A time series prediction method and system based on a learnable Koopman spectral operator and dynamic time-lag graph Laplacian constraint, an electronic device and a storage medium

Abstract

The present application relates to the technical field of multivariate time series prediction tasks, and discloses a time series prediction method and system based on a learnable Koopman spectrum operator and dynamic time-lag graph Laplace constraints, an electronic device and a storage medium; multi-source multivariate time series data input and unified state sequence construction; historical sequence embedding and multi-scale time-lag candidate state construction; learnable Koopman observation function and hidden space state promotion, corresponding spectrum operator construction and stable dynamics promotion; dynamic time-lag graph construction and corresponding Laplace constraint construction for asynchronous influence between variables; Koopman spectrum prior and dynamic time-lag graph message fusion prediction; joint loss function construction and end-to-end training; prediction reasoning and other processes learn interpretable dynamic spectrum operators in the hidden space to depict long-term evolution laws, and the present application improves the precision, robustness, stability and interpretability of complex non-stationary multivariate time series prediction.

Description

Technical Field

[0001] This invention relates to the field of multivariate time series forecasting technology, and can be used for wind power forecasting, photovoltaic power forecasting, source-load net load forecasting, traffic flow forecasting, industrial process forecasting, robot state forecasting, equipment health status forecasting, etc. Specifically, it relates to a time series forecasting method and system based on a learnable Koopman spectral operator and dynamic time delay graph Laplace constraints. Background Technology

[0002] In recent years, with the widespread deployment of sensors and information acquisition systems in scenarios such as smart grids, integrated energy systems, industrial process monitoring, traffic network control, meteorological and environmental forecasting, financial risk management, and complex equipment health management, multivariate time series forecasting has become an important foundation for supporting system state perception, risk warning, operation optimization, and decision control. Compared with univariate time series, multivariate time series not only include the trends, periods, abrupt changes, and noise disturbances of each variable, but also the coupling effects, lag propagation, structural switching, and cross-scale linkages between variables. For example, in energy systems, there are complex correlations between load, wind power, photovoltaics, meteorological variables, energy storage status, and grid operation quantities; in transportation systems, changes in traffic flow in upstream sections affect downstream sections after a certain time delay; in industrial systems, there are often asynchronous responses between control quantities, process status quantities, and quality indicators. These complex systems typically exhibit significant nonlinearity, nonstationarity, multi-scale characteristics, and time-varying coupling, requiring forecasting models not only to learn the mapping relationship from history to the future but also to characterize the underlying dynamic evolution laws and lag propagation structures between variables.

[0003] Existing technologies for predicting complex multivariate time series can be broadly categorized as follows.

[0004] The first category comprises prediction methods based on traditional statistical models, state-space models, and physical mechanism models. Statistical models typically include autoregressive models, moving average models, ARIMA, VAR, Kalman filtering, state-space models, Gaussian process regression, and support vector regression. These methods usually establish linear or weakly nonlinear predictive relationships based on historical observation sequences, offering advantages such as clear model structure, low computational complexity, and statistical interpretability of some parameters. For time series with stable periods, weak coupling, and low noise, these methods can achieve certain predictive results. However, in real-world complex systems, variables often exhibit nonlinear interactions, time-varying correlations, and uncertain disturbances. Traditional statistical models typically require pre-specifying lag orders, variable relationships, or noise assumptions, making them difficult to adapt to scenarios with strong non-stationarity, strong coupling, and long prediction step sizes. While physical mechanism models can improve interpretability by utilizing system equations, conservation relationships, and operational mechanisms, they usually rely on accurate system parameters, boundary conditions, and external inputs. When the system has unmodeled disturbances, parameter drift, or observation errors, prediction accuracy and generalization ability tend to decrease.

[0005] The second category is time series prediction methods based on deep learning. With the development of recurrent neural networks, long short-term memory networks, gated recurrent units, temporal convolutional networks, Transformers and their variants, deep learning methods have gradually become an important technical route for multivariate time series prediction. These methods learn complex nonlinear mapping relationships from historical sequences through end-to-end training, capturing certain long- and short-term dependencies and multivariate interaction features. In recent years, Autoformer has addressed long sequence prediction problems through sequence decomposition and autocorrelation mechanisms; PatchTST improves the utilization of long historical windows by dividing time series into subsequence patches and adopting a channel-independent structure; TimesNet extends one-dimensional time changes to two-dimensional multi-period change modeling; and iTransformer uses the variable dimension as the attention modeling object to enhance the expression of multivariate correlations. The above methods have made significant progress in long-term time series prediction, but most of them still mainly adopt a black-box mapping method from historical windows to future windows, lacking explicit constraints on the system's inherent dynamic operators, spectral modes, growth and decay characteristics, and long-term stability. At the same time, attention weights or hidden layer correlations are not necessarily equivalent to variable coupling relationships with physical or dynamic significance, and it is also difficult to accurately describe the time-delayed transmission process between variables.

