Industrial data prediction method and device based on robust cyclic random configuration network

Abstract

The invention discloses an industrial data prediction method and device based on a robust cyclic random configuration network, and aims to solve the problem that the modeling precision and robustness are insufficient when an existing model processes industrial data which is uncertain in input order, strong in time sequence and contains noise and abnormal values. Through combination of a robust learning mechanism and a recursive structure, order identification is not needed, reserve pool nodes are randomly generated based on a supervision mechanism, a network structure is constructed, and the global approximation capability of any nonlinear mapping function is ensured. The method not only inherits the efficient modeling capability of a cyclic network for time sequence characteristics, but also significantly enhances the stability and generalization capability of the model in a complex interference environment, and is suitable for accurate prediction and intelligent decision support scenes in a complex dynamic industrial process.

Description

Technical Field

[0001] This application relates to the field of industrial modeling and prediction technology, specifically to an industrial data prediction method and apparatus based on a robust cyclic random configuration network. Background Technology

[0002] In actual industrial processes, various systems typically exhibit strong temporal and non-stationary characteristics. The distribution of process data changes continuously over time, and the system's behavior may differ across different time periods. These dynamic changes not only prevent models from accurately capturing the order information of the data but are also frequently affected by noise and outliers. These interfering factors pose a serious threat to the accuracy and stability of modeling. First, the uncertainty of the input order makes it difficult for traditional modeling methods to accurately capture and process complex temporal relationships, thus failing to fully reflect the dynamic behavior of the system and leading to reduced prediction accuracy. Second, the presence of noise and outliers further exacerbates the difficulties in the modeling process, interfering with the normal behavior of the system. This prevents traditional methods from effectively identifying and handling these interfering factors, thereby affecting the model's accuracy, stability, and generalization ability. The shortcomings and deficiencies of existing technologies:

[0003] 1. Recurrent neural networks (RNNs) and echo-state networks store historical order variables in a reservoir and recursively calculate the state information at each step, eliminating the need for order identification. They are commonly used to solve modeling problems involving time-series process data. However, both methods are highly sensitive to parameter settings and cannot adaptively adjust the network structure based on input data, lacking the global approximation property for arbitrary target mappings. Furthermore, traditional RNNs train network parameters through error backpropagation, which is prone to problems such as vanishing or exploding gradients, and have a long learning time.

[0004] 2. Recurrent Stochastic Configuration Networks (RSCNs) utilize supervised mechanisms to generate reserve pool nodes, possessing excellent global approximation capabilities and theoretically able to approximate the objective function infinitely. As a type of stochastic learning model, RSCNs do not rely on gradient descent algorithms to update weights, thus offering faster training speeds and lower computational costs. However, when faced with complex data containing noise and outliers, RSCNs exhibit weak robustness and are easily affected by interference factors, leading to decreased modeling performance and limiting their application effectiveness in real-world industrial environments.

[0005] 3. Robust learning-based network models can effectively cope with common noise interference and outlier effects in industrial data, improving the model's stability and generalization ability in complex environments. These models typically introduce robust loss functions or regularization strategies to reduce the interference of outliers on model training, thereby enhancing the model's adaptability to uncertainty and abnormal situations. However, when facing scenarios with strong non-stationarity, complex temporal relationships, and dynamically changing input orders, these methods still have certain limitations, making it difficult to fully capture the dynamic characteristics of the system, thus affecting their modeling accuracy and practical application value in industrial process modeling. Summary of the Invention

[0006] The main objective of this application is to provide an industrial data prediction method based on a robust cyclic randomized network, comprising the following steps:

[0007] Step S10: Acquire industrial data with time series characteristics, construct the input sample set and the target output set, and initialize the model parameters;

[0008] Step S20: Initialize the output error vector, set the scaling factor of the output error, and build the initial network model;

[0009] Step S30: Randomly generate candidate neuron parameters and select candidate neurons that meet the requirements based on the discrimination conditions introduced by the manifold regularization constraint;

[0010] Step S40: Select the best candidate neuron from the selected candidates and add it to the network model. Calculate the output weights and model error of the current network model. Iterate through the above steps until the termination condition is met.

[0011] Step S50: Based on the kernel density estimation algorithm, calculate the probability density estimate of each training sample in the feature space;

[0012] Step S60: Using the sample density estimate as weights, construct a weighting matrix, and re-solve the output weights of the network model using the weighted least squares method;

[0013] Step S70: Based on the projection algorithm, update the trained output weights online according to real-time data, and output the prediction results in real time.

[0014] In one embodiment, the steps of acquiring industrial data with time-series characteristics, constructing an input sample set and a target output set, and initializing model parameters include:

[0015] Given a set of industrial process data with time-series characteristics, construct the input sample set:

[0016]

[0017] in, and They represent the system at time 10:00 and 11:00 respectively. The output and input of are defined, and the target output set is set as follows:

[0018]

[0019] That is, outputting the next time step based on the input at the current time step. To make predictions, among which This represents the total number of samples;

[0020] Initialize model-related parameters, including: the maximum allowed number of neurons in the reserve pool. Maximum permissible output error Maximum number of candidate neurons generated per round Input weight distribution parameter set ,in , Enter the upper and lower bounds of the weight range. It is a change; manifold regularization coefficient It is used to constrain the maintenance of manifold structure during the generation and selection of neurons.

[0021] In one embodiment, the steps of initializing the output error vector, setting the scaling factor of the output error, and constructing the initial network model include:

[0022] Initialize the output error vector as And set the scaling factor for the output error of the network model. ;

[0023] Assuming a system has already been built by A pool of neurons forms a reserve pool, and the network model at this point can be represented as:

[0024]

[0025]

[0026]

[0027] in, , and They respectively represent the current contents The input weight matrix, reservoir connection weight matrix, and output weight matrix for each neuron. For the corresponding bias term;

[0028] Indicates the first The status of the reserve pool at any given time. Let the set of the reservoir states at all times be a matrix. The selected activation function is... Represents the trace of a matrix;

[0029] Represents the graph Laplacian matrix of the input sample. It is a diagonal matrix. These are similarity matrices, and the two matrices are defined as follows:

[0030]

[0031]

[0032] in Represents an exponential function. For kernel coefficients.

