Industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline edge function

Abstract

The invention provides an industrial process fault diagnosis method based on a Kolmogorov-Arnold network piecewise linear spline edge function, and belongs to the technical field of industrial artificial intelligence and process safety monitoring. The method comprises the following steps: performing data preprocessing on original industrial sensing data through standardization and a sliding window to form a fixed-length sample sequence, and inputting the fixed-length sample sequence into a KAN edge function layer; the method comprises the following steps: adding a linear transformation item and a first-order piecewise linear spline item to obtain a low-dimensional implicit representation; then, outputting a fault category through a lightweight classification network; sparse regularization is introduced in the model training process to promote part of edge function coefficients to converge to zero, after training is completed, the importance ranking of all variables can be obtained through edge function coefficient analysis, and interpretable analysis of a fault judgment basis is achieved in combination with a representative edge function response curve. The method is high in diagnosis precision, the model is light and efficient, the result is transparent and explainable, and variable-level and interval-level accurate contribution degree tracing can be achieved.

Description

Technical Field

[0001] This invention relates to an industrial process fault diagnosis method based on piecewise linear spline side functions of Kolmogorov-Arnold network, belonging to the field of industrial artificial intelligence and process safety monitoring technology. Background Technology

[0002] Modern process industries heavily rely on data-driven process monitoring. While deep learning methods (such as CNN and LSTM) can achieve high classification accuracy, they face two major bottlenecks in actual industrial deployments: First, the models, as "black boxes," cannot provide explanations of the root causes of failures corresponding to the process mechanisms, making it difficult for engineers to trust them and make process adjustments accordingly; second, the increased model complexity to improve performance contradicts the stringent requirements of real-time performance and low power consumption in industrial settings.

[0003] The recently proposed Kolmogorov-Arnold Network (KAN) places learnable univariate functions on the network edges, providing a new approach to building interpretable models. However, directly applying the original KAN to industrial fault diagnosis has significant shortcomings: 1) the computational cost of its high-order splines or other complex functions is high, which is not conducive to online deployment; 2) its initial design is based on generalized function approximation, failing to consider the problems of high noise in industrial data and the need for a stable baseline; 3) it does not combine with the "variable importance" and "interval contribution" analysis requirements most relevant to industrial scenarios. Therefore, there is an urgent need for a KAN variant that is deeply customized and optimized for the specific problems of industrial diagnosis. Summary of the Invention

[0004] The purpose of this invention is to solve the problems existing in the prior art, which are as follows:

[0005] 1. The problem of untraceable diagnostic decisions: Existing deep learning models cannot clearly reveal the specific value range (corresponding to which operating condition) of a certain sensor variable, which plays a decisive role in fault identification, thus hindering rapid fault tracing and process optimization.

[0006] 2. The difficulty of balancing model efficiency and interpretability: Complex models may have high accuracy but lack interpretability, while simple models are interpretable but have low accuracy. There is a lack of lightweight model architectures that can maintain high accuracy while naturally providing variable-level and interval-level interpretations.

[0007] 3. Poor adaptability to continuous changes in operating conditions: Traditional models use fixed nonlinear activation functions, which make it difficult to accurately characterize the characteristics of industrial processes where the contribution of variables changes continuously and piecewise with their values ​​(for example, the effect of a certain temperature variable on the fault may be opposite in the low temperature region and the high temperature region).

[0008] Furthermore, this paper presents an industrial process fault diagnosis method based on piecewise linear spline side functions of Kolmogorov-Arnold networks. Piecewise linear spline side functions refer to the side function implementation method used in Kolmogorov-Arnold networks. The input-output connection is parameterized by first-order B-spline functions (piecewise linear), making the side function output vary piecewise linearly with the input interval. This method employs a lightweight and interpretable industrial process fault diagnosis framework. This framework maintains high diagnostic accuracy while improving the transparency of model results, making it easier for engineers to understand and utilize the diagnostic information.

[0009] The objective of this invention is achieved through the following technical solution:

[0010] A method for fault diagnosis of industrial processes based on piecewise linear spline side functions of Kolmogorov-Arnold networks includes the following steps:

[0011] Step 1: Data preprocessing: Normalize the collected industrial multivariate sensor data and perform sliding window sampling to obtain the input vector;

[0012] Step 2: Construct the KAN side function layer: The Kolmogorov-Arnold network structure is adopted. The network uses a single hidden layer, which contains several hidden units. The connection between each input dimension of the input vector in Step 1 and each hidden unit is represented by a learnable first-order B-spline basis function.