[0006] The third category comprises prediction methods based on frequency domain analysis, multi-scale decomposition, and spectral feature representation. These methods typically utilize Fourier transform, wavelet decomposition, empirical mode decomposition, variational mode decomposition, and seasonal trend decomposition to decompose the original time series into trend, periodic, and high-frequency disturbance terms, then model the variation patterns at different frequencies or scales. Multi-scale methods can alleviate, to some extent, the problems of strong noise, frequency aliasing, and complex periodic variations in the original sequence, and also help improve the model's ability to express periodicity, trends, and local fluctuations. For example, StemGNN attempts to combine graphical Fourier transform and discrete Fourier transform to jointly characterize the correlation and time dependence between variables in the spectral domain. The shortcomings of this type of method are that its frequency or spectral domain representation usually focuses more on signal decomposition and feature enhancement, and has not fully reflected the evolutionary operator structure of the nonlinear dynamic system in the latent space; at the same time, most methods do not explicitly construct the time-varying lag relationships between variables as dynamic graphical constraints, thus still having shortcomings in cross-variable transmission, state switching, and long-term evolutionary consistency.

[0007] The fourth category comprises prediction methods based on the Koopman operator, dynamic mode decomposition, and their deep extensions. The core idea of ​​Koopman operator theory is to map nonlinear dynamic systems to a high-dimensional or infinite-dimensional space composed of observation functions, allowing the originally nonlinear state evolution to be represented by linear operators. Existing research shows that linearizing nonlinear systems using the Koopman operator facilitates prediction, estimation, and control using linear system theory; however, obtaining a finite-dimensional, stable, and expressive Koopman approximation remains a challenge in practical applications. Dynamic mode decomposition and extended dynamic mode decomposition are typically used to estimate finite-dimensional approximations of the Koopman operator from data, extracting the system's modal structure, eigenvalues, frequencies, and growth / decay information. In recent years, deep Koopman methods have further utilized neural networks to learn observation functions or improve mappings, enabling the model to learn linear evolution structures more suitable for prediction in the latent space; neural dynamic mode decomposition also attempts to backpropagate prediction errors through spectral decomposition, achieving end-to-end Koopman spectral analysis learning. These Koopman-type methods provide a dynamically interpretable technical foundation for predicting complex nonlinear time series, but several shortcomings remain. First, traditional DMD or EDMD typically rely on pre-defined dictionaries, kernel functions, or fixed observation functions, which can lead to insufficient approximation capabilities or excessive dimensionality when dealing with complex high-dimensional systems. Second, while deep Koopman methods can learn latent space mappings, most focus primarily on the overall evolution of a single system state, failing to adequately characterize the time-varying graphical dependencies between multiple variables. Third, Koopman spectral operators usually emphasize linear progression and eigenmode decomposition in the latent space, but lack natural constraints on lag propagation paths between variables, making it difficult to guarantee the smoothness, consistency, and interpretability of intervariate transmission relationships in the graphical structure. Fourth, in long-term time-domain predictions, if Koopman eigenvalues, spectral radii, or latent space recursion processes lack constraints, error accumulation, mode drift, or unstable prediction trajectories can easily occur. Existing Koopman prediction models, such as Koopa, have attempted to distinguish between time-varying and time-invariant dynamics for non-stationary time series and advance different dynamic components through Koopman predictors, but their focus remains on time series dynamics decomposition and prediction, without systematically combining dynamic time-lag graphical structures and graphical Laplace constraints.

[0008] The fifth category comprises multivariate time series prediction methods based on graph neural networks and spatiotemporal graph modeling. These methods typically treat variables, sensors, spatial locations, or system components in a multivariate time series as graph nodes and relationships between variables as graph edges, then combine graph convolution, message passing, gated recurrent units, temporal convolution, or attention mechanisms for prediction. Typical methods include spatiotemporal graph convolutional networks based on predefined topologies, and graph time series prediction models based on adaptive adjacency matrices or implicit relationship learning. GraphWaveNet learns hidden spatial dependencies through adaptive dependency matrices, MTGNN proposes a graph neural network framework for multivariate time series and adaptively learns the graph structure between variables, while latent variable graph inference methods attempt to simultaneously infer and utilize relationships between sequences in the latent space. Graph neural network methods can explicitly model dependencies between variables, exhibiting stronger structural expressive power compared to ordinary deep time series models. However, existing graph time series prediction methods still have significant limitations. First, some methods rely on physical distance, topological connectivity, human experience, or static correlation coefficients to construct the graph structure, making it difficult to adapt to situations where variable relationships in real systems change with operating conditions, time, season, disturbances, and control strategies. Existing research has also pointed out that the correlations of variables in real multivariate time series exhibit time-varying characteristics, making static graph predictions prone to bias. Methods such as TPGNN characterize dynamic variable correlations using time multinomial graphs, and TimeGNN attempts to learn dynamic time series graph representations. However, these dynamic graphs mostly focus on immediate or window-based correlations, failing to fully express the lagged propagation relationship where "variable A influences variable B after several time steps." Secondly, while many graph neural network methods can learn the strength of associations between nodes, this strength often lacks a dynamic spectrum interpretation, making it difficult to explain which modes dominate long-term trends, which correspond to periodic oscillations, and which correspond to decay or growth processes in system evolution. Thirdly, existing dynamic graph methods typically generate adjacency matrices through attention or node embedding, making them susceptible to noise, outliers, and local window perturbations, resulting in significant graph structure fluctuations and a lack of stable graph Laplace smoothing constraints. Finally, the message passing mechanism of graph neural networks usually occurs within the same time slice or fixed time window, while the real impacts in complex systems often exhibit asynchronous and time-delayed characteristics; using only an immediate adjacency matrix is ​​insufficient to characterize propagation paths across time and variables.

