Koopman time series prediction system based on chebyshev kan dimensionality and method thereof

Abstract

The application discloses a Koopman time series prediction system based on Chebyshev KAN dimensionality and a method thereof, and the system comprises: a time series data preprocessing module, which synchronously collects multi-source original time series signals of various devices, and pre-processes the multi-source original time series signals to obtain input time series; a linearization embedding module, which is used for acquiring the input time series, and mapping the input time series to a linear space based on a Chebyshev KAN encoder; a dynamic operator generation module, which is used for generating a dynamic operator according to the acquired time series data mapped to the linear space; a segmented prediction module, which is used for dividing a prediction time domain into a plurality of continuous time periods, and performing state recursion and prediction in each time period through the dynamic operator; and a training module, which trains the system in an end-to-end manner, and shares an encoder and a decoder, wherein a training process adopts a weighted combination of a reconstruction loss, a prediction loss and a spectral radius constraint loss.

Description

Technical Field

[0001] This invention relates to the field of industrial time series forecasting technology, and in particular to a Koopman time series forecasting system and method based on Chebyshev KAN dimensionality upscaling. Background Technology

[0002] In modern industrial automation and intelligent monitoring systems, time series forecasting plays a crucial role in ensuring stable equipment operation, optimizing production control, and providing early warnings of anomalies. Especially in applications such as rotary kiln condition monitoring (e.g., kiln head temperature, rotation speed, feed rate) in industries like cement and metallurgy, and transformer load temperature prediction in power systems, industrial time series data generally exhibit strong nonlinearity, significant time variability, high noise levels, and obvious long-term dependencies, placing higher demands on the expressive power, stability, and adaptability of prediction models. In recent years, deep learning methods (such as LSTM and Transformer) have been widely used in nonlinear time series modeling, improving prediction accuracy to some extent. However, these methods typically rely on complex network structures whose internal mechanisms are difficult to explain, leading to performance degradation when operating conditions change or data distribution shifts. Furthermore, the high cost of model training and inference makes it difficult to meet the real-time and deployability requirements of industrial scenarios. The Koopman operator theory offers a new approach to nonlinear system modeling. This theory maps the original nonlinear dynamical system to a high-dimensional linear space, allowing the system's evolution to be characterized by linear operators, thus balancing the model's interpretability with computational analysis advantages. However, existing Koopman-based methods still suffer from key bottlenecks:

[0003] On the one hand, in the mapping stage from nonlinear to linear space, mainstream methods usually rely on predefined basis functions (such as polynomials and trigonometric functions) or use multilayer perceptrons (MLPs) for dimensionality enhancement. The former has limited expressive power and is difficult to cover the multi-scale nonlinear characteristics of complex industrial data; the latter, although it has strong fitting ability, the performance of the model depends on the quantity and quality of the training data, and the resulting model often becomes an uninterpretable black box, thus limiting the interpretability and generalization performance of linear operators.

[0004] On the other hand, existing methods for Koopman operator modeling mostly employ fixed or static operators, meaning that operator parameters are no longer updated with changes in input after model training. This assumption is often difficult to hold in long-term prediction tasks, especially in industrial systems with significant operating condition changes and non-stationary characteristics (such as rotary kiln load fluctuations and transformer thermal inertia changes). Static operators struggle to accurately characterize the dynamic evolution of the system, easily leading to error accumulation and prediction drift. Although some studies have attempted local modeling using piecewise operators, an effective mechanism to adaptively generate dynamic operators based on input characteristics is still lacking, limiting the ability to characterize the time-varying dependencies of the system.

[0005] Therefore, existing technologies still have shortcomings in terms of nonlinear mapping accuracy, the dynamic adaptability of the Koopman operator, and long-term prediction stability. There is an urgent need for a time series prediction method that can achieve high-quality linear embedding and support dynamic evolutionary modeling to improve the prediction performance and robustness of the Koopman framework in complex industrial scenarios. Simultaneously, this method should also possess high computational efficiency to meet the practical needs of real-time performance and resource-constrained deployment in industrial environments. Summary of the Invention

[0006] In view of this, the present invention provides a Koopman time series prediction system and method based on Chebyshev KAN dimensionality upgrade, which can at least solve the problems that still exist in the prior art in terms of nonlinear mapping accuracy, dynamic adaptability of Koopman operator and long-term prediction stability.

[0007] This invention achieves refined modeling and high-precision long-term prediction of non-stationary time series through the above structure, significantly enhancing the accuracy and robustness of Koopman theory in long-term time series prediction scenarios.

[0008] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0009] The Koopman time series forecasting system based on Chebyshev KAN dimensionality upscaling includes:

[0010] Time series data preprocessing module: synchronously acquires multi-source raw time series signals from various devices, and preprocesses the multi-source raw time series signals to obtain the input time series;

[0011] The linearization embedding module is used to acquire the input time series and map the input time series to a linear space based on the Chebyshev KAN encoder;

[0012] The dynamic operator generation module is used to generate dynamic operators based on the acquired time-series data mapped to the linear space.

[0013] The segmented prediction module is used to divide the prediction time domain into multiple consecutive time periods, and perform state recursion and prediction in each time period through dynamic operators.

[0014] The training module trains the system in an end-to-end manner and shares the encoder and decoder. The training process uses a weighted combination of reconstruction loss, prediction loss and spectral radius constraint loss.

[0015] Preferably, the time-series data preprocessing module includes a synchronous historical data acquisition unit, an outlier handling unit, and a dataset partitioning unit; wherein,

[0016] The synchronous historical data acquisition unit uses a unified timestamp to synchronously sample each variable channel to ensure the consistency of time-series data from each device in the time dimension, thereby acquiring multi-source raw time-series signals. ,in The vector represents the multivariate observation signal at time t, where C is the number of variables in the collected data and p is the total length of the collected data.