[0013] Step 3: Combining linear benchmark and spline residual: In the KAN side function layer constructed in step 2, the final output of the hidden layer is represented by adding a linear transformation term and a first-order piecewise linear spline term;

[0014] Step 4: Lightweight classifier output: Normalize the final output of the hidden layer generated by the KAN side function layer in Step 3, and then input it into the classifier network to output the fault category;

[0015] Step 5: Model Training and Sparse Regularization: Train the KAN side function layer and the classifier. During the training process, use a joint loss function that includes cross-entropy loss and a sparse penalty term to optimize the parameters in the KAN side function layer.

[0016] Step Six: Result Interpretation and Variable Importance Analysis: Using the sparse model parameters obtained after training in Step Five, the interpretability of the diagnostic decision is analyzed.

[0017] Preferably, the specific steps for normalization and sliding window sampling in step one are as follows:

[0018] Step 11: Standardize each measurement variable in the collected industrial multivariate sensor data using training set statistics to eliminate scale differences between different units of measurement;

[0019] Steps 1 and 2: Slice the time series into fixed-length windows to form a sample window containing data from the most recent time steps;

[0020] Step 13: Expand each window obtained in Step 12 into a one-dimensional feature vector, which is used as the input vector of the KAN side function layer.

[0021] Preferably, the specific steps in step two of representing the connection between each input dimension of the input vector and each hidden unit using a learnable first-order B-spline basis function are as follows:

[0022] Input vector The Each component and hidden units Between these, a side function is learned through the spline function model defined on each input-hidden connection in the KAN side function layer. The model is composed of a linear combination of spline basis functions, and the coefficients of each set of spline functions are learnable parameters, making the hidden unit... Receive from the Each component The contribution is The edge function takes the value of the hidden unit. ;

[0023] The edge functions are expanded using a first-order piecewise linear B-spline basis, that is, uniformly discrete nodes are defined over the input value range, and linear interpolation is performed using a hat function. Thus, the function for each edge... Parameterized by a set of trainable coefficients, it is possible to assign different response values ​​to input variables in different numerical ranges.

[0024] Preferably, the final output formula of the hidden layer in step three is:

[0025]

[0026] in, For linear transformation terms, For the input vector, For the input vector of the th scalar components of dimension The dimension of the input vector. This is the weight matrix. For bias vectors, It is a first-order piecewise linear spline term. , The index of the B-spline basis function, with values ​​ranging from 0 to... ; The number of first-order B-spline basis functions used for each edge function is reduced by one, i.e., using One control point; For input number Wei and Di The first hidden unit between the edge functions The parameters can be learned by expanding a single B-spline coefficient; For the first The position of the center node of each B-spline; h is the fixed interval width between adjacent B-spline nodes.

[0027] Preferably, the specific steps in step four of inputting the classifier network to output fault categories are as follows:

[0028] The classifier network is a two-layer fully connected feedforward network. The final output of the hidden layer after normalization is activated by an activation function and dropout regularization is applied to output the required number of fault categories. Finally, the probability distribution of each category is obtained through the Softmax layer, and the category with the highest probability is selected as the diagnostic result output.

[0029] Preferably, the sparse penalty term in step five applies to the coefficient matrix C of the spline edge functions, and the coefficient matrix C is composed of all edge functions. B-spline expansion coefficients The structure is designed to encourage most coefficients to converge to zero, thereby reducing the connection weights of unimportant variables;

[0030] The model employs L1 norm penalty to drive the coefficient matrix C to become sparse element by element, or L2,1 norm-based group sparsity penalty to reduce insignificant input-hidden unit connections on a whole-edge-function basis. By adjusting the weights of the sparse regularization coefficients λ, the model simplifies the connection structure while maintaining classification accuracy. The training optimization algorithm uses the AdamW adaptive optimizer and combines it with an early stopping strategy to select the optimal model.

[0031] Preferably, the specific steps for performing interpretability analysis of diagnostic decisions using sparse model parameters in step six are as follows:

[0032] Step 61: Collect statistical data on the side functions of each sensor variable in the industrial multivariate sensor data, i.e., each dimension of the input vector, over all hidden units. B-spline expansion coefficients The sum of absolute values ​​yields the global ranking of variable importance; the larger the sum of the absolute values ​​of spline coefficients, the higher the overall contribution of the sensing variable to the model's discrimination results, thus enabling the identification of the few most critical variables for fault identification.

[0033] Step 62: For each key minority variable obtained in Step 61, its corresponding learning side function It can be plotted as an input-output response curve, which can intuitively show the degree of influence of the variable on the fault diagnosis result when it takes different values;

[0034] If the learning side function of a certain variable exhibits non-monotonic piecewise characteristics, it indicates that the model assigns different weights to different value ranges of the variable, which corresponds to different operating conditions or nonlinear mechanisms in industrial processes, thus providing valuable clues for fault diagnosis for engineers.