[0033] In one embodiment, the step of randomly generating candidate neuron parameters and selecting candidate neurons that meet the requirements based on the discriminant conditions introduced by the manifold regularization constraint includes:

[0034] From the set of input weight distributions In the process, the input weight matrix is ​​randomly generated. , Reserve pool connection matrix and bias vector And substitute it into the activation function formula:

[0035]

[0036] To generate Group candidate neurons , Indicates having The first reserve pool node One candidate neuron;

[0037] The above Substitute the candidate neurons into the randomized configuration criteria with manifold regularization constraints to select neurons that meet the requirements:

[0038]

[0039] in It is the output dimension, from Candidate neurons that satisfy the inequality constraints are selected from the samples.

[0040] In one embodiment, the step of selecting the optimal candidate neuron from the selected candidates and adding it to the network model, calculating the output weights and model error of the current network model, and iteratively executing the above steps until the termination condition is met includes:

[0041] To determine the optimal candidate neuron, we first define a set of variables:

[0042]

[0043] in Among the generated candidate neurons, those that satisfy both the inequality constraints and... Neurons that reach their maximum value are selected as candidate nodes for the optimal reserve pool.

[0044] Add the node to the current network model and directly solve for the new output weight matrix based on the target output. Then, the standard root mean square error of the network model's output is calculated and used as the loss function;

[0045] Update output error At the same time, update the number of nodes in the reserve pool to If the model outputs the root mean square error at this time... Still greater than the set error tolerance value And the total number of neurons Not exceeding the maximum limit If so, continue with steps S30 and S40;

[0046] If the standard root mean square error of the network model's output decreases to The following, or the number of nodes in the reserve pool has exceeded If the training terminates, a neural network model that conforms to the conditions of the manifold regularized stochastic configuration theory is finally obtained.

[0047] In one embodiment, the step of calculating the probability density estimate of each training sample in the feature space based on the kernel density estimation algorithm includes:

[0048] Based on the kernel density estimation algorithm, probability density modeling is performed on the sample data. For each sample data, the probability density value in the feature space is calculated using a density estimation function of the following form:

[0049]

[0050]

[0051] in and They are the first and A sample in having The network output error of the reserve pool neurons; It is a bandwidth parameter used to adjust the smoothness of density estimation.

[0052] In one embodiment, the step of constructing a weighted matrix using the sample density estimate as weights and resolving the output weights of the network model using weighted least squares includes:

[0053] The estimated density values ​​of each obtained sample Introduced as a weighting factor in the weighted least squares optimization process, it enables the weighted solution of the output weights:

[0054]

[0055] Therefore, we can obtain:

[0056]

[0057] in This represents the Moore-Penrose pseudo-inverse matrix. It is a diagonal matrix composed of the estimated density values ​​of each sample.

[0058] In one embodiment, the step of updating the trained output weights online based on real-time data according to the projection algorithm and outputting the prediction result in real time includes:

[0059] Based on the projection algorithm, the trained output weights are updated online according to real-time data, and the real-time output is calculated:

[0060]

[0061]

[0062] in , They are two constants. yes The output of the reserve pool at any time, It is the output error before correction.

[0063] Therefore, this application has the following beneficial effects:

[0064] This application provides an industrial data prediction method based on a robust recurrent stochastic network, comprising the following steps: acquiring industrial data with time-series characteristics, constructing an input sample set and a target output set, and initializing model parameters; initializing the output error vector, setting a scaling factor for the output error, and constructing an initial network model; randomly generating candidate neuron parameters, and selecting qualified candidate neurons based on a discriminant condition incorporating manifold regularization constraints; selecting the optimal candidate neuron from the selected candidate neurons and adding it to the network model, and calculating the output weights and model error of the current network model, iteratively executing the above steps until a termination condition is met; calculating the probability density estimate of each training sample in the feature space based on a kernel density estimation algorithm; constructing a weighting matrix using the sample density estimate as weights, and resolving the output weights of the network model using a weighted least squares method; updating the trained output weights online based on real-time data using a projection algorithm, and outputting the prediction results in real time. This application aims to address the shortcomings of existing models in terms of modeling accuracy and robustness when processing industrial data with uncertain input order, strong temporal sequence, and noise and outliers. By combining robust learning mechanisms with recursive structures, and without the need for order identification, this application ensures global approximation of arbitrary nonlinear mapping functions by randomly generating reserve pool nodes and constructing the network structure based on a supervised mechanism. This not only inherits the efficient modeling capabilities of recurrent networks for temporal features but also significantly enhances the model's stability and generalization ability under complex disturbance environments, making it suitable for accurate prediction and intelligent decision support scenarios in complex dynamic industrial processes. Attached Figure Description

[0065] To more clearly illustrate the technical solutions in this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0066] Figure 1 This is a flowchart illustrating an industrial data prediction method based on a robust cyclic randomized network.

[0067] Figure 2 This is a graph showing the fitting results of different network models in predicting power plant load data;

[0068] Figure 3 This is a comparison of the power load prediction errors of different network models under different noise ratios. Detailed Implementation

[0069] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0070] It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of this application.

[0071] To address the shortcomings of existing technologies, this application constructs a novel robust recurrent stochastic configuration network model. First, industrial time-series data is acquired and preprocessed to construct a sample set for single-step forward prediction. Then, a network reserve pool is built incrementally: in each iteration, several candidate neurons are randomly generated and selected based on a supervised discriminant condition incorporating manifold regularization constraints. This ensures that the selected neurons can both rapidly reduce prediction errors and maintain the inherent manifold geometry of the data, thereby enhancing the model's inherent resistance to noise. The selected optimal neurons are added to the network, and their output weights are directly calculated. This process is iterated until the model error meets the requirements.

[0072] After the network structure is constructed, kernel density estimation is further introduced for sample analysis to calculate the density weight of each training sample, thereby distinguishing normal samples from potentially abnormal samples. Subsequently, weighted least squares is used to recalculate the output weights, significantly reducing the impact of low-density abnormal samples on the final model parameters—a crucial step in achieving robustness. Finally, to adapt to the dynamic changes in industrial processes, this method integrates an online update mechanism based on a projection algorithm, enabling rapid and stable fine-tuning of the output weights using real-time data streams, thus giving the model continuous adaptability.