[0017] An outlier processing unit is used to identify outlier data in multi-source raw time-series signals based on dynamic thresholds: for each channel Calculate the local mean within the sliding window w. with standard deviation and set dynamic threshold range ,in This is the adaptive sensitivity coefficient; if the observed value at any time t is... If the data exceeds this range, it is identified as an outlier. Subsequently, the identified outliers are repaired using a moving average interpolation strategy based on the non-outlier values ​​in their immediate vicinity, generating complete time-series data after interpolation. ;

[0018] ;

[0019] in This is the data from channel c after interpolation at time t. This is the raw acquisition data of channel c at time t;

[0020] Dividing the dataset into units is used to process the complete time-series data after interpolation. The time series data is divided into training, validation, and test sets according to time sequence and preset division ratios, and used as input time series data.

[0021] Preferably, the linearization embedding module includes a sequence embedding unit and a Chebyshev KAN encoder; wherein,

[0022] The sequence embedding unit is used to capture local dynamic features in the neighborhood of each time point in the input time series using a deep convolutional model. By modeling the temporal dependencies before and after a time point, it extracts an initial representation including contextual information.

[0023] Input time series Dimensional transformation is performed to obtain At this point, each channel corresponds to a time series of length L;

[0024] Apply a one-dimensional depthwise convolution in the time dimension, for each channel Timing represents independent action and maps it to The high-dimensional representation is obtained by embedding to obtain the initial representation. :

[0025] ;

[0026] Where C is the number of variables in the time series data, L is the time step of the input model, and k is the dimension of the high-dimensional linear space. Indicates a depthwise convolution operation;

[0027] The Chebyshev KAN encoder, based on the initial representation, utilizes the function approximation capability based on Chebyshev polynomials to further map the original nonlinear state space into a linear space suitable for linear dynamics modeling, thus obtaining the upgraded time series. :

[0028] ;

[0029] in, It is a Chebyshev KAN encoder.

[0030] Preferably, the specific contents of the Chebyshev KAN encoder include:

[0031] The Chebyshev KAN encoder first calculates the Chebyshev polynomial value for each input position in the batch. Tensor:

[0032] ;

[0033] ;

[0034] in, It is a Chebyshev polynomial of the first kind. The encoder consists of Chebyshev polynomials, where n is the order of the Chebyshev polynomial;

[0035] To enable data-driven learning, a trainable coefficient tensor is included. It stores the Chebyshev weights updated during training:

[0036] ;

[0037] The final output is calculated by applying a weighted combination of basis tensors and coefficients. .

[0038] Preferably, the dynamic operator generation module includes:

[0039] The dynamic operator generation module consists of a multilayer sensing mechanism and is used to generate time series data in up-dimensional format. Learn and generate the corresponding dynamic operators ;

[0040] ;

[0041] Among them, dynamic operators Used to characterize the linear evolution of time series states in a high-dimensional linear space.

[0042] Preferably, the segmented prediction module includes the following specific components:

[0043] Predict step size Enter time step Divided into Section, among which ;

[0044] In the i-th time period, based on dynamic operators Construct the corresponding piecewise prediction operator , where i represents the i-th power of the dynamic operator;

[0045] And perform state recursion in a high-dimensional linear space:

[0046] ;

[0047] Representing the current high dimension and the high-dimensional representations predicted for each time period By performing a nonlinear mapping using a decoder composed of a multilayer perceptron (MLP), we obtain:

[0048] ;

[0049] ;

[0050] in, express Directly decode the reconstructed data mapped back to the original space; This represents the prediction result for the i-th segment;

[0051] The prediction results from each time period are concatenated in chronological order to obtain the final prediction sequence:

[0052] ;

[0053] in, For the overall prediction results, Indicates splicing.

[0054] Preferably, the loss function L of the training module is:

[0055] ;

[0056] in, To reconstruct the loss, To predict losses, Let be the spectral radius constraint loss, and a and b represent the weights of the reconstruction loss and the spectral radius constraint loss, respectively.

[0057] The reconstruction loss is specifically as follows:

[0058] ;

[0059] in, For the time series of the input model, This involves reconstructing the original data obtained by directly decoding the time series data back into the original space after mapping it to a linear space. The constraint model establishes a stable mapping relationship between nonlinear and linear spaces;

[0060] The predicted loss is as follows:

[0061] ;

[0062] in, Output predictions for the model. For the true value that needs to be predicted, Used to constrain models and improve prediction accuracy;

[0063] The spectral radius constraint loss is defined as:

[0064] ;

[0065] Where k is the dimension of the high-dimensional linear space. The modulus of the eigenvalues ​​of the dynamic operator is given. When the modulus is greater than 1, the system state exhibits a divergent trend. When the modulus is less than 1, the system evolution gradually weakens. When the modulus is equal to 1, the system is in a stable evolutionary state. The eigenvalues ​​of the constraint operator enable the model to learn stable components of the data.

[0066] Preferably, it includes the following steps:

[0067] Simultaneously acquire multi-source raw time-series signals from various devices, and preprocess the multi-source raw time-series signals to obtain the input time series;

[0068] The input time series is obtained and mapped to a linear space based on the Chebyshev KAN encoder.

[0069] Dynamic operators are generated based on the obtained time-series data mapped to the linear space;

[0070] The prediction time domain is divided into multiple continuous time periods, and state recursion and prediction are performed in each time period through dynamic operators.

[0071] The system is trained in an end-to-end manner, with the encoder and decoder shared. The training process uses a weighted combination of reconstruction loss, prediction loss, and spectral radius constraint loss.

[0072] The beneficial effects of this invention are:

[0073] This invention proposes a Koopman time series prediction system and method based on Chebyshev KAN dimensionality enhancement, constructing an end-to-end prediction framework that integrates time series data preprocessing, linearization embedding, dynamic operator generation, and piecewise prediction mechanisms. The time series data preprocessing identifies outliers in each channel based on dynamic thresholds and repairs them using a moving average interpolation strategy for non-outliers in the preceding and following neighborhoods, providing high-quality input for subsequent modeling. The linearization embedding module parametrically models the nonlinear mapping using a learnable family of univariate functions. Combined with the excellent numerical stability and minimax approximation properties of Chebyshev polynomials, it effectively reduces parameter redundancy and numerical instability while ensuring the expressive power of the dimensionality enhancement mapping, thereby improving the representativeness and interpretability of the system dynamics in the Koopman linear space. The dynamic operator generation module dynamically adjusts operators according to changes in system state, overcoming the limitation of fixed operators in traditional models that cannot reflect the time-varying characteristics of the system, significantly improving the model's adaptability to non-stationary time series and multi-condition changing scenarios.