[0035] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0036] 1. High Diagnostic Accuracy: The method of this invention achieves excellent performance on the typical TEP multi-fault diagnostic benchmark, and is applicable to process industries such as petrochemicals, pharmaceuticals, and energy production. Based on multivariate sensor time series anomaly detection and fault classification, this method achieves an overall classification accuracy of approximately 90% in a test set containing 21 operating conditions (normal + 20 faults), with a macro-average F1 score exceeding 0.905. This demonstrates that the diagnostic method can reliably distinguish the vast majority of fault types. Figure 2 As shown in the confusion matrix, except for a few operating conditions (such as normal operating conditions and some minor faults) which are somewhat confused, most fault categories present a clear diagonal structure on the matrix. This means that the model has correctly identified the vast majority of faults, and the diagnostic results have high reliability.

[0037] 2. Lightweight and Efficient Model: The neural network structure constructed in this invention is highly streamlined, requiring only a single hidden layer (e.g., 64 hidden dimensions) to fit complex nonlinear relationships. Compared to traditional methods that require multiple deep networks to achieve the same accuracy, this significantly reduces the size of model parameters and computational overhead. Furthermore, the model parameters themselves serve as an interpretive report, offering the advantages of low cost and unbiasedness, whereas existing technologies rely on computationally expensive SHAP for post-hoc approximation. In addition, the introduction of first-order B-spline basis functions provides nonlinear expressive power while maintaining low computational cost and exhibiting stronger robustness to noise disturbances. Experiments show that the model of this invention can quickly complete training and inference in a standard GPU / CPU environment, meeting the real-time requirements of industrial online monitoring. The simplified model structure also makes it easier to deploy on resource-constrained field equipment.

[0038] 3. Employing an Edge-wise Linear + Spline Decomposition Structure: This invention explicitly models the mapping from input variables to hidden layer units as an edge function consisting of a linear transformation term and a first-order piecewise linear spline (order-1 B-spline / hat basis) correction term, rather than simply using fixed activations or black-box attention weights. The key to this edge function structure is that the linear transformation term provides a stable global trend (beneficial for robustness and generalization of industrial data); the first-order piecewise linear spline term provides local nonlinear compensation only in necessary intervals (beneficial for interpretation and lightweighting), representing a customized design consideration for industrial scenarios (e.g., interval response, computational efficiency, and synergy with sparse regularization); the coupling of these two terms allows the model to output visualized edge response curves while maintaining performance, achieving "visualization of the discrimination criteria," unlike conventional solutions that only output weight matrices or posterior attribution.

[0039] 4. Implementation details of "Input clipping + uniform grid hat basis (piecewise linear first-order B-spline basis)" for standardized industrial signals (engineering reproducible): The standardized input is clipped in the side function layer, and uniform nodes are set in the clipped domain. Piecewise linear interpolation is performed using the hat basis to obtain the learnable piecewise response of each side function. This invention clips the z-score standardized input to a fixed range (e.g., [-4,4]) to avoid the spline basis function from failing when it goes out of bounds. This process matches the common distribution range of industrial measurements and is an engineering improvement for process industrial noise and outliers. Furthermore, a low-order hat basis is used in the side functions, which significantly reduces the computational cost compared to high-order splines / complex attention modules, making it suitable for real-time / edge deployment.

[0040] 5. The training phase introduces "structured Group sparsity" to achieve stable variable selection (different from simple posterior interpretation): Unlike posterior interpretation methods such as attention weights / gradient attribution, this invention applies structured Group sparsity constraints to the side function parameters during the training phase, making the model structure itself form interpretable sparse connections, thereby obtaining a stable and reproducible set of Top-K key variables. Comparative experiments show that the variable importance graphs exhibit observable differences under different regularization strategies, and Group sparsity makes the ranking of important variables more stable and easier for engineering verification, thus forming a verifiable chain of evidence for the technical effect.

[0041] Verifiable technical effects include: overall diagnostic performance reaching engineering usability levels and being on par with typical deep learning models; controlled risk of false negatives (Fault→Normal) and verifiable safety indicators; outputting the importance and side response curves of stable variables, forming an explanatory evidence chain of "variable-interval-contribution" to support process experts in reviewing and making maintenance decisions.

[0042] 6. A Safety-Oriented Evaluation and Optimization Approach Prioritizing Risk of Missed Reporting: This invention not only reports the evaluation indicator Acc / F1, but also explicitly introduces Fault→Normal Rate (misreporting as normal) as a key safety indicator. Comparative experiments demonstrate that this indicator can be kept at a low level, thereby reducing the safety risk of "faults being misjudged as normal." This safety-oriented indicator system and evidence presentation method differ from conventional academic solutions that only pursue overall accuracy, and are more aligned with the needs of industrial implementation.