[0073] Corresponding to the above methods, this application also provides an industrial data prediction device based on a robust cyclic randomized configuration network. This device can be implemented in software, hardware, or a combination of both, for example, integrated into industrial edge computing devices, servers, or dedicated industrial control computers. This application creatively solves the pain point of existing technologies in processing complex industrial time-series data—the difficulty of simultaneously achieving robustness, adaptability, and computational efficiency—by embedding manifold regularization constraints into the incremental construction process, utilizing data density-weighted reshaping of the loss function, and combining it with an online projection update mechanism. The method and device can maintain high prediction accuracy and stability in environments with strong noise and outliers, require no complex order identification, have fast training speeds, and can continuously track dynamic changes in the system. It provides a reliable technical tool for accurate prediction, optimized control, and intelligent decision-making in industrial processes, and has significant industrial application value.

[0074] This application provides an industrial data prediction method based on a robust cyclic randomized network, including steps S10-S70, referring to... Figure 1 , Figure 1 This is a flowchart illustrating an industrial data prediction method based on a robust cyclic randomized network.

[0075] Step S10: Acquire industrial data with time series characteristics, construct the input sample set and the target output set, and initialize the model parameters;

[0076] Step S20: Initialize the output error vector, set the scaling factor of the output error, and build the initial network model;

[0077] Step S30: Randomly generate candidate neuron parameters and select candidate neurons that meet the requirements based on the discrimination conditions introduced by the manifold regularization constraint;

[0078] Step S40: Select the best candidate neuron from the selected candidates and add it to the network model. Calculate the output weights and model error of the current network model. Iterate through the above steps until the termination condition is met.

[0079] Step S50: Based on the kernel density estimation algorithm, calculate the probability density estimate of each training sample in the feature space;

[0080] Step S60: Using the sample density estimate as weights, construct a weighting matrix, and re-solve the output weights of the network model using the weighted least squares method;

[0081] Step S70: Based on the projection algorithm, update the trained output weights online according to real-time data, and output the prediction results in real time.

[0082] Specifically, this embodiment provides an industrial data prediction method based on a robust cyclic stochastic configuration network. This method aims to effectively handle strong time-series data common in industrial environments, containing noise and outliers, and improve the robustness, accuracy, and adaptability of the prediction model. The core of this method lies in constructing a stochastic configuration network with a cyclic structure, and integrating manifold regularization constraints, a data density-based weighted learning mechanism, and an online update strategy within this framework. The following is combined with... Figure 1 The system flowchart shown illustrates the main steps of this method:

[0083] Step S10: Data preparation and model parameter initialization

[0084] First, historical operational data with time-series characteristics are acquired from the target industrial process (e.g., a power plant monitoring system). This data typically includes multiple process variables (such as temperature, pressure, and flow rate) and key indicators to be predicted (such as load and quality parameters). To construct supervised learning samples, the values ​​of multiple process variables collected at each time step are combined into an input vector, and the key indicator value at the next time step is used as the prediction target. All historical time steps are paired according to this rule, forming the input sample set and target output set of the model. Next, necessary preprocessing is performed on the raw data, such as cleaning invalid values ​​and standardizing each variable to eliminate the influence of dimensions. Based on this, the key hyperparameters of the model are initialized, including the maximum number of neurons allowed in the network pool, the expected prediction error tolerance during training, the regularization coefficient used to control model complexity and manifold preservation, and the distribution range setting for randomly generating neuron parameters. This step lays the data foundation and framework constraints for subsequent network construction.

[0085] Step S20: Robust Cyclic Random Configuration Network Basic Initialization

[0086] This step begins by constructing the basic framework of the network model. First, an error vector with the same dimension as the target output set is initialized. The initial value of this vector is the target output value itself, representing the prediction residual in the initial state of the model. Simultaneously, a sequence of error scaling factors is defined, gradually decreasing as the network is built. This sequence is used to dynamically adjust the threshold during subsequent neuron selection, ensuring the effectiveness of incremental network construction. Next, an initial network model framework with a recurrent connection structure is defined. This framework predefines the connection relationships between the input layer, the recurrent buffer, and the output layer, but at this point, the buffer contains no neurons, and its state is zero. Furthermore, to facilitate the subsequent introduction of manifold regularization, this step also computes a graph Laplacian matrix based on the input sample set. This matrix encodes the local proximity relationships of all training samples in high-dimensional space and serves as the mathematical basis for applying manifold constraints.

[0087] Step S30: Generation and selection of candidate neurons based on manifold regularization

[0088] This step is a key component in achieving robustness in this method, aiming to select suitable neurons for the network pool in a supervised manner. First, based on a pre-defined parameter distribution range, a set of candidate neuron internal parameters is randomly generated, including their weights connected to the input layer, their weights cyclically connected to historical states in the pool, and their bias terms. For each set of candidate parameters, they are substituted into a pre-defined activation function, and combined with the current network state (initially empty or with some neurons), the output response sequence of the candidate neuron to all training samples is calculated. Subsequently, all candidate neurons are screened based on a discriminant condition that incorporates manifold regularization constraints. This condition requires that the addition of candidate neurons not only significantly reduce the model's current overall prediction residual but also that they introduce a geometric manifold structure inherent in the feature space without excessively perturbing the data. Through this dual constraint, the selected candidate neurons can effectively learn the input-output mapping and are more resistant to noise and anomalous perturbations in the data, thereby enhancing the structural stability of the model.

[0089] Step S40: Optimal neuron selection and incremental network construction

[0090] From the candidate neurons selected in step S30 that meet the constraints, an "optimal" neuron is selected and formally added to the network reserve pool. Optimality is typically defined as maximizing model performance, and can be quantified using an evaluation metric that comprehensively considers both error reduction contribution and manifold preservation. The selected neuron's parameters (input weights, recurrent weights, biases) are permanently added to the network. Subsequently, the entire network's internal state representation of the training samples is updated, and based on the output states of all neurons in the current reserve pool, the connection weights of the output layer are directly calculated using the least squares method, ensuring that the network's predicted output under the current structure is as close as possible to the target value. The overall prediction error of the network at this point (e.g., root mean square error) is calculated. If this error exceeds a preset tolerance and the number of neurons in the reserve pool has not reached its limit, step S30 is returned to continue generating and selecting new candidate neurons for the next iteration. This "generation-selection-addition-evaluation" cycle continues until the prediction error meets the requirements or the number of neurons reaches its limit. Finally, a compact and high-performance initial network model is obtained.