[0074] Furthermore, the segmented prediction mechanism divides the time domain into multiple consecutive time periods during long-term prediction and adaptively selects corresponding dynamic operators within each segment for system state recursion and prediction. This strategy effectively reduces time-step iterative calculations, significantly alleviates the problems of long-term prediction error accumulation and computational delay in traditional methods, thereby improving the stability and efficiency of prediction.

[0075] The synergistic effect of these technologies enables the prediction model to significantly improve the modeling accuracy and long-term prediction stability of complex nonlinear systems while maintaining the computational advantages of the Koopman linear framework. In practical industrial applications, this invention has been validated on industrial time-series datasets such as rotary kiln temperature and power transformers. Experimental results show that this invention can significantly improve prediction accuracy and robustness when processing complex, nonlinear, multivariate, and strongly disturbed industrial time-series data, demonstrating good scalability and engineering application value. Attached Figure Description

[0076] To more clearly illustrate the technical solutions in the embodiments of the present invention 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 only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0077] Figure 1 This is a model structure diagram of the Koopman network based on Chebyshev KAN dimensionality upgrade;

[0078] Figure 2 This is a structural diagram of the linearized embedded module;

[0079] Figure 3 This is a structural diagram of the dynamic operator generation module;

[0080] Figure 4 This is a structural diagram of the segmented prediction module. Detailed Implementation

[0081] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0082] Example 1:

[0083] In the industrial setting of rotary kilns, for tasks such as predicting the future state of H steps using historical time-series data of length L, Figure 1 As shown, the present invention provides a Koopman network for long-term time series prediction, specifically comprising the following steps:

[0084] Time-series data preprocessing module: In the rotary kiln industrial production system, through multi-source sensors and control systems deployed on the kiln body and auxiliary equipment, time-series signals of eight key process variables, such as kiln head temperature and sintering zone temperature, are synchronously collected based on a unified time reference to form multi-source raw time-series signals. Subsequently, anomaly identification based on dynamic thresholds is performed on the multi-source raw time-series signals, and the identified abnormal data is repaired using a moving average interpolation method to obtain high-quality interpolated data. Finally, the interpolated data is divided into training set, validation set, and test set in a 7:2:1 ratio to provide data support for subsequent model training and evaluation.

[0085] Linearization Embedding Module: The linearization embedding module maps the input time series data to a high-dimensional linear space. The linearization embedding module consists of two parts: a sequence embedding unit and a Chebyshev KAN structure. The sequence embedding unit is used to extract the temporal features of the time series. The Chebyshev KAN structure uses the function approximation ability of Chebyshev polynomials and the nonlinear fitting ability of KAN network to perform dimensionality increase processing on the embedded features, so as to realize the nonlinear dynamic linearization representation.

[0086] Dynamic operator generation module: Input the time series data in the high-dimensional linear space into the dynamic operator generation module, and use the multilayer perceptron (MLP) structure to generate the dynamic Koopman operator corresponding to the current input; the operator is adaptively adjusted as the input data changes, used to characterize the dynamic evolution law of the system state under different inputs, and to provide operator parameters for subsequent piecewise prediction.

[0087] Segmented prediction module: Performs segmented prediction, dividing the prediction time domain into multiple continuous time periods; within each time period, performs linear recursion and prediction of the system state based on the dynamic Koopman operator generated in step three; the decoder restores the recursed linear space state to the original time series space, and splices the segmented results in chronological order to obtain the complete long-term prediction output.

[0088] Training module: The entire network is trained in an end-to-end manner. The decoder structure is shared during the training process, and a weighted combination of reconstruction loss, prediction loss and spectral radius constraint loss is used as the total loss function to ensure the stability and accuracy of the model in long-term prediction.

[0089] The specific content of data acquisition in the time series data preprocessing module includes:

[0090] In rotary kiln industrial production systems, a multi-source sensor and control system deployed on the kiln body and auxiliary equipment synchronously collects time-series signals of eight key process variables reflecting the thermal and mechanical operating conditions within the kiln. These variables include: kiln head temperature, sintering zone temperature, kiln tail temperature, main motor current, kiln negative pressure (air pressure), kiln rotation speed, coal feed rate, and cooling fan current. All signals are synchronously sampled at the millisecond level using a unified time reference to obtain multi-source raw time-series signals. ,in The vector of multivariate observed signals at time t provides a high-quality raw data foundation for subsequent linearization embedding and dynamic operator generation. p is the total length of the collected data. Each element in the data matrix corresponds to the monitoring value of a key variable at a certain time.

[0091] Anomaly processing is crucial because the rotary kiln production environment is complex, and sensors are susceptible to interference from high temperatures, vibrations, and dust. The raw time-series data may contain outliers (such as sudden temperature rises and falls, abrupt current drops to zero, and abnormal pressure fluctuations), requiring anomaly processing to ensure data quality. A dynamic threshold-based anomaly identification method is employed. The specific process involves setting sliding windows for the time-series data of eight key process variables. The window length *w* is set to 15 time steps based on the acquisition frequency (covering a 15-minute production status fluctuation range). The sliding windows iterate through the time-series data of each variable, calculating the local mean of the data within each window *w*. with standard deviation and set dynamic threshold range ,in 2.5 represents the number of data channels, and 2.5 is the adaptive sensitivity coefficient, adapting to the normal distribution characteristics of rotary kiln production data and the fault tolerance requirements of industrial scenarios; data points exceeding the dynamic threshold range of the corresponding variable are judged as abnormal data; if the observed value at a certain time t... If the data exceeds this range, it is identified as an outlier. Subsequently, the identified outliers are repaired using a moving average interpolation strategy based on the non-outlier values ​​in their immediate vicinity, generating complete time-series data after interpolation. ;

[0092] ;

[0093] in This is the data from channel c, after interpolation at time t. This is the raw data acquired at time t from channel c.