[0043] 7. Transparent and Interpretable Results: This invention achieves transparency in the fault diagnosis decision-making process through sparse regularization strategies and side function mechanisms. In the model obtained after training, only a few sensor variables retain significant connection weights, while the side function coefficients of most irrelevant variables are weakened to zero, thus forming a clear and easily understandable subset of important variables. For example... Figure 3 As shown, the model can extract the global variable importance distribution based on the learned side function parameters. In the instance test, the top-ranking variables include V22, V51, and V19, indicating that the model primarily relies on a few key measurement point signals for fault identification. These high-contribution variables are often closely related to the process mechanism, and the importance ranking given by the model is consistent with engineering experience, making it directly applicable to engineering root cause analysis and providing strong support for subsequent fault cause analysis.

[0044] 8. By examining the shape of the side functions corresponding to each important variable, we can further understand the decision-making basis of the model. For example... Figure 4 As shown, the method of this invention exhibits nonlinear selectivity in the side function response curve of a highly important variable (e.g., V22) across different value ranges: when the variable is within a certain range, its contribution to fault detection is positive, while in other ranges its contribution may turn negative or approach zero. This piecewise linear response reflects that the variable's impact on fault occurrence differs under different operating conditions, verifying that the model can capture the condition-dependent characteristics of industrial processes. This visualized consistency clue helps engineers understand the potential causes of faults and increases their confidence in the model's diagnostic results. Attached Figure Description

[0045] Figure 1 This is a flowchart illustrating the overall workflow of the NeuroSpline-TEP method of the present invention.

[0046] Figure 2 This is a schematic diagram of the confusion matrix of the NeuroSpline-TEP model of the present invention on the TEP multi-fault test set.

[0047] Figure 3This is a global variable importance distribution diagram obtained from the joint model of linear terms and spline terms of the present invention.

[0048] Figure 4 The graph shows the response curve of the spline function corresponding to the intermediate time portion of the key variable V22 in this invention.

[0049] Figure 5 This is a diagram showing the scores of each comparative model for the evaluation metrics Acc, Macro-F1, and Weighted-F1 in the comparative experiment.

[0050] Figure 6 This is a schematic diagram of the row-normalized confusion matrix of the trained model on the TEP dataset.

[0051] Figure 7 This is a schematic diagram of the response curve of a certain connected edge in the KAN edge function layer (input dimension = 541, output node = 53).

[0052] Figure 8 This is a schematic diagram of the response curve of a certain connected edge in the KAN edge function layer (input dimension = 570, output node = 53).

[0053] Figure 9 This is a schematic diagram showing the normal operating conditions and false alarm / missed alarm rates of each comparative model in the comparative experiment.

[0054] Figure 10 This is a schematic diagram of the single-class F1 score for difficult-to-classify faults in the TEP test set.

[0055] Figure 11 A comparison chart of the importance of the Top-15 variables under different sparse regularization strategies.

[0056] Figure 12 This is a diagram showing the ranking of the importance of the Top-15 variables. Detailed Implementation

[0057] The present invention will be further described in detail below with reference to the accompanying drawings: This embodiment is implemented under the premise of the technical solution of the present invention, and detailed implementation methods are given, but the protection scope of the present invention is not limited to the following embodiments.

[0058] like Figure 1 and Figure 2 As shown, the industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions involved in this embodiment includes the following steps:

[0059] Step 1: Data Preprocessing: Normalize and perform sliding window sampling on the collected industrial multivariate sensor data. Standardize each measured variable using training set statistics (e.g., z-score standardization) to eliminate scale differences between different units. Then, slice the time series data (i.e., the multivariate process data stream collected from the TEP simulation platform) into fixed-length windows (e.g., window length T=20) to form sample windows containing data from the most recent time steps. Each window is flattened into a one-dimensional feature vector, which serves as the input to the neural network model. In this step, the one-dimensional feature vector precisely matches the side function layer in Step 2, ensuring that each input dimension directly and uniquely corresponds to an original sensor variable, laying the foundation for subsequent accurate contribution tracking at the variable and interval levels.

[0060] Step 2: KAN-style Side Function Layer: The preprocessed input vector is fed into a side function layer based on the Kolmogorov–Arnold network concept for feature extraction. Specifically, this layer contains several hidden units, and the connection between each input dimension and each hidden unit is modeled by a univariate nonlinear function represented by a learnable first-order B-spline (hat function) basis, rather than the traditional fixed-weight summation. This means that for the input vector... The Each component and hidden units Between them, the model learns a side function. , Indicates the first The input component to the first The side function of the hidden unit, such that the hidden unit receives from the . Each component The contribution is , Indicates the input number The component to the first A univariate learnable mapping of hidden units It is the contribution of the edge function to the hidden unit (the function value).

[0061] The edge functions are expanded using a first-order (piecewise linear) B-spline basis, that is, uniformly discrete nodes are defined over the input value range (e.g., 10 control nodes are set within the normalized signal range [-4,4]), and linear interpolation is performed using a hat function. Thus, the function of each edge... Parameterized by a set of trainable coefficients, different response values ​​can be assigned to the input variable across different numerical ranges. Employing a piecewise linear function ensures sufficient nonlinear expressiveness while limiting the function's complexity, resulting in computational efficiency and robustness to noise.