[0091] Step S50: Calculation of sample weights based on kernel density estimation

[0092] After obtaining the initial network model in step S40, this step aims to further identify the sample distribution characteristics in the training data, preparing for subsequent robust optimization. First, using the trained initial model, the prediction error for each training sample is calculated. Then, a kernel density estimation algorithm is used to analyze the distribution of these prediction errors in the feature space. This algorithm calculates a probability density estimate for each sample, reflecting the number of "neighbor" samples with similar prediction errors. Samples with high density values ​​are located in the main region of the data distribution, usually representing a "normal" pattern; while samples with low density values ​​may be on the edge of the distribution, often representing noise points, outliers, or rare conditions. Through this step, each sample is assigned a quantified density weight, providing a basis for the next step of differential learning.

[0093] Step S60: Calculation of robust output weights based on sample weighting

[0094] This step aims to correct the initial model and reduce the negative impact of outliers. The density estimates for each sample calculated in step S50 are transformed to generate corresponding penalty weights. Typically, low-density samples (potential outliers) are assigned smaller weights to weaken their impact on the model, while high-density samples (mainly normal data) are assigned larger weights. A diagonal weighted matrix is ​​constructed using these weights. Then, weighted least squares is used to resolve the weights of the network's output layer. Unlike the ordinary least squares method used in step S40, weighted least squares considers the sample weight matrix during the solution process, making the optimization objective more focused on fitting the high-weight normal samples and less affected by low-weight outliers. Through this step, a robustly enhanced network model is finally obtained, which exhibits more stable and accurate predictive performance when facing data containing noise and outliers.

[0095] Step S70: Online adaptive update based on projection algorithm

[0096] To enable the model to adapt to the dynamic changes in industrial processes, this step provides an online learning mechanism. The robust model trained offline is deployed to the actual system for real-time prediction. When new real-time data arrives, the model first makes a prediction based on existing parameters. Once the actual observations for that moment are obtained, the prediction error can be calculated. Subsequently, a computationally efficient projection algorithm is used to perform small, targeted online updates to the output layer weights based on the current prediction error and the network's internal state. This algorithm adapts to the new data with minimal weight adjustments, tracking both slow and abrupt changes in the system while avoiding the destruction of the robust model foundation acquired through offline training due to excessively large single updates. After the update, the model continues to predict for the next moment with the latest weights. This step forms an online closed loop of "prediction-observation-update," ensuring the model's continuous adaptability and predictive reliability during long-term operation.

[0097] In one embodiment, the steps of acquiring industrial data with time-series characteristics, constructing an input sample set and a target output set, and initializing model parameters include:

[0098] Given a set of industrial process data with time-series characteristics, construct the input sample set:

[0099]

[0100] in, and They represent the system at time 10:00 and 11:00 respectively. The output and input of are defined, and the target output set is set as follows:

[0101]

[0102] That is, outputting the next time step based on the input at the current time step. To make predictions, among which This represents the total number of samples;

[0103] Initialize model-related parameters, including: the maximum allowed number of neurons in the reserve pool. Maximum permissible output error Maximum number of candidate neurons generated per round Input weight distribution parameter set ,in , Enter the upper and lower bounds of the weight range. It is a change; manifold regularization coefficient It is used to constrain the maintenance of manifold structure during the generation and selection of neurons.

[0104] Specifically, in this embodiment, the data preparation and parameter initialization steps are the foundation for constructing the entire prediction model. Specifically, firstly, the system acquires a set of industrial process data collected in chronological order. Based on this data, the input sample set and target output set of the model are constructed. Each sample in the input sample set consists of a multi-dimensional input feature vector (such as process variables like temperature and pressure) collected at a certain time t; simultaneously, the corresponding target output is not the measurement value at the same time, but is set as the key system output value to be predicted at the next time (t+1) (such as product quality or load). This means that the model's learning objective is to establish a dynamic mapping relationship from the current operating condition to the state at the next time. All historical moments are paired according to this rule to form a complete supervised learning dataset. Given a set of industrial process data with time-series characteristics, the input sample set is constructed as follows:

[0105]

[0106] in, and They represent the system at time 10:00 and 11:00 respectively. The output and input of are defined, and the target output set is set as follows:

[0107]

[0108] That is, outputting the next time step based on the input at the current time step. To make predictions, among which This represents the total number of samples;

[0109] Initialize model-related parameters, including: the maximum allowed number of neurons in the reserve pool. Maximum permissible output error Maximum number of candidate neurons generated per round Input weight distribution parameter set ,in , Enter the upper and lower bounds of the weight range. It is a change; manifold regularization coefficient It is used to constrain the maintenance of manifold structure during the generation and selection of neurons.

[0110] After defining the data structure, this embodiment initializes the key operating parameters of the model. These parameters can be divided into two categories: one category consists of macroscopic parameters that control the model structure and accuracy, including the upper limit of the number of neurons in the reserve pool, the error tolerance threshold required during training, and the upper limit of the number of randomly generated candidate neurons in each iteration; the other category consists of generation and constraint parameters that guide the core algorithm, mainly including a set of parameters defining the range of random generation of input weights and biases, and a crucial manifold regularization coefficient. This regularization coefficient is used to apply geometric constraints during the subsequent neuron selection process, ensuring that the network growth process can perceive and maintain the inherent manifold structure of the data, which is one of the important design considerations for improving model robustness. Through the above steps, a clear data specification and parameter framework are provided for the subsequent construction of adaptive networks.