[0094] Dataset partitioning: To meet the different needs of model training, parameter tuning, and performance evaluation, the high-quality data after imputation will be partitioned. The dataset is divided into training, validation, and test sets in a 7:2:1 ratio. Simultaneously, based on subsequent model input requirements, sliding sampling can be performed on the transformed samples according to a preset temporal window length L, generating sample sequences containing continuous temporal information. This ensures that each sample can fully represent the operating state of the rotary kiln over a continuous period of time. A random stratified partitioning strategy is adopted to ensure that the variable distribution of each dataset after partitioning is consistent with the original imputed data, thus avoiding the impact of data distribution deviation on model training performance.

[0095] like Figure 2 As shown, the linearized embedding consists of two steps: sequence embedding and Chebyshev KAN. First, the sequence embedding module captures the local dynamic features in the neighborhood of each time point in the time series. By modeling the temporal dependencies before and after the time point, it extracts an initial representation rich in contextual information, thus effectively capturing short-term time patterns and local change trends. Subsequently, the Chebyshev KAN module, based on this high-dimensional nonlinear representation, utilizes the function approximation capability based on Chebyshev polynomials to further map the original nonlinear state space into a space suitable for linear dynamics modeling. This process not only achieves efficient linearization representation of nonlinear dynamics but also enhances the model's expressive ability for complex nonlinear transformations through learnable basis functions. These two stages complement each other; the former emphasizes the awareness of temporal context, while the latter focuses on the smooth transition from nonlinear to linear, jointly constructing a robust and interpretable linear space representation, providing a solid foundation for subsequent dynamic Koopman operator modeling.

[0096] Sequence embedding: First, the input time series is... Dimensional transformation is performed to obtain The model employs a channel-independent approach, allowing it to focus on capturing temporal patterns without being affected by inter-channel relationships. Specifically, each channel corresponds to a time series of length L. Subsequently, a one-dimensional depthwise convolution is applied along the time dimension. For each channel Timing represents independent action and maps it to A high-dimensional representation. The initial representation is obtained after embedding. :

[0097] ;

[0098] Where L is the time step of the input model, and k is the dimension of the high-dimensional linear space. This indicates a depthwise convolution operation.

[0099] Chebyshev KAN: This model parameterizes the learnable function using Chebyshev polynomials based on the initial representation. By leveraging its powerful nonlinear approximation capabilities and local activation behavior, it can more effectively extract nonlinear structures from time series data. The specific calculation formula is as follows:

[0100] ;

[0101] ;

[0102] ;

[0103] It is a Chebyshev polynomial of the first kind. The module consists of Chebyshev polynomials, where n is the order of the Chebyshev polynomial and k is the dimension of the high-dimensional linear space. This is the initial representation of the input data after embedding. The Chebyshev KAN encoder first computes the Chebyshev polynomial value for each input position in the batch. A tensor is used to encode the input based on Chebyshev basis functions. To enable data-driven learning, a trainable coefficient tensor is included. It stores the Chebyshev weights updated during training. The final output is then computed by applying a weighted combination of the basis tensors and coefficients. Therefore, the overall mapping process of the linearized embedding module can be uniformly represented as:

[0104] ;

[0105] This invention designs a dynamic evolution operator generation mechanism based on input feature-driven approach to overcome the fundamental limitations of existing time series modeling methods in dealing with non-stationary, time-varying dynamic systems.

[0106] Although Koopman theory can represent nonlinear systems as linear systems in high-dimensional spaces, its static evolution assumption makes it difficult to model dynamic processes over long periods. Therefore, this invention introduces dynamic operators to improve the model's adaptability to non-stationary time-series patterns. The structure of the dynamic operator generation module is as follows: Figure 3 As shown.

[0107] Upgraded time series Input the MLP module to obtain the dynamic operator. The resulting dynamic operators are not fixed. After the model is trained, they are adaptively generated by the fixed parameters of the MLP model according to the changes in the input data, so that the generated operators can be dynamically adjusted with the input features, which is different from the fixed operators in traditional models that remain unchanged after training.

[0108] like Figure 4 As shown, to address the problem of insufficient modeling capability and decreased prediction accuracy in existing time series forecasting techniques when dealing with non-stationary, long-term dependent systems due to the use of fixed or static evolution operators, this invention employs a segmented prediction strategy, dividing the entire prediction time domain into multiple continuous time intervals. When the prediction step size is... At that time, Divided into Section, satisfying in The input time length is used; the segmented prediction module is based on the improved Koopman theory and performs a one-step prediction on each segment, replacing the point-by-point iteration of the original Koopman theory.

[0109] Specifically, in the i-th time period, based on the dynamic operator obtained in the third step... Construct the corresponding piecewise prediction operator , where i represents the i-th power of the dynamic operator;

[0110] And perform state recursion in a high-dimensional linear space:

[0111] ;

[0112] Representing the current high dimension and the high-dimensional representations predicted for each time period By performing a nonlinear mapping using a decoder composed of a multilayer perceptron (MLP), we obtain:

[0113] ;

[0114] ;

[0115] in, express Directly decode the reconstructed data mapped back to the original space; This represents the prediction result for the i-th segment;

[0116] The prediction results from each time period are concatenated in chronological order to obtain the final prediction sequence:

[0117] ;

[0118] in, For the overall prediction results, Indicates splicing.

[0119] In order to maintain the consistency of Koopman embeddings across different segments, this invention shares a decoder. Training is performed end-to-end, and three loss functions are designed, with the specific calculation formulas as follows:

[0120] ;

[0121] in, To reconstruct the loss, To predict losses, Let be the spectral radius constraint loss, and a and b represent the weights of the reconstruction loss and the spectral radius constraint loss, respectively.