[0062] Step 3: Combining Linear Benchmark and Spline Residual: To maintain the model's representation of the global linear trend while allowing for local nonlinear deviations, this invention introduces a "linear + spline residual" structure into the aforementioned edge function layer. The hidden layer output is obtained by adding the two parts:

[0063]

[0064] in, For the input vector, For the input vector of the th scalar components of dimension The dimension of the input vector. This is the weight matrix. For bias vectors, The index of the B-spline basis function, with values ​​ranging from 0 to... ; The number of first-order B-spline basis functions used for each edge function is reduced by one, i.e., using One control point; For input number Wei and Di The first hidden unit between the edge functions The parameters can be learned by expanding the B-spline coefficients.

[0065] One part is the traditional linear weighted summation. (That is, multiplying the input vector by the weight matrix W and adding the bias b to obtain a linear output), providing a global linear baseline to ensure model stability and fitting to mainstream normal data. The other part consists of all side functions. ,in The sum of the outputs (i.e., each hidden unit sums the spline transform contributions from each input), For the first The B-spline contains the center node positions, and h is the fixed interval width between adjacent B-spline nodes. It is specifically designed to characterize the "interval dependence" effect of industrial variables, and its physical meaning is clear, such as: "Within interval X, this variable contributes positively." Here, "interval X" refers to a sub-interval within the standardized (and optionally pruned) range of the input variable, i.e., the range of values ​​for that variable. The range; when using a first-order piecewise linear B-spline (hat basis) expansion, this interval can correspond to the segmented intervals between adjacent nodes, such as: Or, corresponding to the support domain of a certain hat basis function, such as: When the learned edge function When a variable takes a positive value within this interval, it indicates that the variable contributes positively to the implicit representation / fault discrimination within that value range. The computational cost is extremely low, and it works exceptionally well with group sparsity, enabling the training of an extremely sparse subset of key variables with clear connections. The final implicit representation is obtained by decomposing the hidden unit output into linear baseline components and nonlinear residual components, which are then added together. This design allows the model to retain the global interpretability of a linear model while using spline functions to capture local range-dependent nonlinear effects, thus balancing the model's interpretability and flexibility.

[0066] Step 4: Lightweight Classifier Output: The hidden representations generated by the side function layers are normalized (e.g., by applying a LayerNorm layer to balance the scale of different hidden units), and then input into a simplified classifier network to output the fault categories. The classifier is a two-layer fully connected feedforward network, using activation functions such as SiLU and appropriate dropout regularization to output the required number of fault categories (e.g., 21 categories in the TEP scenario, including 1 normal condition and 20 faults). Finally, the probability distribution of each category is obtained through a Softmax layer, and the category with the highest probability is selected as the diagnostic result output.

[0067] Step 5: Model Training and Sparse Regularization: During training, a joint loss function including cross-entropy loss and a sparsity penalty term is used to optimize the parameters in the KAN side function layer. The parameters in the KAN side function layer are the model parameters, including the piecewise linear spline coefficients of each connection in the side function layer, the linear basis weights, and the weight parameters in the classifier network. That is, the spline coefficients + linear transformation term + classifier weights in the KAN side function layer. The sparsity penalty term applies to the coefficient matrix C of the spline side function, and the coefficient matrix C consists of all the side functions. B-spline expansion coefficients The sparsity regularization term λ represents the strength of nonlinear connections between the input dimension and hidden nodes, aiming to reduce the connection weights of unimportant variables by causing most coefficients to converge to zero. Specifically, L1 norm penalty can be used to drive element-wise sparsity of coefficients, or group sparsity penalty based on L2,1 norm can be used to reduce insignificant input-hidden unit connections on a whole-edge-function basis. By adjusting the weights of the sparse regularization coefficients λ, the model simplifies the connection structure as much as possible while maintaining classification accuracy. The training optimization algorithm can use adaptive optimizers such as AdamW, combined with early stopping strategies (such as using macro F1 scores on the validation set) to select the optimal model.

[0068] Step Six: Result Interpretation and Variable Importance Analysis: Utilizing the sparse model parameters obtained after training, interpretability analysis of diagnostic decisions is achieved. On one hand, the side functions of each sensor variable in the collected industrial multivariate sensor data—that is, each dimension of the input vector—on all hidden units are statistically analyzed. B-spline expansion coefficients The sum of absolute values, by taking the norm of these coefficients and aggregating them over hidden units, can estimate the global importance of the input variable to the model output, thus obtaining a global ranking of variable importance; the larger the sum of coefficients, the higher the overall contribution of the sensing variable to the model's discrimination result. This method can identify the few variables most critical for fault identification. On the other hand, for each important variable, its corresponding learning side function... Input-output response curves can be plotted to visually demonstrate the impact of the variable on fault diagnosis results at different values. If the side function of a variable exhibits non-monotonic piecewise characteristics, it indicates that the model assigns different weights to different value ranges of the variable. This often corresponds to different operating conditions or nonlinear mechanisms in industrial processes, thus providing valuable diagnostic clues for engineers.