[0009] The sixth category comprises prediction methods based on time-lag correlation, causal discovery, and delayed embedding. These methods typically identify time-lag relationships between variables through cross-correlation analysis, lagged correlation coefficients, Granger causality tests, transfer entropy, delayed coordinate embedding, lagged attention mechanisms, or time-lag VAR models, using variables of different lag orders as input features for prediction. These methods can reveal, to some extent, the "change first, response later" propagation pattern between variables, and have practical value for scenarios such as industrial processes, traffic propagation, meteorological impacts, and energy conversion. However, most time-lag modeling methods still suffer from problems such as fixed lag orders, offline relationship estimation, and insufficient adaptability to nonlinear and non-stationary relationships. While some deep learning methods can implicitly select historical time points through attention, their time-lag weights often do not form an interpretable dynamic graph structure, nor do they utilize graph Laplace constraints to ensure the smoothness, sparsity, and structural consistency of the lagged propagation relationship.

[0010] The seventh category comprises structural constraint methods based on graph signal processing and graph Laplace regularization. Graph signal processing treats data defined on irregular graph nodes as graph signals and utilizes tools such as graph spectral analysis, graph filtering, graph smoothing, and graph Laplace regularization to handle the structural relationships between nodes. The graph Laplace matrix can be used to describe graph topology, measure the smoothness of graph signals, and constrain the consistency of signal changes among connected nodes in graph structure learning; related research has also proposed learning Laplace matrices from observed signals to ensure that graph signals satisfy smoothing priors. However, existing graph Laplace regularization methods are mostly used for static graph signal smoothing, semi-supervised learning, graph structure learning, or spatial correlation constraints, and are rarely used in conjunction with Koopman spectral operators for nonlinear dynamical systems. For multivariate time series prediction, applying graph Laplace smoothing constraints only on the same time slice is insufficient to characterize the lag propagation relationships between variables across time steps; similarly, applying ordinary smoothing constraints only to input features or output prediction results is insufficient to constrain the long-term evolution stability of latent space dynamic modes. Therefore, there is still a lack of complete and feasible technical solutions for constructing dynamic time-delay graphs and imposing graph Laplace constraints consistent with time-delay propagation in the Koopman latent space or prediction space.

[0011] In summary, while existing technologies have made some progress in statistical modeling, deep time series modeling, multi-scale frequency domain representation, Koopman dynamics modeling, graph neural networks, dynamic graph learning, time delay relationship identification, and graph Laplace regularization, a unified time series prediction mechanism capable of simultaneously characterizing both "nonlinear dynamic spectral evolution" and "dynamic time delay graph structure between variables" has yet to be established. Specifically, existing methods typically suffer from the following technical gaps: First, there is a lack of learnable Koopman spectral operators to explicitly express the long-term evolution modes, oscillation frequencies, and growth and decay patterns of complex sequences; second, there is a lack of dynamic time delay graph construction mechanisms for multivariable systems, making it difficult to describe the delayed effects and propagation paths of variables over time; third, there is a lack of structural regularization mechanisms that combine dynamic time delay graphs with graph Laplace constraints, making it difficult to guarantee the consistency and smoothness of cross-variable propagation during the prediction process; and fourth, there is a lack of a systematic solution that unifies spectral dynamics prediction, dynamic graph structure learning, time delay dependency modeling, and long-term stability constraints into a single prediction framework.

[0012] To address the aforementioned technological gaps, while currently available multivariate time series forecasting methods have made some improvements in prediction accuracy and feature representation capabilities, they still suffer from the following shortcomings in complex nonlinear and strongly nonstationary scenarios: First, existing deep forecasting models mostly employ a black-box approach of direct mapping from historical windows to future windows, lacking explicit modeling of the system's intrinsic dynamic spectral structure and long-term evolution modes; second, existing Koopman operator methods primarily focus on the linear progression of the latent space, failing to fully integrate the dynamic graph structure and time-delay propagation relationships between variables, making it difficult to describe the asynchronous coupling effects in multivariate systems; third, existing graph time series forecasting methods mostly use static adjacency matrices or instantaneous correlation graphs, making it difficult to adapt to situations where variable relationships change over time and there are delayed responses between different variables; fourth, existing time-delay modeling methods mostly rely on fixed lag orders or offline correlation analysis, lacking a dynamic, end-to-end, and interpretable time-delay graph construction mechanism; fifth, existing graph Laplace constraints are mostly used for static graph smoothing, and have not been fully utilized to constrain dynamic time-delay propagation processes and Koopman latent space evolution processes; sixth, existing methods lack sufficient constraints on error accumulation, mode drift, and prediction trajectory stability in long-term forecasts.