[0122] The reconstruction loss is used to constrain the self-consistency of the dimensionality upscaling-linearization-decoding link within the observation space, ensuring the invertibility of the linear space and the completeness of information. The input time series is... The input data is projected onto a linear space and then directly decoded back to the original space to obtain the result. The specific formula for reconstruction loss is:

[0123] ;

[0124] The prediction loss is used to constrain the final prediction accuracy of piecewise predictions; the model output prediction is... The true value The specific formula for predicting loss is:

[0125] ;

[0126] Since the learned Koopman operator can typically have arbitrary spectra, while time-based models usually only capture dynamic, stable components such as seasonal and trend terms, this invention introduces a spectral loss term to effectively constrain the Koopman operator and prevent its spectral radius from becoming too large. It explicitly controls the structure of the Koopman spectrum. To constrain system stability and prevent divergence or error explosion caused by excessively large spectral radius, this invention introduces an eigenvalue deviation penalty. Since the modulus of the eigenvalues ​​of the Koopman operator is a key attribute of the dynamic operator, it directly determines the evolution trend of the data: when the modulus is greater than 1, the system state exhibits a divergent trend; when the modulus is less than 1, the system evolution gradually weakens; when the modulus is equal to 1, the system is in a stable evolution state. Therefore, the spectral radius constraint loss is defined as:

[0127] ;

[0128] Where k is the dimension of the high-dimensional linear space. This represents the modulus of the dynamic operator's eigenvalues. During neural network training, the Adam optimizer is used to calculate and update the gradients of the model weight parameters to minimize the prediction error. The Adam optimizer can provide an independent adaptive learning rate for each parameter, effectively improving optimization efficiency. By iteratively updating the model weights, the loss function gradually decreases until convergence, ultimately achieving optimal prediction performance. The overall training algorithm is as follows:

[0129] (1) Input multivariate time series data L is the data length.

[0130] (2) Set parameters: learning rate η, batch size B, input historical sequence length L, and prediction sequence length. And the number of training epochs, etc.

[0131] (3) Iterative training network:

[0132] 1. Initialize model parameters, input data into the linearization embedding module to obtain linear space data, then obtain dynamic operators through the dynamic operator module, complete the prediction in the linear space, and finally decode back to the original state space;

[0133] 2. Concatenate the model's output predictions in the order of prediction to obtain the predicted values. ;

[0134] 3. Calculate the loss and update the model parameters using the backward gradient;

[0135] 4. If the training count is reached, stop training; otherwise, repeat steps 2-4 above.

[0136] Example 2:

[0137] In the industrial setting of aluminum electrolysis cells, for tasks such as predicting the future state of H steps using historical time-series data of length L, Figure 1 As shown, the present invention provides a Koopman network for long-term time series prediction, specifically comprising the following steps:

[0138] Time-series data preprocessing module: In the aluminum electrolysis cell industrial production system, through multi-source sensors and control systems deployed on the electrolysis cell body and auxiliary equipment, time-series signals of nine key process variables, such as the outer wall temperature and inner temperature of the cell, are synchronously collected based on a unified time reference to form multi-source raw time-series signals. Subsequently, anomaly identification based on dynamic thresholds is performed on the raw data, and the identified abnormal data is repaired using a moving average interpolation method to obtain high-quality interpolated data. Finally, the interpolated data is divided into training set, validation set and test set in a 7:2:1 ratio to provide data support for subsequent model training and evaluation.

[0139] Linearization Embedding Module: The linearization embedding module maps the input time series data to a high-dimensional linear space. The linearization embedding module consists of two parts: a sequence embedding unit and a Chebyshev KAN structure. The sequence embedding unit is used to extract the temporal features of the time series. The Chebyshev KAN structure uses the function approximation ability of Chebyshev polynomials and the nonlinear fitting ability of KAN network to perform dimensionality increase processing on the embedded features, thereby realizing nonlinear dynamic linearization representation.

[0140] Dynamic operator generation module: Input the time series data in the high-dimensional linear space into the dynamic operator generation module, and use the multilayer perceptron (MLP) structure to generate the dynamic Koopman operator corresponding to the current input; the operator is adaptively adjusted as the input data changes, used to characterize the dynamic evolution law of the system state under different inputs, and to provide operator parameters for subsequent piecewise prediction.

[0141] Segmented prediction module: Performs segmented prediction, dividing the prediction time domain into multiple continuous time periods; within each time period, the system state is linearly recursively evolved based on the dynamic Koopman operator generated by the dynamic operator generation module; the decoder restores the recursively linear spatial state to the original time series space, and splices the segmented results in chronological order to obtain the complete long-term prediction output.

[0142] Training module: The entire network is trained in an end-to-end manner. The training process shares the decoder structure and uses a weighted combination of reconstruction loss, prediction loss and spectral radius constraint loss as the total loss function to ensure the stability and accuracy of the model in long-term prediction.

[0143] The specific content of data acquisition in the time series data preprocessing module is as follows:

[0144] In the industrial production system of aluminum electrolysis cells, a multi-source sensor and control system deployed on the electrolysis cell body and auxiliary equipment synchronously collects time-series signals of nine key process variables reflecting the thermal state, electrochemical reaction state, and equipment operation status of the electrolysis cell. These include: synchronously collected data based on a unified time reference, including cell outer wall temperature, cell inner temperature, cell current, cell voltage, aluminum level, electrolyte level, alumina feeding frequency, number of operating days of the aluminum electrolysis cell, and fluoride salt addition. All signals are synchronously sampled at the millisecond level using a unified time reference to obtain multi-source raw time-series signals. ,in The vector of multivariate observed signals at time t provides a high-quality raw data foundation for subsequent linearization embedding and dynamic operator generation. p is the total length of the collected data. Each element in the data matrix corresponds to the monitoring value of a key variable at a certain time.

[0145] Anomaly processing is crucial because the aluminum electrolysis cell production environment is characterized by high temperatures, strong electromagnetic interference, electrolyte volatile corrosion, and mechanical vibration. Sensors are susceptible to environmental interference, leading to outliers in the raw time-series data (such as sudden temperature rises and falls, voltage spikes, abnormal current fluctuations, abrupt changes in aluminum / electrolyte levels, and abnormally high or zero alumina feeding frequency). Anomaly processing is necessary to ensure data quality. A dynamic threshold-based anomaly identification method is employed. The specific process involves setting sliding windows for the time-series data of nine key process variables. The window length *w* is set to 15 time steps based on the acquisition frequency (covering a 15-minute production status fluctuation range). The sliding windows iterate through the time-series data of each variable, calculating the local mean of the data within each window *w*. with standard deviation and set dynamic threshold range ,in 1.5 represents the number of data channels, and 1.5 is the adaptive sensitivity coefficient, adapting to the fault tolerance requirements of the aluminum electrolysis cell industrial scenario; data points exceeding the dynamic threshold range of the corresponding variable are judged as abnormal data; if the observed value at a certain time t... If the data exceeds this range, it is identified as an outlier. Subsequently, the identified outliers are repaired using a moving average interpolation strategy based on the non-outlier values ​​in their immediate vicinity, generating complete time-series data after interpolation. ;

[0146] ;

[0147] in This is the data from channel c, after interpolation at time t. This is the raw data acquired at time t from channel c.