[0111] In one embodiment, the steps of initializing the output error vector, setting the scaling factor of the output error, and constructing the initial network model include:

[0112] Initialize the output error vector as And set the scaling factor for the output error of the network model. ;

[0113] Assuming a system has already been built by A pool of neurons forms a reserve pool, and the network model at this point can be represented as:

[0114]

[0115]

[0116]

[0117] in, , and They respectively represent the current contents The input weight matrix, reservoir connection weight matrix, and output weight matrix for each neuron. For the corresponding bias term;

[0118] Indicates the first The status of the reserve pool at any given time. Let the set of the reservoir states at all times be a matrix. The selected activation function is... Represents the trace of a matrix;

[0119] Represents the graph Laplacian matrix of the input sample. It is a diagonal matrix. These are similarity matrices, and the two matrices are defined as follows:

[0120]

[0121]

[0122] in Represents an exponential function. For kernel coefficients.

[0123] Specifically, in this embodiment, the network initialization step aims to set a starting point and establish a mathematical expression framework for the incremental construction of the model. Specifically, firstly, the network's output error vector is initialized, with its initial value directly set to the complete target output vector, representing the model's original prediction error before any neurons are included. Simultaneously, a sequence of output error scaling factors between 0 and 1 is set, which decays successively during the network construction process, dynamically adjusting the error reduction threshold during subsequent neuron selection.

[0124] Initialize the output error vector as And set the scaling factor for the output error of the network model. in The range of values ​​is .

[0125] Subsequently, the network model state is defined when the current reservoir contains a specific number of neurons. This model is characterized by several core matrices: the input weight matrix maps external data to the reservoir, the recurrent connection weight matrix defines the recursive dynamics between neurons within the reservoir, and the output weight matrix synthesizes the complex states of the reservoir into the final predicted value. The real-time dynamics of the network are described by the reservoir state vector, which recursively depends on historical states, current input, and bias terms through a non-linear activation function. The state vectors at all time points together constitute the state set matrix, which connects the reservoir dynamics with the model output.

[0126] Specifically, it can be represented as a... A pool of neurons forms a reserve pool, and the network model at this point can be represented as:

[0127]

[0128]

[0129]

[0130] in, , and They respectively represent the current contents The input weight matrix, reservoir connection weight matrix, and output weight matrix for each neuron. This is the corresponding bias term.

[0131] Indicates the first The status of the reserve pool at any given time. Let the set of the reservoir states at all times be a matrix. The selected activation function is... This represents the trace of the matrix. Furthermore, to embed manifold regularization constraints later, this step also requires pre-compiling a graph Laplacian matrix that reflects the geometry of the training data itself. This matrix is ​​constructed based on the similarity matrix between the input samples, where the sample similarity is typically calculated using a Gaussian kernel function, and its value decays exponentially with increasing Euclidean distance between samples. The diagonal matrix is ​​the sum of the rows (or columns) of the similarity matrix. (Graph Laplacian matrix) This will be used to impose constraints during subsequent neuron generation, forcing newly added neurons to maintain local proximity relationships of data in the feature space, thereby enhancing the model's generalization and noise resistance.

[0132] Represents the graph Laplacian matrix of the input sample. It is a diagonal matrix. These are similarity matrices, and the two matrices are defined as follows:

[0133]

[0134]

[0135] in Represents an exponential function. For kernel coefficients.

[0136] In one embodiment, the step of randomly generating candidate neuron parameters and selecting candidate neurons that meet the requirements based on the discriminant conditions introduced by the manifold regularization constraint includes:

[0137] From the set of input weight distributions In the process, the input weight matrix is ​​randomly generated. , Reserve pool connection matrix and bias vector And substitute it into the activation function formula:

[0138]

[0139] To generate Group candidate neurons , Indicates having The first reserve pool node One candidate neuron;

[0140] The above Substitute the candidate neurons into the randomized configuration criteria with manifold regularization constraints to select neurons that meet the requirements:

[0141]

[0142] in It is the output dimension, from Candidate neurons that satisfy the inequality constraints are selected from the samples.

[0143] Specifically, in this embodiment, the candidate neuron generation and selection step based on manifold regularization constraints is the core recurrent unit of incremental model construction. This step aims to efficiently select neurons from a large number of randomly generated candidate neurons that can significantly improve model performance and meet structural constraints. The process can be divided into two consecutive stages:

[0144] First, candidate neurons are randomly generated. Based on a pre-defined parameter distribution range, multiple sets of neuron internal parameters are randomly generated. Each set of parameters includes an input weight vector, a recurrent weight vector connected to the historical states in the buffer pool, and a bias value. Substituting these parameters into a pre-defined non-linear activation function and combining them with the current network's input data and historical states, the dynamic output sequence of each candidate neuron for all training samples can be calculated, thus forming a set of candidate neuron feature representations.

[0145] From the set of input weight distributions In the process, the input weight matrix is ​​randomly generated. , Reserve pool connection matrix and bias vector And substitute it into the activation function formula:

[0146]

[0147] To generate Group candidate neurons , Indicates having The first reserve pool node 1 candidate neuron.

[0148] The above Substitute the candidate neurons into the randomized configuration criteria with manifold regularization constraints to select neurons that meet the requirements:

[0149]

[0150] in It is the output dimension (a sequence of non-negative real numbers). satisfy ,and ),from Candidate neurons that satisfy the inequality constraints are selected from the samples.

[0151] Subsequently, a crucial constraint screening process is performed. The outputs of all candidate neurons are substituted into a supervised discriminant inequality incorporating a manifold regularization term for testing. The left side of this inequality measures the correlation between the candidate neuron's output and the current model residuals, representing its potential to reduce prediction error; the right side is a dynamically decaying error tolerance threshold, a manifold regularization term related to the graph Laplacian matrix, which quantifies the potential perturbation to the intrinsic manifold structure of the data if this neuron is added. Only candidate neurons with sufficiently strong correlations and whose structural perturbations are limited to a controllable range can pass this inequality test. This mechanism ensures that each newly added neuron represents an optimal balance between improving accuracy and preserving the inherent geometric structure of the data, fundamentally endowing the model with structural robustness to noise and outliers.

[0152] In one embodiment, the step of selecting the optimal candidate neuron from the selected candidates and adding it to the network model, calculating the output weights and model error of the current network model, and iteratively executing the above steps until the termination condition is met includes:

[0153] To determine the optimal candidate neuron, we first define a set of variables:

[0154]

[0155] in Among the generated candidate neurons, those that satisfy both the inequality constraints and... Neurons that reach their maximum value are selected as candidate nodes for the optimal reserve pool.