[0069] In summary, this invention constructs a neural network fault diagnosis model based on learnable side functions (which can be called the "NeuroSpline-TEP" framework). For example... Figure 1 As shown, the framework comprises three parts: data preprocessing, a side function extraction layer, and a classification decision layer. The side function extraction layer uses first-order splines to perform interval nonlinear transformations on the input signal. Combined with sparse regularization training of the output layer, a concise and easily interpretable model structure can be obtained while maintaining accuracy. This invention achieves extreme simplification through a one-dimensional input + first-order spline structure, enabling lightweight design for industrial deployment. While maintaining high interpretability, it optimizes computational efficiency and memory usage, making it particularly suitable for embedded or edge deployments.

[0070] like Figure 1 The diagram illustrates the overall architecture flowchart of the method of this invention. Raw industrial sensor data is first standardized and processed using a sliding window to form a fixed-length sample sequence, which is then input into the KAN (Kinshall Analytic Network) layer. Through feature extraction using piecewise linear spline functions and the superposition of linear benchmarks, a low-dimensional latent representation is obtained. Subsequently, a lightweight classification network outputs the fault category. During model training, sparse regularization is introduced to cause some side function coefficients to converge to zero. After training, the importance ranking of each variable can be obtained through side function coefficient analysis, and combined with representative side function response curves, an interpretable analysis of the fault discrimination criteria can be achieved.

[0071] like Figure 2The diagram shows the prediction results of the method of this invention for classifying 21 operating conditions (normal + 20 faults IDV1-IDV20), normalized by row. Most fault categories show high classification accuracy in the diagonal region, indicating that the model has good recognition ability.

[0072] like Figure 3 As shown, the importance scores assigned to each sensor variable (V1-V52) during the training process of the model (based on linear + spline contribution aggregation) show that variables V22, V19 and V51 are the most critical diagnostic variables in this operation and have significant engineering reference value.

[0073] like Figure 4 As shown, the spline response shape of variable V22 within the standardized input interval is displayed, reflecting its nonlinear and interval-selective influence on the model's discriminative output, which helps to understand the behavior mechanism of the variable under different operating states.

[0074] Comparative experiment

[0075] To verify the effectiveness and interpretability of the proposed solution in process industry fault diagnosis, a system comparison experiment was conducted under completely identical preprocessing procedures (52 variables, sliding window length T=20, training set statistics z-score standardization, same data partitioning and random seed). The comparison models included:

[0076] (1) Logistic Regression (linear lower limit baseline);

[0077] (2) A two-layer MLP with aligned parameter and computational costs;

[0078] (3) Typical temporal deep model CNN1D;

[0079] And the KAN edge function model of the present invention and its ablation version (without regularization / L1 sparsity / Group structured sparsity, with or without linear terms).

[0080] kan_base1_regnone_lam0 represents: a KAN edge function model containing linear terms (base1) and without sparse regularization (regnone) and regularization coefficients λ=0 (lam0);

[0081] kan_base1_regl1_lam0.0001 represents: a KAN side function model containing linear terms (base1) and L1 sparse regularization (regl1) coefficients λ=0.0001;

[0082] kan_base1_reggroup_lam0.0001 represents: a KAN side function model containing linear terms (base1) Group(L2,1) structured sparse regularization (reggroup) coefficients λ=0.0001;

[0083] kan_base0_reggroup_lam0.0001 represents: a KAN edge function model with a structured sparse regularization (reggroup) coefficient λ=0.0001 that does not contain linear terms (base0, only spline edge function terms);

[0084] Wherein, base0 / base1 is used to distinguish whether a linear baseline term is included. regnone / regl1 / reggroup are used to distinguish regularity types, and lam is a coefficient of a regularity weight.

[0085] Deep learning training involves randomness, such as random initialization and mini-batch sampling. The same code and configuration can exhibit slight fluctuations in metrics across different runs. For example... Figure 5 As shown, the scores of each comparative model for the evaluation metrics Acc, Macro-F1, and Weighted-F1 are displayed. Macro-F1 = 0.9057 represents a representative result from a single run. The results of the ablation experiments (default Order-1+Group) show that the overall Macro-F1 of this invention is approximately 0.90, fluctuating within the range of approximately 0.89–0.91 in repeated runs. This difference is within the range of random fluctuation and does not affect the conclusion that it is "of the same magnitude as CNN / MLP and superior to linear models." This invention can maintain high accuracy (within the same magnitude of performance) while achieving interpretability and lightweight deployment, offering equivalent accuracy (of the same magnitude as CNN / MLP) + increased importance / interval contribution of output variables (interpretable) + a more lightweight model / more suitable for engineering deployment.