[0148] Dataset partitioning: To meet the different needs of model training, parameter tuning, and performance evaluation, the high-quality data after imputation will be partitioned. The dataset is divided into training, validation, and test sets in a 7:2:1 ratio. Simultaneously, based on subsequent model input requirements, sliding sampling can be performed on the transformed samples according to a preset temporal window length L, generating sample sequences containing continuous temporal information. This ensures that each sample can fully represent the operating state of the rotary kiln over a continuous period of time. A random stratified partitioning strategy is adopted to ensure that the variable distribution of each dataset after partitioning is consistent with the original imputed data, thus avoiding the impact of data distribution deviation on model training performance.

[0149] like Figure 2As shown, the linearized embedding consists of two steps: sequence embedding and Chebyshev KAN. First, the sequence embedding module captures the local dynamic features in the neighborhood of each time point in the time series. By modeling the temporal dependencies before and after the time point, it extracts an initial representation rich in contextual information, thus effectively capturing short-term time patterns and local change trends. Subsequently, the Chebyshev KAN module, based on this high-dimensional nonlinear representation, utilizes the function approximation capability based on Chebyshev polynomials to further map the original nonlinear state space into a space suitable for linear dynamics modeling. This process not only achieves efficient linearization representation of nonlinear dynamics but also enhances the model's expressive ability for complex nonlinear transformations through learnable basis functions. These two stages complement each other; the former emphasizes the awareness of temporal context, while the latter focuses on the smooth transition from nonlinear to linear, jointly constructing a robust and interpretable linear space representation, providing a solid foundation for subsequent dynamic Koopman operator modeling.

[0150] Sequence embedding: First, the input time series is... Dimensional transformation is performed to obtain The model employs a channel-independent approach, allowing it to focus on capturing temporal patterns without being affected by inter-channel relationships. Specifically, each channel corresponds to a time series of length L. Subsequently, a one-dimensional depthwise convolution is applied along the time dimension. For each channel Timing represents independent action and maps it to A high-dimensional representation. The initial representation is obtained after embedding. :

[0151] ;

[0152] Where C is the number of variables in the time series data, L is the time step of the input model, and k is the dimension of the high-dimensional linear space. This indicates a depthwise convolution operation.

[0153] Chebyshev KAN: This model parameterizes the learnable function using Chebyshev polynomials based on the initial representation. By leveraging its powerful nonlinear approximation capabilities and local activation behavior, it can more effectively extract nonlinear structures from time series data. The specific calculation formula is as follows:

[0154] ;

[0155] ;

[0156] ;

[0157] It is a Chebyshev polynomial of the first kind. The module consists of Chebyshev polynomials, where n is the order of the Chebyshev polynomial and k is the dimension of the high-dimensional linear space. This is the initial representation of the input data after embedding. The Chebyshev KAN encoder first computes the Chebyshev polynomial value for each input position in the batch. A tensor is used to encode the input based on Chebyshev basis functions. To enable data-driven learning, a trainable coefficient tensor is included. It stores the Chebyshev weights updated during training. The final output is then computed by applying a weighted combination of the basis tensors and coefficients. Therefore, the overall mapping process of the linearized embedding module can be uniformly represented as:

[0158] ;

[0159] This invention designs a dynamic evolution operator generation mechanism based on input feature-driven approach to overcome the fundamental limitations of existing time series modeling methods in dealing with non-stationary, time-varying dynamic systems.

[0160] Although Koopman theory can represent nonlinear systems as linear systems in high-dimensional spaces, its static evolution assumption makes it difficult to model dynamic processes over long periods. Therefore, this invention introduces dynamic operators to improve the model's adaptability to non-stationary time-series patterns.

[0161] Upgraded time series Input the MLP module to obtain the dynamic operator. The resulting dynamic operators are not fixed. After the model is trained, they are adaptively generated by the fixed parameters of the MLP model according to the changes in the input data, so that the generated operators can be dynamically adjusted with the input features, which is different from the fixed operators in traditional models that remain unchanged after training.

[0162] like Figure 4 As shown, to address the problem of insufficient modeling capability and decreased prediction accuracy in existing time series forecasting techniques when dealing with non-stationary, long-term dependent systems due to the use of fixed or static evolution operators, this invention employs a segmented prediction strategy, dividing the entire prediction time domain into multiple continuous time intervals. When the prediction step size is... At that time, Divided into Section, satisfying in The input time length is used; the segmented prediction module is based on the improved Koopman theory and performs a one-step prediction on each segment, replacing the point-by-point iteration of the original Koopman theory.

[0163] Specifically, in the i-th time period, based on the dynamic operator obtained in the third step... Construct the corresponding piecewise prediction operator , where i represents the i-th power of the dynamic operator;

[0164] And perform state recursion in a high-dimensional linear space:

[0165] ;

[0166] Representing the current high dimension and the high-dimensional representations predicted for each time period By performing a nonlinear mapping using a decoder composed of a multilayer perceptron (MLP), we obtain:

[0167] ;

[0168] ;

[0169] in, express Directly decode the reconstructed data mapped back to the original space; This represents the prediction result for the i-th segment;

[0170] The prediction results from each time period are concatenated in chronological order to obtain the final prediction sequence:

[0171] ;

[0172] in, For the overall prediction results, Indicates splicing.

[0173] In order to maintain the consistency of Koopman embeddings across different segments, this invention shares a decoder. Training is performed end-to-end, and three loss functions are designed, with the specific calculation formulas as follows:

[0174] ;

[0175] in, To reconstruct the loss, To predict losses, Let be the spectral radius constraint loss, and a and b represent the weights of the reconstruction loss and the spectral radius constraint loss, respectively.