[0013] In summary, existing technologies cannot simultaneously meet the comprehensive requirements of dynamic spectrum modeling, dynamic time delay relationship identification, graph structure smoothing constraints, long-term stable prediction, and prediction interpretability in complex multivariate time series prediction. Summary of the Invention

[0014] The technical problem to be solved by this invention is how to perform spectral domain dynamic modeling of nonlinear time series using learnable Koopman spectral operators, and adaptively characterize the time-varying lag coupling relationship between variables by combining dynamic time-delay graph Laplace constraints, thereby improving the accuracy, stability, long-term consistency and interpretability of prediction of complex multivariate time series.

[0015] To address the aforementioned technical problems, this invention provides a time series prediction method based on a learnable Koopman spectral operator and a dynamic time delay graph Laplace constraint, comprising the following steps: Step 1: Inputting multi-source, multi-variable time-series data and constructing a unified state sequence; Step 2: Historical sequence embedding and multi-scale time-delay candidate state construction; Step 3 involves learning the Koopman observation function and latent space state lifting; Step four allows you to learn about the construction of Koopman spectral operators and the advancement of stable dynamics; Step 5: Constructing a dynamic time delay graph for asynchronous effects between variables; Step 6: Constructing Laplace constraints for dynamic time delay graphs; Step 7: Fusion prediction of Koopman spectral prior and dynamic time-delay graph messages; Step 8: Decoding the prediction results and mapping them to physical constraints; Step 9: Construction of joint loss function and end-to-end training; Step 10: Predictive reasoning and output of interpretable information.

[0016] Preferably, step one specifically refers to acquiring multi-source, multivariate time series data of the object to be predicted and organizing it into a unified time series state representation; the object to be predicted is a time series system with multivariate coupling relationships; the specific operation method is as follows: assuming the system contains... There are 1 variable nodes, and each variable node is at time 1 / 2. have If we consider the observed features, then the original multivariate input can be represented as: , in, Indicates the first Each variable node at time... The observation features may include one or more of the following: the target variable's own historical values, external driving variables, environmental variables, time context variables, equipment state variables, and spatial location variables; if there are known external driving variables in the future, they are denoted as: , in, Indicates the prediction step size. Indicates the future number Known external conditions at each predicted time point; To address the inconsistencies in temporal resolution, missing values, outliers, and dimensional differences in the original multi-source data, the first step is to perform time alignment, missing value completion, outlier removal, and smoothing normalization. For the second step... The first variable node The following normalization method is used for each feature: , in, and Let represent the mean and standard deviation of the corresponding feature in the training set, respectively. To prevent extremely small positive numbers with a denominator of zero.

[0017] As a preferred option, step two is performed as follows: First, an embedding transformation is performed on the input features of each variable node at each time step: , in, and For learnable linear transformation matrices, Indicates the first Node embedding of a variable node, This indicates either time location embedding or calendar context embedding. This represents the temporal characteristics of the embedded nodes; Subsequently, a temporal encoder is used to extract long-short-term dependencies within the historical window; the temporal encoder employs a Transformer encoder, a temporal convolutional network, a gated recurrent network, a multi-scale convolutional network, or a combination thereof; represented by a general encoder as follows: , in, Represents an embedded sequence. The parameter is The timing encoder, Represents a sequence of historical implicit states; To explicitly describe the potential delayed responses between different variables, a multi-scale time delay candidate set is further constructed: , in, Represents the maximum candidate lag order; for any candidate lag order... Construct time-delay candidate states: , in, Indicates the first Each variable node is lagging. The historical implicit state at each time step.

[0018] As a preferred option, step three is executed as follows: For the first Each variable node at time... The historical encoded state is used to define a learnable Koopman observation function: , in, This indicates a learnable Koopman observation function. This indicates the short history length used to construct the local observation function. Indicates time context or operating condition context. Indicates the first The state of each variable node in the Koopman latent space; The Koopman hidden state of all nodes can be represented as: , To ensure that the Koopman latent space retains the necessary information of the original temporal state, a further reconstruction decoder is configured: , in, This represents the latent space reconstruction function; the corresponding reconstruction loss can be expressed as: , Through the above processing, the state of complex nonlinear and multivariate coupled time series is elevated to the Koopman latent space, providing a state foundation for subsequent modeling of learnable spectral operators.