[0156] Add the node to the current network model and directly solve for the new output weight matrix based on the target output. Then, the standard root mean square error of the network model's output is calculated and used as the loss function;

[0157] Update output error At the same time, update the number of nodes in the reserve pool to If the model outputs the root mean square error at this time... Still greater than the set error tolerance value And the total number of neurons Not exceeding the maximum limit If so, continue with steps S30 and S40;

[0158] If the standard root mean square error of the network model's output decreases to The following, or the number of nodes in the reserve pool has exceeded If the training terminates, a neural network model that conforms to the conditions of the manifold regularized stochastic configuration theory is finally obtained.

[0159] Specifically, in this embodiment, the incremental construction and termination control steps of the network model complete the crucial transition from theoretical screening to actual model growth and ensure the orderly convergence of the entire construction process. This step begins with optimal selection from the pool of candidate neurons selected in the previous step that meet the manifold regularization constraints. For this purpose, a comprehensive evaluation index ξ is defined, which quantifies the two contributions of each candidate neuron: its numerator characterizes the correlation between the neuron's output and the current model residual, reflecting its ability to directly reduce prediction errors; its denominator and manifold regularization term assess the energy scale of its output and its potential impact on the data manifold structure. Selecting the candidate neuron that maximizes this index ξ means selecting the node that achieves the best balance between improving accuracy and maintaining structural stability.

[0160] Specifically, it is expressed as follows:

[0161]

[0162] in Among the generated candidate neurons, those that satisfy both the inequality constraints and... Neurons that reach their maximum value are selected as candidate nodes for the optimal reserve pool.

[0163] Add the node to the current network model and directly solve for the new output weight matrix based on the target output. Then, the standard root mean square error of the network model's output is calculated and used as the loss function.

[0164] Model growth and parameter updates follow. The parameters of the selected optimal neurons are formally incorporated into the network's state pool, thereby expanding the network's state representation capabilities. Subsequently, based on the expanded pool state matrix, a new output weight matrix is ​​directly calculated by solving a least-squares problem. This analytical solution avoids iterative optimization and is highly efficient. Next, the model's predictions on all current training samples are calculated using the new weights, with the standard root mean square error used as the loss function for performance evaluation.

[0165] Finally, the iterative loop and termination check are performed. The model updates the residual vector based on the latest error and increments the neuron count. Two termination conditions are then checked: first, whether the model accuracy has reached the preset tolerance (error less than ε); second, whether the network pool size has reached its upper limit (number of neurons exceeds ε). If any condition is met, the model is built and the final model is output; otherwise, the process returns to the candidate neuron generation step, starting the next round of the "filter-add-update" loop. Update the output error. At the same time, update the number of nodes in the reserve pool to If the model outputs the root mean square error at this time... Still greater than the set error tolerance value And the total number of neurons Not exceeding the maximum limit Then continue with steps S30 and S40. If the standard root mean square error of the network model's output decreases to... The following, or the number of nodes in the reserve pool has exceeded If the training terminates, a neural network model that conforms to the conditions of the manifold regularized stochastic configuration theory is finally obtained.

[0166] In one embodiment, the step of calculating the probability density estimate of each training sample in the feature space based on the kernel density estimation algorithm includes:

[0167] Based on the kernel density estimation algorithm, probability density modeling is performed on the sample data. For each sample data, the probability density value in the feature space is calculated using a density estimation function of the following form:

[0168]

[0169]

[0170] in and They are the first and A sample in having The network output error of the reserve pool neurons; It is a bandwidth parameter used to adjust the smoothness of density estimation.

[0171] Specifically, in this embodiment, the sample density calculation step based on kernel density estimation is an important preprocessing step for robust optimization of the model. This step aims to quantify the distribution position of each training sample in the model error space, thereby distinguishing between "normal" samples and potentially "abnormal" samples, and providing a data foundation for subsequent differential weighted learning.

[0172] After obtaining the initial network model through incremental construction, the network prediction error for all training samples is first calculated. These error values ​​constitute a feature space, reflecting the model's fit to each sample. Subsequently, kernel density estimation, a nonparametric statistical method, is used to model the probability distribution of this error space. For each specific sample... The calculation of its probability density estimate essentially involves examining the "proximity" of the sample's error to the errors of all other samples in the dataset using a smoothed kernel function (usually a Gaussian kernel). The errors of all other samples are related to the sample's error. The density contribution is calculated by weighting the sum of the two samples based on their distance in the error space using a kernel function. Samples that are closer in distance have a greater contribution weight.

[0173] Among them, bandwidth parameter This is a key hyperparameter that controls the smoothness of the density estimate. A larger bandwidth results in a smoother, more generalized density distribution, which may mask local details; a smaller bandwidth makes the density estimate focus more on local fluctuations but is more sensitive to noise. By setting this parameter appropriately, density estimates that truly reflect the inherent distribution of the data can be obtained. Ultimately, each sample will obtain a continuous probability density estimate. The level of this value has clear significance: a high density value indicates that the sample is in the core region of the error distribution, belonging to the "mainstream" pattern that the model can fit well; while a low density value suggests that the sample is on the edge of the distribution, and its error pattern differs significantly from most samples, likely corresponding to noise, outliers, or rare conditions. This quantified density information will become the direct basis for the next step of adaptive sample weighting.

[0174] In one embodiment, the step of constructing a weighted matrix using the sample density estimate as weights and resolving the output weights of the network model using weighted least squares includes:

[0175] The estimated density values ​​of each obtained sample Introduced as a weighting factor in the weighted least squares optimization process, it enables the weighted solution of the output weights:

[0176]

[0177] Therefore, we can obtain:

[0178]

[0179] in This represents the Moore-Penrose pseudo-inverse matrix. It is a diagonal matrix composed of the estimated density values ​​of each sample.

[0180] Specifically, in this embodiment, the robust optimization step based on sample density weighting is the core step in achieving the final robustness of the model. This step aims to use the sample density knowledge obtained in the previous stage to finely reconstruct the output weights of the initial model, thereby systematically correcting the negative impact of abnormal samples on the model parameters.