[0176] The reconstruction loss is used to constrain the self-consistency of the dimensionality upscaling-linearization-decoding link within the observation space, ensuring the invertibility of the linear space and the completeness of information. The input time series is... The input data is projected onto a linear space and then directly decoded back to the original space to obtain the result. The specific formula for reconstruction loss is:

[0177] ;

[0178] The prediction loss is used to constrain the final prediction accuracy of piecewise predictions; the model output prediction is... The true value The specific formula for predicting loss is:

[0179] ;

[0180] Since the learned Koopman operator can typically have arbitrary spectra, while time-based models usually only capture dynamic, stable components such as seasonal and trend terms, this invention introduces a spectral loss term to effectively constrain the Koopman operator and prevent its spectral radius from becoming too large. It explicitly controls the structure of the Koopman spectrum. To constrain system stability and prevent divergence or error explosion caused by excessively large spectral radius, this invention introduces an eigenvalue deviation penalty. Since the modulus of the eigenvalues ​​of the Koopman operator is a key attribute of the dynamic operator, it directly determines the evolution trend of the data: when the modulus is greater than 1, the system state exhibits a divergent trend; when the modulus is less than 1, the system evolution gradually weakens; when the modulus is equal to 1, the system is in a stable evolution state. Therefore, the spectral radius constraint loss is defined as:

[0181] ;

[0182] Where k is the dimension of the high-dimensional linear space. This represents the modulus of the dynamic operator's eigenvalues. During neural network training, the Adam optimizer is used to calculate and update the gradients of the model weight parameters to minimize the prediction error. The Adam optimizer can provide an independent adaptive learning rate for each parameter, effectively improving optimization efficiency. By iteratively updating the model weights, the loss function gradually decreases until convergence, ultimately achieving optimal prediction performance. The overall training algorithm is as follows:

[0183] (1) Input multivariate time series data L is the data length.

[0184] (2) Set parameters: learning rate η, batch size B, input historical sequence length L, and prediction sequence length. And the number of training epochs, etc.

[0185] (3) Iterative training network:

[0186] 1. Initialize model parameters, input data into the linearization embedding module to obtain linear space data, then obtain dynamic operators through the dynamic operator module, complete the prediction in the linear space, and finally decode back to the original state space;

[0187] 2. Concatenate the model's output predictions in the order of prediction to obtain the predicted values. ;

[0188] 3. Calculate the loss and update the model parameters using the backward gradient;

[0189] 4. If the training count is reached, stop training; otherwise, repeat steps 2-4 above.

[0190] This invention proposes a Koopman time series prediction system and method based on Chebyshev KAN dimensionality enhancement, and constructs an end-to-end sequence modeling and prediction framework, which effectively improves the modeling ability and prediction performance of traditional Koopman theory in handling long-term prediction tasks of non-stationary and complex nonlinear systems. The framework comprises four core technical mechanisms: First, it identifies outliers in each channel based on dynamic thresholds and repairs them using a moving average interpolation strategy for non-outliers in the preceding and following neighborhoods, providing high-quality input for subsequent modeling. Second, it introduces a nonlinear-to-linear mapping module based on Chebyshev KAN, leveraging the function approximation capability of Chebyshev polynomials and the nonlinear fitting advantage of KAN networks to efficiently embed the original nonlinear time-series data into a high-dimensional linear space, achieving accurate linearization of complex dynamic processes. This reduces multi-layer deep computation in nonlinear feature extraction, lowers model computational complexity, and improves network training and inference efficiency. Third, it designs a dynamic operator generation module, using a multilayer perceptron (MLP) structure to enable the Koopman operator to adaptively change with input features, overcoming the limitation of traditional fixed operator models in capturing the time-varying dynamic characteristics of the system, and enhancing the model's generalization ability and modeling accuracy of non-stationary modes under various operating conditions. Finally, it employs a segmented prediction strategy, dividing the time domain into multiple continuous time periods during long-term prediction, and independently calling the corresponding dynamic Koopman operator within each segment for state recursion and evolution. This strategy effectively reduces point-by-point iterative calculations per time step, alleviates the problems of long-term prediction error accumulation and computational delay, and significantly improves prediction stability and computational efficiency. Through the synergistic effect of the above key modules, this invention achieves high-precision modeling and efficient long-term prediction of complex nonlinear time series, significantly enhancing the applicability, stability, and robustness of the Koopman framework in complex application scenarios such as industrial time series data analysis and operational status prediction.

[0191] The various embodiments in this specification are described in a related manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the system embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0192] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A Koopman time series forecasting system based on Chebyshev KAN dimensionality upscaling, used in rotary kiln industrial scenarios, targeting kilns with a length of [missing information - likely a number]... L Historical time series data predict the future H The task in the step state is characterized by, include: Time series data preprocessing module: synchronously acquires multi-source raw time series signals from various devices, and preprocesses the multi-source raw time series signals to obtain the input time series; The multi-source raw time-series signals include: kiln head temperature, sintering zone temperature, kiln tail temperature, main motor current, kiln internal negative pressure, kiln rotation speed, coal feed rate, and cooling fan current. The linearization embedding module is used to acquire the input time series and maps the input time series to a linear space based on the Chebyshev KAN encoder; The dynamic operator generation module is used to generate dynamic operators based on the acquired time-series data mapped to the linear space. The segmented prediction module divides the prediction time domain into multiple consecutive time periods, and performs state recursion and prediction within each time period using dynamic operators. Specific components include: Predict step size Enter time step Divided into Section, among which ; In the During the time period, based on dynamic operators Construct the corresponding piecewise prediction operator ,in Representing dynamic operators Power; And perform state recursion in a high-dimensional linear space: ； Representing the current high dimension and the high-dimensional representations predicted for each time period By performing a nonlinear mapping using a decoder composed of a multilayer perceptron (MLP), we obtain: ； ； in, express Directly decode the reconstructed data mapped back to the original space; Indicates the first The prediction results for the segment; The prediction results from each time period are concatenated in chronological order to obtain the final prediction sequence: ； in, For the overall prediction results, Indicates splicing; The training module trains the system in an end-to-end manner and shares the encoder and decoder. The training process uses a weighted combination of reconstruction loss, prediction loss and spectral radius constraint loss.