[0019] As a preferred embodiment, step four is executed as follows: After obtaining the Koopman latent space state, a learnable Koopman spectral operator is constructed to characterize the global dynamic evolution mode, periodic frequency, growth and decay characteristics, and long-term propagation law of the system within the prediction window; the construction method is as follows: For the prediction starting point First, the context state is generated based on the historical hidden state and the known external conditions for the future: , in, This represents a pooling or context aggregation function. This indicates the overall state context of the current prediction window; Subsequently, the Koopman spectral parameters are obtained through a spectral parameter generation network: , in, Denotes the Koopman spectral basis transformation matrix. Indicates the first The amplitude or growth / decrease coefficient of each spectral mode. Indicates the first The oscillation frequency of each spectral mode Indicates the number of spectral modes. This represents a learnable spectral parameter generation network; To represent the oscillation modes corresponding to complex eigenvalues ​​in the real space, a two-dimensional real block matrix is ​​used to represent each spectral mode: , in, Indicates the time step; a block diagonal spectral matrix composed of multiple spectral blocks: , This leads to the learnable Koopman spectral operator within the prediction window: , in, This represents the local Koopman linear advance operator within the current prediction window; Through the above processing, the long-term evolution of nonlinear time series can be characterized in the form of spectral operators in the latent space, enabling the model to not only learn the black box mapping from history to the future, but also to explicitly represent the oscillation frequency, growth decay and stable propagation modes of the system.

[0020] Preferably, step five involves constructing a dynamic time-delay plot to describe the lag propagation relationships between different variables over time. This dynamic time-delay plot no longer only describes the instantaneous correlations within the same time slice, but rather describes the source variables. At any moment The state of the target variable At any moment The impact of this is specifically addressed in the following construction method: For the prediction step and candidate lag order Construct the target node query vector: , Construct the source node key vector: , in, Indicates the lag order Embedded representation, and For a learnable projection matrix; when hour, Retrieve historical Koopman hidden states; when hour, Take the hidden states that have been obtained in the previous prediction step; Subsequently, the calculation of the first Individual source variables in lag Next to the Impact rating of each target variable: , in, This represents the attention dimension of the graph. This indicates the strength of prior relationships, such as physical topological distance, spatial proximity, process connectivity, or empirical correlation; if no prior relationship exists, then we can let . This represents the learnable bias for different lag orders. and These are the weighting coefficients; Normalize all source nodes and all candidate lag orders to obtain dynamic time-delay adjacency weights: , in, Indicating in the prediction step Below, variables With lag Influencing variables The strength; To enhance the sparsity and interpretability of graph structure, Execute Top- filter: , Then, the retained edge weights are renormalized, and the equivalent lag time between variable pairs can be calculated: , in, Representing variables For variables The average lag effect time at the current prediction step; through the above processing, the time delay graph structure can be dynamically generated based on the current sample, the current prediction step, and the candidate lag order.

[0021] As a preferred option, step six is ​​specifically executed as follows: After obtaining the dynamic time-delay adjacency matrix, construct the dynamic time-delay graph Laplace constraint to constrain the smoothness, consistency and stability of the lag propagation process between variables in the latent space. For the prediction step and lag order Define the dynamic time-delay adjacency matrix: , Because the dynamic time delay graph represents the source node To the target node Based on the time-directed propagation relationship, an extended bipartite graph adjacency matrix is ​​further constructed: , The corresponding expansion matrix is: , The Laplace matrix of the dynamic time delay graph is: , To ensure that the lagging source node and the current target node remain aligned after Koopman dynamics propagation, an extended graph signal is constructed: , in, Indicates the prediction step The fusion of hidden states, This indicates that the source node's lagging state is advanced by the Koopman spectral operator. The state after one time step; This yields the Laplace regularization term for the dynamic time-delay graph: , Its equivalent form can be expressed as: , The above formula indicates that if the dynamic time delay diagram considers variables... Lag The following pairs of variables If it has a strong influence, then the variable The current hidden state should be related to the variable The state of the hysteretic latent state remains consistent or approximately consistent after being advanced by Koopman dynamics.

[0022] Preferably, in step seven, after obtaining the Koopman prior hidden states and the dynamic time-delay graph structure, the prediction step is performed. The future hidden states are generated recursively; each prediction step utilizes both the global dynamic priors provided by the Koopman spectral operator and the local intervariate propagation information provided by the dynamic time delay graph. First, obtain the result according to step four. Koopman prior states for each prediction step Furthermore, based on the dynamic time-delay adjacency weights generated in step five, time-delay graph messages from different source variables and different lag orders are aggregated: , in, Indicates the lag order The corresponding learnable message transformation matrix; when When this term represents the immediate relationship between variables within the same prediction step; when In this case, the term represents the lag propagation relationship across time steps; Subsequently, the gating fusion coefficients are constructed: , in, This represents the adaptive fusion weights between the Koopman spectral prior and the graph message; Finally, the number was obtained Each node in the prediction step Fusion hidden state: , in, Representation layer normalization, Indicates a feedforward network. This represents the mapping matrix of future external driving variables.

[0023] Preferably, in step eight, after obtaining the future hidden state sequence, the Koopman hidden space state is mapped to the actual predicted value by the prediction decoder. For the The variable node in the future The output of each prediction step is represented as: , in, Indicates the predictive decoder, This indicates the prediction result.