[0181] First, the probability density estimate calculated for each training sample is converted into its importance weight in the optimization process. Typically, the conversion rule is designed to make the weight inversely proportional to the density value; that is, low-density (potentially anomalous) samples are assigned smaller weights, while high-density (mainstream normal) samples are assigned larger weights. Based on this, a diagonal weight matrix is ​​constructed, with each sample's weight as its diagonal element. This matrix is ​​then embedded as a penalty factor into the loss function.

[0182] The estimated density values ​​of each obtained sample Introduced as a weighting factor in the weighted least squares optimization process, it enables the weighted solution of the output weights:

[0183]

[0184] Therefore, we can obtain:

[0185]

[0186] in This represents the Moore-Penrose pseudo-inverse matrix. It is a diagonal matrix composed of the estimated density values ​​of each sample.

[0187] Subsequently, a weighted least squares method is performed. Unlike the standard least squares method, this method, when solving for the output weights, does not aim to minimize the sum of squared prediction errors for all samples, but rather to minimize the sum of squared weighted errors. This means that the optimization process is explicitly guided to prioritize fitting the normal samples that are assigned high weights. Mathematically, the optimal solution for the output weights can be directly obtained by calculating the pseudo-inverse of the weighted reservoir state matrix and multiplying it by the weighted target output matrix. Through this crucial re-solution process, the training of model parameters is placed within a "de-biasing" framework. The high-density, dominant normal sample group effectively guides the determination direction of the final model parameters, while the interference effect of the sparsely distributed, low-density abnormal samples is significantly weakened by the weight matrix.

[0188] In one embodiment, the step of updating the trained output weights online based on real-time data according to the projection algorithm and outputting the prediction result in real time includes:

[0189] Based on the projection algorithm, the trained output weights are updated online according to real-time data, and the real-time output is calculated:

[0190]

[0191]

[0192] in , They are two constants. yes The output of the reserve pool at any time, It is the output error before correction.

[0193] Specifically, in this embodiment, the online adaptive update step based on the projection algorithm aims to enable the offline-trained static model to dynamically adapt to real-time changes in the industrial process, thereby ensuring the accuracy and reliability of long-term predictions. This step constructs an efficient and stable online learning closed loop, the core of which is to continuously fine-tune the model's output weights using a computationally simple projection algorithm. When deployed in a real industrial environment, the model receives real-time input data at each sampling moment. First, based on the current fixed network reservoir structure (input weights, loop connection weights, and biases), combined with historical internal states, the reservoir state vector at the current moment is calculated. Subsequently, the model uses the current output weights to linearly combine this state to generate a preliminary prediction result for the next moment. Once the system obtains the actual observation value corresponding to this moment, the error of the preliminary prediction can be calculated. This error signal, along with the current reservoir state, is sent to the projection algorithm for weight update calculation. As shown in the formula, the algorithm adjusts the output weight vector in a controlled, small step size according to the ratio of error to reservoir state. The formula incorporates a regularization constant in the denominator to ensure numerical computation stability and prevent uncontrolled update step size due to excessively small or large state vector magnitudes. The learning rate controls the overall update magnitude and speed. Mathematically, this update method is equivalent to projecting the new weight vector onto the constraint subspace defined by the new samples, achieving fitting of new data with minimal parameter changes. The updated weights take effect immediately for prediction in the next time step, forming a continuous online adaptive loop that significantly improves the model's practical value and robustness in dynamic industrial environments.

[0194] This embodiment uses a power load forecasting example from a power plant in Liaoning Province. The data used is collected in real time by the automated monitoring system of a 500kV substation in Liaoning Province. This automated monitoring system can record the total load of the substation every hour, ensuring the timeliness and accuracy of the load data. During the study period from January 1, 2023 to February 28, 2023, a total of 59 days of load data were collected. The input variables are four key climate factors: temperature, humidity, precipitation, and wind speed; the output is the power load. The entire dataset contains a large amount of noise and outliers introduced by factors such as measurement errors and environmental changes. This phenomenon is common in actual industrial process data and has a significant impact on modeling and prediction performance.

[0195] from Figure 2It can be seen that the data has obvious outliers. The experiment compared recurrent neural networks, specifically echo state networks, cyclic stochastic configuration networks, robust stochastic configuration networks, robust cyclic stochastic configuration networks without manifold regularization constraints, and the robust cyclic stochastic configuration network method for industrial process data prediction proposed in this application. The results of the method in this application are as follows: Figure 2 As shown in (f), experimental results demonstrate that the proposed method can better fit the target output value. Compared with traditional recurrent neural networks and echo-state networks, this method can better suppress noise interference and outlier effects, significantly improving the model's prediction accuracy. Compared with ordinary recurrent stochastic networks and robust stochastic networks, the addition of manifold regularization constraints further enhances the model's ability to perceive the inherent structure of the input data, effectively strengthening the model's generalization performance and robustness. Especially on samples with many outliers, the proposed method exhibits lower prediction errors and higher stability, verifying the important role of manifold regularization and robust learning mechanisms in complex industrial processes. In summary, the experiments fully demonstrate that the proposed method has excellent prediction performance and strong practical value in industrial scenarios with high noise and frequent outliers.

[0196] Figure 3 The comparison results of prediction errors for six different methods in the prediction task are shown, under different noise sample ratios. The noise sample ratios are used to simulate uncertainties introduced by factors such as sensor malfunctions, data acquisition errors, or external interference in real-world applications. Figure 3 It can be seen that, under different noise sample ratios, the prediction error of the original model without a robust mechanism increases significantly with the increase of the noise ratio, demonstrating high sensitivity to noise interference. In contrast, the model with a robust processing strategy can effectively suppress the growth of prediction error under various noise interference conditions, indicating that the robust mechanism can reduce the impact of noise samples on model parameter updates and prediction results. Further comparison of the performance of various robust models shows that the network model proposed in this application achieves the lowest prediction error under different noise sample ratios, with a small error variation, demonstrating good prediction stability and anti-interference ability. This indicates that, by introducing a robust modeling mechanism into the randomly configured network structure, this application can effectively weaken the interference of noise on the model learning process in the presence of noise samples and outliers, thereby achieving more accurate and stable prediction results.