2. The Koopman time series prediction system based on Chebyshev KAN dimensionality upscaling as described in claim 1, characterized in that, The time-series data preprocessing module includes a synchronous historical data acquisition unit, an outlier handling unit, and a dataset partitioning unit; among which, The synchronous historical data acquisition unit uses a unified timestamp to synchronously sample each variable channel to ensure the consistency of time-series data from each device in the time dimension, thereby acquiring multi-source raw time-series signals. ,in Indicates time The multivariable observed signal vector, The number of variables for which data was collected. The total length of the collected data; An outlier processing unit is used to identify outlier data in multi-source raw time-series signals based on dynamic thresholds: for each channel Calculate the sliding window Local mean within with standard deviation and set dynamic threshold range ,in For any given time, the adaptive sensitivity coefficient is used; Observations If the data exceeds this range, it is identified as an outlier. Subsequently, the identified outliers are repaired using a moving average interpolation strategy based on the non-outlier values ​​in their immediate vicinity, generating complete time-series data after interpolation. ; ； in yes The passage is The data after interpolation at each step, yes The passage is The raw data collected at any given moment; Dividing the dataset into units is used to process the complete time-series data after interpolation. The time series data is divided into training, validation, and test sets according to time sequence and a preset division ratio, and used as input time series data.

3. The Koopman time series prediction system based on Chebyshev KAN dimensionality upscaling according to claim 1, characterized in that, The linearization embedding module includes a sequence embedding unit and a Chebyshev KAN encoder; among which, The sequence embedding unit is used to capture local dynamic features in the neighborhood of each time point in the input time series using a deep convolutional model. By modeling the temporal dependencies before and after a time point, it extracts an initial representation including contextual information. Input time series Dimensional transformation is performed to obtain At this time, each channel corresponds to a length of Time series; Apply a one-dimensional depthwise convolution in the time dimension, for each channel Timing represents independent action and maps it to The high-dimensional representation is obtained by embedding to obtain the initial representation. : ； in, It is the number of variables in the time series data. It is the time step of the input model. It is the dimension of a high-dimensional linear space. Indicates a depthwise convolution operation; The Chebyshev KAN encoder, based on the initial representation, utilizes the function approximation capability based on Chebyshev polynomials to further map the original nonlinear state space into a linear space suitable for linear dynamics modeling, thus obtaining the upgraded time series. : ； in, It is a Chebyshev KAN encoder.

4. The Koopman time series prediction system based on Chebyshev KAN dimensionality upscaling according to claim 3, characterized in that, The Chebyshev KAN encoder includes the following details: The Chebyshev KAN encoder first calculates the Chebyshev polynomial value for each input position in the batch. Tensor: ； ； in, It is a Chebyshev polynomial of the first kind. The encoder consists of Chebyshev polynomials. Let be the order of the Chebyshev polynomial; To enable data-driven learning, a trainable coefficient tensor is included. It stores the Chebyshev weights updated during training: ； The final output is calculated by applying a weighted combination of basis tensors and coefficients. .

5. The Koopman time series prediction system based on Chebyshev KAN dimensionality upscaling according to claim 1, characterized in that, The dynamic operator generation module includes: The dynamic operator generation module consists of a multilayer perceptron and is used to generate time series data in up-dimensional format. Learn and generate the corresponding dynamic operators ; ； Among them, dynamic operators Used to characterize the linear evolution of time series states in a high-dimensional linear space.

6. The Koopman time series prediction system based on Chebyshev KAN dimensionality upscaling according to claim 1, characterized in that, The loss function L for the training module is: ； in, To reconstruct the loss, To predict losses, Let be the spectral radius constraint loss, and a and b represent the weights of the reconstruction loss and the spectral radius constraint loss, respectively. The reconstruction loss is specifically as follows: ； in, For the time series of the input model, This involves reconstructing the original data obtained by directly decoding the time series data back into the original space after mapping it to a linear space. The constraint model establishes a stable mapping relationship between nonlinear and linear spaces; The predicted loss is as follows: ； in, Output predictions for the model. For the true value that needs to be predicted, Used to constrain models and improve prediction accuracy; The spectral radius constraint loss is defined as: ； in, It is the dimension of a high-dimensional linear space. The modulus of the eigenvalues ​​of the dynamic operator is given. When the modulus is greater than 1, the system state exhibits a divergent trend. When the modulus is less than 1, the system evolution gradually weakens. When the modulus is equal to 1, the system is in a stable evolutionary state. The eigenvalues ​​of the constraint operator enable the model to learn stable components of the data.

7. A Koopman time series prediction method based on Chebyshev KAN dimensionality upscaling, characterized in that, Includes the following steps: Simultaneously acquire multi-source raw time-series signals from various devices, and preprocess the multi-source raw time-series signals to obtain the input time series; The input time series is obtained and mapped to a linear space based on the Chebyshev KAN encoder. Dynamic operators are generated based on the obtained time-series data mapped to the linear space; The prediction time domain is divided into multiple continuous time periods, and state recursion and prediction are performed in each time period through dynamic operators. The specific content includes: Predict step size Enter time step Divided into Section, among which ; In the During the time period, based on dynamic operators Construct the corresponding piecewise prediction operator ,in Representing dynamic operators Power; And perform state recursion in a high-dimensional linear space: ； Representing the current high dimension and the high-dimensional representations predicted for each time period By performing a nonlinear mapping using a decoder composed of a multilayer perceptron (MLP), we obtain: ； ； in, express Directly decode the reconstructed data mapped back to the original space; Indicates the first The prediction results for the segment; The prediction results from each time period are concatenated in chronological order to obtain the final prediction sequence: ； in, For the overall prediction results, Indicates splicing; The system is trained in an end-to-end manner, with the encoder and decoder shared. The training process uses a weighted combination of reconstruction loss, prediction loss, and spectral radius constraint loss.

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