[0024] As a preferred option, in step nine, an end-to-end optimization approach is adopted, which simultaneously constrains prediction accuracy, Koopman dynamic consistency, spectral stability, dynamic graph smoothness, time-delay propagation consistency, and latent space information preservation capability. First, construct the prediction error loss. Take the Huber loss as an example: , in, Indicates the number of samples in the batch. Huber's loss function is represented as follows: , Secondly, a Koopman consistency loss is constructed to ensure that the hidden state after the real future state is boosted by the observation function is consistent with the Koopman boosting result: , in, This indicates the prediction step size used for the Koopman consistency constraint; Next, construct the spectral stability loss: , The first term is used to limit the spectral radius, and the second term is used to constrain the numerical stability of the spectral basis matrix, avoiding... Pathological conditions lead to unstable Koopman propulsion; Combining the Laplace regularization term, graph smoothing term, and sparsity term of the dynamic time-delay graph obtained in step six, the overall training objective is: , in, These are non-negative weighting coefficients; The specific training process includes the following sub-steps: First, the temporal encoder, Koopman observation function, and decoder are pre-trained using reconstruction loss and prediction loss; then, Koopman consistency loss and spectral stability constraints are added to make the latent space satisfy the linear dynamics propagation characteristics; finally, dynamic time-delay graph Laplace constraints are added to make the delay propagation relationship between variables consistent with Koopman spectral dynamics.

[0025] Preferably, in step ten, after model training is completed, the latest historical window is input during the inference phase. and known external conditions in the future The process involves sequentially performing historical encoding, Koopman latent space lifting, spectral operator generation, dynamic time-delay graph construction, graph Laplace constraint fusion, and decoding output to obtain the future prediction sequence. , At the same time, the present invention can output interpretable information related to the prediction process; First, output the Koopman dominant spectral modes. Each spectral mode in the prediction step The intensity of contribution can be expressed as: , in, Indicates the first Spectral basis directions Indicates the first Each spectral mode is processed Evolutionary factors after the step; according to The magnitude of the value can be used to determine whether the current forecast is dominated by a trend mode, a cyclical mode, a decay mode, or a growth mode. Secondly, output the contribution of time delay propagation between variables; variables Lag The following pairs of variables In the prediction step The contribution can be expressed as: , according to The magnitude of the value can be used to obtain the dominant source variable, the dominant lag order, and the main propagation path.

[0026] This invention also provides a time series prediction system based on a learnable Koopman spectral operator and dynamic time delay graph Laplace constraint. The system includes: a data input and preprocessing module, a historical sequence encoding module, a Koopman latent space lifting module, a learnable Koopman spectral operator module, a dynamic time delay graph construction module, a dynamic time delay graph Laplace constraint module, a spectral fusion prediction module, a prediction decoding output module, a training optimization module, and an interpretable result output module. The data input and preprocessing module is used to acquire multi-source, multi-variable time series data and perform time alignment, missing data completion, anomaly handling, and normalization operations. The history sequence encoding module is used to map multivariate sequences within a history window to deep temporal hidden states; The Koopman latent space boosting module is used to learn observation functions. Map the nonlinear temporal state to a Koopman latent space suitable for linear dynamics propagation; The learnable Koopman spectral operator module is used to generate the spectral basis matrix, spectral mode frequencies, and growth and decay coefficients based on the current prediction context, and to construct the Koopman spectral operator. ; The dynamic time-delay graph construction module is used to generate dynamic adjacency matrices under different prediction steps and different candidate lag orders. This describes the asynchronous transmission relationship between variables; The dynamic time-delay graph Laplace constraint module is used to construct the extended graph Laplace matrix based on the dynamic time-delay adjacency matrix. and through Constrain the consistency and smoothness of delayed propagation; The spectral fusion prediction module is used to gating and fusing the Koopman spectral prior prediction state with the dynamic graph message propagation result to form the hidden state of each future prediction step. The prediction decoding output module is used to map future hidden states to actual prediction results and apply nonnegativity, capacity limits, power boundaries or other business constraints according to specific application scenarios. The training optimization module is used for end-to-end training based on prediction loss, reconstruction loss, Koopman consistency loss, spectral stability loss, dynamic time-delay graph Laplacian loss, graph smoothing loss, and sparsity loss. The interpretable results output module is used to output the Koopman dominant spectral modes, dynamic time delay plots, variable propagation paths, equivalent lag times, and predicted contributions. Each module can be implemented by software programs, hardware processing units, or a combination of software and hardware, and can be deployed on local servers, edge computing devices, cloud computing platforms, industrial control systems, or power dispatch and analysis platforms.

[0027] The present invention also provides an electronic device, comprising: a processor; and a memory for storing processor-executable instructions; The processor is configured to perform the steps of any of the methods described in the above invention.

[0028] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements any one of the methods described above.

[0029] After adopting the above technical solution, the present invention has the following beneficial effects: The present invention does not simply predict "what the future value will be", but further solves three problems: First, what dynamic mode the system will evolve in the future; second, which variables have influence relationships; and third, whether there is a time lag in this influence, and how many time steps the lag is.

