weights in convolution; The convolution process is to convolve the window on the sample S L Slide up, the input data in the window first undergoes nonlinear transformation of B-spline function, and then extracts features through weighted combination of spline basis functions, and finally obtains the output result of KAN convolution; among them, the convolution kernel K M =[φ1 φ2 … φ M ] T Indicates that the convolution kernel size is M, S L =[a1 a2 … a L ] T Indicates that the length of a single sample is L; the convolution result at a certain position i of the sample is Each sample S L After one layer of KAN convolution, the output is O conv =[b1 b2…b k …b N ] T , where b k =[c1 c2…c l ], there are N convolution kernels in total, and the output length after convolution is l = L-N+1; Step 4: The output feature O obtained by the KAN convolution operation in step 3 conv Flatten it and convert it into a one-dimensional feature vector O in =[d1 d2…d FL ], where d FL =N×l; Step 5: Create a KANLinear layer and transform the one-dimensional feature vector O obtained in step 4 in Input to the KANLinear layer to map high-dimensional features to different fault types; the output consists of the base linear layer output (BaseOutput) and the B-spline output (SplineOutput); Step 6: Calculate the training loss, which includes cross entropy loss and regularization loss; Step 7: Substitute the total loss L obtained in step 6 reg As the optimization target in the training phase, the model parameter η is updated using the gradient descent method: Where α represents the learning rate; Step 8: Repeat steps 3 to 7 to iteratively optimize the fault diagnosis model until the maximum number of iterations is reached; Step 9: Use the test set E divided in step 2 m ,The fault diagnosis model is tested by accuracy, precision, recall and F1-score indicators.

2. The method according to claim 1, characterized in that The definition of each element φ in the KAN convolution kernel in step 3 is: φ=w1·spline(x)+w2·silu(x) (1) Where x is the input data, spline(x) is the B-spline function, silu(x) is the SiLU activation function, w1 and w2 are the weights of the B-spline function and the SiLU activation function respectively; the calculation of the B-spline function depends on the input data x and the predefined grid grid, which is defined as follows: Among them, R represents the order of the B-spline function, grid k Represents the kth grid point in the grid; the grid is defined as follows: grid=[{k·h+GLIM0|k∈{−R,...,D+R}}] (3) where GLIM is the specified grid range, GLIM1 is the upper limit, and GLIM0 is the lower limit; D is the grid size, and h is the step size, which is defined as:

3. The method according to claim 1, characterized in that The output part in step 5 is composed of the base linear layer output (BaseOutput) and the B-spline output (SplineOutput), and the specific calculation method is as follows: BaseOutput=W base ·SiLU(O in (5)SplineOutput=W spline ·spline(O in )(6)In the formula, The weight matrix corresponding to the base linear layer, The weight matrix corresponding to the B-spline; the final output is Output = BaseOutput + SplineOutput.

4. The method according to claim 1, wherein The specific calculation formula for the loss in step 6 is as follows: Where, L ce is the cross entropy loss, B is the number of data samples in the batch, y i,j represents the true label value of sample i under category c, p i,j represents the predicted probability that sample i belongs to category j; L reg is the total loss including cross entropy loss and regularization term, where λ is the regularization coefficient, μ1 and μ2 are the weights of μ2 regularization direction and entropy regularization term respectively, M represents the number of network layers, Φ m is the weight matrix of the mth layer.

5. The method according to claim 1, wherein The specific calculation formulas for the accuracy, precision, recall, and F1-score indicators in step 9 are as follows: Where TP represents true positives, FP represents false positives, and FN represents false negatives.

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