[0197] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0198] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0199] It should be particularly noted that, through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, or of course, by hardware. Based on this understanding, the above technical solutions, in essence or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0200] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. An industrial data prediction method based on a robust cyclic randomized network, characterized in that, Includes the following steps: Step S10: Acquire industrial data with time series characteristics, construct the input sample set and the target output set, and initialize the model parameters; Step S20: Initialize the output error vector, set the scaling factor of the output error, and build the initial network model; Step S30: Randomly generate candidate neuron parameters and select candidate neurons that meet the requirements based on the discrimination conditions introduced by the manifold regularization constraint; Step S40: Select the best candidate neuron from the selected candidates and add it to the network model. Calculate the output weights and model error of the current network model. Iterate through the above steps until the termination condition is met. Step S50: Based on the kernel density estimation algorithm, calculate the probability density estimate of each training sample in the feature space; Step S60: Using the sample density estimate as weights, construct a weighting matrix, and re-solve the output weights of the network model using the weighted least squares method; Step S70: Based on the projection algorithm, update the trained output weights online according to real-time data, and output the prediction results in real time.

2. The method according to claim 1, characterized in that, The steps of acquiring industrial data with time-series characteristics, constructing an input sample set and a target output set, and initializing model parameters include: Given a set of industrial process data with time-series characteristics, construct the input sample set: ； in, and They represent the system at time 10:00 and 11:00 respectively. The output and input of are defined, and the target output set is set as follows: ； That is, outputting the next time step based on the input at the current time step. To make predictions, among which This represents the total number of samples; Initialize model-related parameters, including: the maximum allowed number of neurons in the reserve pool. Maximum permissible output error Maximum number of candidate neurons generated per round Input weight distribution parameter set ,in , Enter the upper and lower bounds of the weight range. It is a change; manifold regularization coefficient It is used to constrain the maintenance of manifold structure during the generation and selection of neurons.

3. The method according to claim 1, characterized in that, The steps of initializing the output error vector, setting the scaling factor of the output error, and constructing the initial network model include: Initialize the output error vector as And set the scaling factor for the output error of the network model. ; Assuming a system has already been built by A pool of neurons forms a reserve pool, and the network model at this point can be represented as: ； ； ； in, , and They respectively represent the current contents The input weight matrix, reservoir connection weight matrix, and output weight matrix for each neuron. For the corresponding bias term; Indicates the first The status of the reserve pool at any given time. Let the set of the reservoir states at all times be a matrix. The selected activation function is... Represents the trace of a matrix; Represents the graph Laplacian matrix of the input sample. It is a diagonal matrix. These are similarity matrices, and the two matrices are defined as follows: ； ； ； in Represents an exponential function. For kernel coefficients.

4. The method according to claim 1, characterized in that, The step of randomly generating candidate neuron parameters and selecting candidate neurons that meet the requirements based on the discriminant conditions introduced by the manifold regularization constraint includes: From the set of input weight distributions In the process, the input weight matrix is ​​randomly generated. , Reserve pool connection matrix and bias vector And substitute it into the activation function formula: ； To generate Group candidate neurons , Indicates having The first reserve pool node One candidate neuron; The above Substitute the candidate neurons into the randomized configuration criteria with manifold regularization constraints to select neurons that meet the requirements: ； in It is the output dimension, from Candidate neurons that satisfy the inequality constraints are selected from the samples.

5. The method according to claim 1, characterized in that, The step of selecting the optimal candidate neuron from the selected candidates and adding it to the network model, calculating the output weights and model error of the current network model, and iteratively executing the above steps until the termination condition is met includes: To determine the optimal candidate neuron, we first define a set of variables: ； in Among the generated candidate neurons, those that satisfy both the inequality constraints and... Neurons that reach their maximum value are selected as candidate nodes for the optimal reserve pool. Add the node to the current network model and directly solve for the new output weight matrix based on the target output. Then, the standard root mean square error of the network model's output is calculated and used as the loss function; Update output error At the same time, update the number of nodes in the reserve pool to If the model outputs the root mean square error at this time... Still greater than the set error tolerance value And the total number of neurons Not exceeding the maximum limit If so, continue with steps S30 and S40; If the standard root mean square error of the network model's output decreases to The following, or the number of nodes in the reserve pool has exceeded If the training terminates, a neural network model that conforms to the conditions of the manifold regularized stochastic configuration theory is finally obtained.

6. The method according to claim 1, characterized in that, The step of calculating the probability density estimate of each training sample in the feature space based on the kernel density estimation algorithm includes: Based on the kernel density estimation algorithm, probability density modeling is performed on the sample data. For each sample data, the probability density value in the feature space is calculated using a density estimation function of the following form: ； ； in and They are the first and A sample in having The network output error of the reserve pool neurons; It is a bandwidth parameter used to adjust the smoothness of density estimation.

7. The method according to claim 1, characterized in that, The step of using the sample density estimate as weights to construct a weighted matrix and then resolving the output weights of the network model using the weighted least squares method includes: The estimated density values ​​of each obtained sample Introduced as a weighting factor in the weighted least squares optimization process, it enables the weighted solution of the output weights: ； Therefore, we can obtain: ； in This represents the Moore-Penrose pseudo-inverse matrix. It is a diagonal matrix composed of the estimated density values ​​of each sample.

8. The method according to claim 1, characterized in that, The step of updating the trained output weights online based on real-time data according to the projection algorithm and outputting the prediction result in real time includes: Based on the projection algorithm, the trained output weights are updated online according to real-time data, and the real-time output is calculated: ； ； in , They are two constants. yes The output of the reserve pool at any time, It is the output error before correction.

9. An industrial data prediction device based on a robust cyclic randomized network, characterized in that, include: The preprocessing module is used to clean and standardize industrial data; The initialization module is used to initialize the network model structure and introduces a manifold regularization strategy to enhance the system robustness. A reservoir construction module is used to generate candidate neurons based on a supervision mechanism with manifold regularization. The neuron selection module is used to select the best candidate neurons to add to the reserve pool based on constraints. The kernel density estimation module is used to calculate the estimated density value for each sample, providing a basis for subsequent weighting. The weighted solution module is used to calculate the output weights using the weighted least squares method combined with the pseudo-inverse matrix, thereby suppressing the influence of outliers. The online update module is used to update the output weights in real time based on the projection algorithm, further improving the adaptability and robustness of the network model.

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