[0030] Compared with existing technologies such as the better Transformer-type prediction models, graph neural network prediction models, and ordinary Koopman prediction models, the main advantages of this invention are as follows.

[0031] The prediction process is more dynamically interpretable.

[0032] Existing deep time series models typically map historical sequences directly to future sequences in a black-box manner. However, this invention uses a learnable Koopman spectral operator to characterize the long-term evolution modes, oscillation frequencies, and growth and decay trends of the system, enabling the prediction results to not only provide numerical outputs but also explain the underlying dynamic changes.

[0033] It is more suitable for complex, multi-variable coupled scenarios.

[0034] Existing graph neural network methods mostly use static graphs or instantaneous correlation graphs, while this invention constructs a dynamic time-delay graph, which can express "variables". Influencing variables at a certain lag time in the past The relationship between "current state" and "current state" makes it more suitable for complex systems with delayed propagation, such as energy, transportation, industry, and meteorology.

[0035] Long-term forecasts are more stable.

[0036] This invention controls the growth or decay of latent space states by using Koopman spectral radius constraints, which can reduce the risk of error accumulation, trend drift and abnormal divergence in long step size prediction.

[0037] Graph structures are more stable and reliable.

[0038] This invention does not simply learn an adjacency matrix, but uses dynamic time-delay graph Laplace constraints to ensure that strongly correlated variables remain consistent after being driven by Koopman dynamics, thereby improving the smoothness, stability and noise resistance of variable propagation relationships.

[0039] It can output richer interpretation results.

[0040] This invention not only outputs future predicted sequences, but also the dominant Koopman spectral mode, the time-delay propagation path between variables, the equivalent lag time, and the contribution of variables, which is beneficial for reviewing prediction results, risk analysis, and scheduling decisions.

[0041] In summary, this invention primarily addresses the challenges of complex multivariate time series forecasting, including significant nonlinear dynamics, difficulty in capturing long-term evolution patterns, time-varying correlations with lag propagation, inability of fixed graph structures to adapt to non-stationary coupling relationships, and severe accumulation of prediction errors over long time domains. It proposes a time series forecasting method that jointly models the spectral evolution of dynamics and the constraints of dynamic time-delay graph structures. This method and its corresponding systems, electronic devices, and storage media utilize learnable Koopman spectral operators to perform spectral dynamic modeling of nonlinear time series and adaptively characterize the time-varying lag coupling relationships between variables using dynamic time-delay graph Laplace constraints, thereby improving the accuracy, stability, long-term consistency, and interpretability of forecasting complex multivariate time series. Attached Figure Description

[0042] Figure 1 This is a network structure diagram of the time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph in this invention.

[0043] Figure 2 This is a flowchart of the time series prediction system based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph, which is the present invention. Detailed Implementation

[0044] The present invention will now be described in further detail with reference to the accompanying drawings.

[0045] Referring to the accompanying drawings, this invention provides a time series prediction method based on a learnable Koopman spectral operator and a dynamic time delay graph Laplace constraint, which includes the following steps: Step 1: Inputting Multi-Source, Multi-Variable Time Series Data and Constructing a Unified State Sequence This invention first acquires multi-source, multi-variable time series data of the object to be predicted and organizes it into a unified time-series state representation. The object to be predicted includes, but is not limited to, time series systems with multi-variable coupling relationships, such as wind power, photovoltaic power, net source-load, traffic flow, industrial process variables, meteorological environmental variables, equipment operating status quantities, and financial risk indicators.

[0046] Suppose the system contains There are 1 variable nodes, and each variable node is at time 1 / 2. have If we consider the observed features, then the original multivariate input can be represented as: , in, Indicates the first Each variable node at time... The observation features may include one or more of the following: the target variable's own historical values, external driving variables, environmental variables, time context variables, device state variables, and spatial location variables.

[0047] If there are known external driving variables in the future, such as weather forecasts, calendar information, planned control quantities, or operating condition instructions, then it is denoted as: , in, Indicates the prediction step size. Indicates the future number Known external conditions at each prediction time.

[0048] To address the inconsistencies in temporal resolution, missing values, outliers, and dimensional differences in the original multi-source data, this invention first performs time alignment, missing value completion, outlier removal, and smoothing normalization. For the first... The first variable node These features can be normalized using the following method: , in, and Let represent the mean and standard deviation of the corresponding feature in the training set, respectively. To prevent extremely small positive numbers with a denominator of zero.

[0049] In one implementation, a missing value mask can be introduced. The completed input is obtained through forward padding, interpolation, neighbor variable estimation, or neural network completion. , in, This indicates that the estimated value needs to be completed. This indicates element-wise multiplication.

[0050] Subsequently, with length as The historical window and its length are Constructing training samples using future prediction windows: , in, Represents the historical input sequence. This represents the true target sequence to be predicted. This represents the prediction target dimension for each node.

[0051] Through the above processing, this invention unifies time series data from different sources, scales, and physical meanings into a standardized multivariate state sequence, providing an input basis for subsequent Koopman latent space lifting, dynamic time delay graph construction, and graph Laplace constraint prediction.