[0086] In addition to Acc, Macro-F1, and Weighted-F1, the evaluation metrics further focus on industrial alarm scenarios: Normal Recall (health condition recognition rate) and FAR (Normal→Fault) (false alarm rate), reflecting the false alarm burden; Fault→Normal Rate (the proportion of faults mistaken for normal conditions), reflecting the risk of missed alarms / missed detections (more important in safe production scenarios), such as... Figure 9 As shown; and with a focus on reporting the single-class F1 and confusion matrix of difficult-to-classify faults (such as IDV9, IDV15), as shown. Figure 10 and Figure 6 As shown.

[0087] like Figure 7As shown, the input-output response curve of a representative connection edge (output unit out=53, input dimension in=541) of the KAN edge function layer is displayed, where the edge function output... It can be decomposed into linear reference terms. Piecewise linear spline residuals The superposition of values; the horizontal axis represents the range of input feature values ​​after z-score standardization and truncation. The curve reflects the direction and intensity of the influence of this input dimension on the output of the connecting edge in different value ranges, and can be used for interval-level contribution tracing and interpretability analysis. For example... Figure 8 As shown, the input-output response curve of another representative connection edge (output unit out=53, input dimension in=570) of the KAN edge function layer is presented, and the results are also given. Linear terms and spline residuals The decomposition relationship; by comparing the response patterns of different connecting edges, the differences in the role of different input dimensions in the internal representation of the model in each value range can be intuitively depicted, providing interpretable support for fault identification.

[0088] Experimental results show that:

[0089] 1. The KAN side function model of this invention achieves overall performance (Acc / Macro-F1 / Weighted-F1) comparable to CNN / MLP, and significantly outperforms linear models;

[0090] 2. Regarding key safety indicators, this invention maintains a low level (approximately 0.03) in the Fault→Normal Rate (false negatives are considered normal), demonstrating effective control over the risk of false negatives;

[0091] 3. The F1 score remains usable for difficult-to-classify faults (IDV9 / IDV15), and more stable variable selection and interpretation results can be obtained through structured sparse regularization (see the variable importance comparison chart, e.g.). Figure 11 and Figure 12 (As shown).

[0092] Example 1

[0093] Multivariate time series data from the TEP (Training Experience Program) were collected as model input, with a variable dimension of 52. Samples were constructed using a sliding window process, with each window having a length of 20 time steps. After standardization (z-score normalization using the training set mean and standard deviation), each sample was flattened into a 1040-dimensional vector as input features.

[0094] 1. Side Function Layer Design: Construct a side function extraction layer containing 64 output units. This layer adopts the Kolmogorov–Arnold architecture, with each input feature... With each output unit The connection between them is provided by a first-order B-spline basis function. This function is expanded from a hat function basis consisting of 10 equidistant nodes and exhibits a definite interval response.

[0095] 2. Linear benchmark and residual combination: The final output of each output unit consists of linear terms. The residual term is the residual of all spline edge functions. The components are added together to enhance nonlinear modeling capabilities while preserving an interpretable basis.

[0096] 3. Classifier structure: The output of the side function layer is fed into LayerNorm and then into a two-layer lightweight fully connected network, which outputs 21-dimensional softmax probabilities, representing 21 types of operating conditions (normal + 20 types of faults).

[0097] 4. Model Training and Regularization: The AdamW optimizer was used for 50 rounds of training. A group sparsity regularization term was introduced to compress edge function connections and improve interpretability. The model was tested on the validation set using a macro F1 score for early stopping and optimal parameter saving.

[0098] Diagnostic Results and Interpretation Output: After training, the model can not only output the fault category of each sample, but also derive the global importance ranking of each sensor variable and the response curve of key variables (such as variable V22 exhibiting a nonlinear multi-peak response) by analyzing the side function parameters, providing engineers with visual diagnostic clues.

[0099] The above description is merely a preferred embodiment of the present invention. These specific embodiments are different implementations based on the overall concept of the present invention, and the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for fault diagnosis of industrial processes based on piecewise linear spline side functions of Kolmogorov-Arnold networks, characterized in that, Includes the following steps: Step 1: Data preprocessing: Normalize the collected industrial multivariate sensor data and perform sliding window sampling to obtain the input vector; Step 2: Construct the KAN side function layer: The Kolmogorov-Arnold network structure is adopted. The network uses a single hidden layer, which contains several hidden units. The connection between each input dimension of the input vector in Step 1 and each hidden unit is represented by a learnable first-order B-spline basis function. Step 3: Combining linear benchmark and spline residual: In the KAN side function layer constructed in step 2, the final output of the hidden layer is represented by adding a linear transformation term and a first-order piecewise linear spline term; Step 4: Lightweight classifier output: Normalize the final output of the hidden layer generated by the KAN side function layer in Step 3, and then input it into the classifier network to output the fault category; Step 5: Model Training and Sparse Regularization: Train the KAN side function layer and the classifier. During the training process, use a joint loss function that includes cross-entropy loss and a sparse penalty term to optimize the parameters in the KAN side function layer. Step Six: Result Interpretation and Variable Importance Analysis: Using the sparse model parameters obtained after training in Step Five, the interpretability of the diagnostic decision is analyzed.

2. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 1, characterized in that, The specific steps for normalization and sliding window sampling described in step one are as follows: Step 11: Standardize each measurement variable in the collected industrial multivariate sensor data using training set statistics to eliminate scale differences between different units of measurement; Steps 1 and 2: Slice the time series into fixed-length windows to form a sample window containing data from the most recent time steps; Step 13: Expand each window obtained in Step 12 into a one-dimensional feature vector, which is used as the input vector of the KAN side function layer.

3. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 1, characterized in that, The specific steps in step two, which involve representing the connection between each input dimension of the input vector and each hidden unit using a learnable first-order B-spline basis function, are as follows: Input vector The Each component and hidden units Between these, a side function is learned through the spline function model defined on each input-hidden connection in the KAN side function layer. The model is composed of a linear combination of spline basis functions, and the coefficients of each set of spline functions are learnable parameters, making the hidden unit... Receive from the Each component The contribution is The edge function takes the value of the hidden unit. ; The edge functions are expanded using a first-order piecewise linear B-spline basis, that is, uniformly discrete nodes are defined over the input value range, and linear interpolation is performed using a hat function. Thus, the function for each edge... Parameterized by a set of trainable coefficients, it is possible to assign different response values ​​to input variables in different numerical ranges.

4. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 3, characterized in that, The final output formula of the hidden layer in step three is: in, For linear transformation terms, For the input vector, For the input vector of the th scalar components of dimension The dimension of the input vector. This is the weight matrix. For bias vectors, It is a first-order piecewise linear spline term. , The index of the B-spline basis function, with values ​​ranging from 0 to... ; The number of first-order B-spline basis functions used for each edge function is reduced by one, i.e., using One control point; For input number Wei and Di The first hidden unit between the edge functions The parameters can be learned by expanding a single B-spline coefficient; For the first The position of the center node of each B-spline; h is the fixed interval width between adjacent B-spline nodes.

5. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 1, characterized in that, The specific steps for inputting the classifier network into step four to output fault categories are as follows: The classifier network is a two-layer fully connected feedforward network. The final output of the hidden layer after normalization is activated by an activation function and dropout regularization is applied to output the required number of fault categories. Finally, the probability distribution of each category is obtained through the Softmax layer, and the category with the highest probability is selected as the diagnostic result output.

6. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 4, characterized in that, The sparse penalty term mentioned in step five applies to the coefficient matrix C of the spline edge functions. The coefficient matrix C is composed of all edge functions. B-spline expansion coefficients The structure is designed to encourage most coefficients to converge to zero, thereby reducing the connection weights of unimportant variables; The model employs L1 norm penalty to drive the coefficient matrix C to become sparse element by element, or L2,1 norm-based group sparsity penalty to reduce insignificant input-hidden unit connections on a whole-edge-function basis. By adjusting the weights of the sparse regularization coefficients λ, the model simplifies the connection structure while maintaining classification accuracy. The training optimization algorithm uses the AdamW adaptive optimizer and combines it with an early stopping strategy to select the optimal model.

7. The industrial process fault diagnosis method based on Kolmogorov-Arnold network piecewise linear spline side functions according to claim 6, characterized in that, The specific steps for using sparse model parameters to perform interpretability analysis of diagnostic decisions, as described in step six, are as follows: Step 61: Collect statistical data on the side functions of each sensor variable in the industrial multivariate sensor data, i.e., each dimension of the input vector, over all hidden units. B-spline expansion coefficients The sum of absolute values ​​yields the global ranking of variable importance; the larger the sum of the absolute values ​​of spline coefficients, the higher the overall contribution of the sensing variable to the model's discrimination results, thus enabling the identification of the few most critical variables for fault identification. Step 62: For each key minority variable obtained in Step 61, its corresponding learning side function It can be plotted as an input-output response curve, which can intuitively show the degree of influence of the variable on the fault diagnosis result when it takes different values; If the learning side function of a certain variable exhibits non-monotonic piecewise characteristics, it indicates that the model assigns different weights to different value ranges of the variable, which corresponds to different operating conditions or nonlinear mechanisms in industrial processes, thus providing valuable clues for fault diagnosis for engineers.

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