[0052] Step 2: Historical sequence embedding and multi-scale time-delay candidate state construction After obtaining a unified state sequence, this invention embeds features into the historical sequence to obtain an implicit representation that combines node attributes, time position, and local dynamic changes.

[0053] First, an embedding transformation is performed on the input features of each variable node at each time step: , in, and For learnable linear transformation matrices, Indicates the first Node embedding of a variable node, This indicates either time location embedding or calendar context embedding. This represents the temporal characteristics of the embedded nodes.

[0054] Subsequently, a temporal encoder is used to extract long-short-term dependencies within the historical window. The temporal encoder can be a Transformer encoder, a temporal convolutional network, a gated recurrent network, a multi-scale convolutional network, or a combination thereof. A general encoder is represented as: , in, Represents an embedded sequence. The parameter is The timing encoder, It represents the historical implicit state sequence.

[0055] To explicitly describe the potential delayed responses between different variables, this invention further constructs a multi-scale time delay candidate set: , in, This represents the maximum candidate lag order. For any candidate lag order... Construct time-delay candidate states: , in, Indicates the first Each variable node is lagging. The historical implicit state at each time step.

[0056] Through the above processing, the present invention not only extracts the deep temporal features of historical time series, but also provides candidate lag states for the subsequent construction of dynamic time lag diagrams, enabling the model to adaptively determine the delay influence relationship between variables from multiple lag orders.

[0057] Step 3: Learn the Koopman observation function and latent space state lifting. After obtaining the historical latent state, this invention constructs a learnable Koopman observation function to map the original nonlinear time series state to a high-dimensional Koopman latent space, so that the evolution of the nonlinear system approximately satisfies the linear dynamic progression relationship in the latent space.

[0058] For the Each variable node at time... The historical encoded state is used to define a learnable Koopman observation function: , in, This indicates a learnable Koopman observation function. This indicates the short history length used to construct the local observation function. Indicates time context or operating condition context. Indicates the first The state of each variable node in the Koopman latent space.

[0059] The Koopman hidden state of all nodes can be represented as: , To ensure that the Koopman latent space retains the necessary information of the original temporal state, this invention can further configure the reconstruction decoder: , in, Let represent the latent space reconstruction function. The corresponding reconstruction loss can be expressed as: , In one implementation, the Koopman observation function It can be implemented using multilayer perceptrons, convolutional networks, Transformer encoders, graph neural networks, or hybrid neural networks. Its function is not to directly output predicted values, but rather to learn a latent space representation suitable for linear dynamics.

[0060] Through the above processing, this invention elevates the state of complex nonlinear, multivariate coupled time series to the Koopman latent space, providing a state foundation for subsequent modeling of learnable spectral operators.

[0061] Step 4: Learn how to construct Koopman spectral operators and advance stable dynamics. After obtaining the Koopman latent space state, this invention constructs a learnable Koopman spectral operator to characterize the global dynamic evolution mode, periodic frequency, growth and decay characteristics, and long-term propagation law of the system within the prediction window.

[0062] For the prediction starting point First, the context state is generated based on the historical hidden state and the known external conditions for the future: , in, This represents a pooling or context aggregation function. This indicates the overall state context of the current prediction window.

[0063] Subsequently, the Koopman spectral parameters are obtained through a spectral parameter generation network: , in, Denotes the Koopman spectral basis transformation matrix. Indicates the first The amplitude or growth / decrease coefficient of each spectral mode. Indicates the first The oscillation frequency of each spectral mode Indicates the number of spectral modes. This represents a learnable spectral parameter generation network.

[0064] To represent the oscillation modes corresponding to complex eigenvalues ​​in the real space, this invention uses a two-dimensional real block matrix to represent each spectral mode: , in, Indicates the time step. A block diagonal spectral matrix is ​​composed of multiple spectral blocks: , This leads to the learnable Koopman spectral operator within the prediction window: , in, This represents the local Koopman linear advance operator within the current prediction window.

[0065] To improve long-term prediction stability, this invention constrains the spectral radius. The following parameterization method can be used: , in, This represents the Sigmoid function. These are learnable parameters or parameters generated by the context network. This is the upper bound of the spectral radius. When it is necessary to allow for slightly growing modes, we can set... When strict stability prediction is required, it can be set as follows: .

[0066] Taking into account future external driving variables, the future... The Koopman prior hidden states for each prediction step can be represented as: , in, This represents the mapping of future external driving variables in the Koopman latent space. This is the external driver encoding function.

[0067] Through the above processing, the present invention can characterize the long-term evolution of nonlinear time series in the form of spectral operators in the latent space, enabling the model to not only learn the black box mapping from history to the future, but also to explicitly represent the oscillation frequency, growth decay and stable propagation modes of the system.

[0068] Step 5: Constructing a dynamic time delay graph for asynchronous effects between variables After obtaining the Koopman prior prediction state, this invention further constructs a dynamic time-lag plot to describe the time-dependent lag propagation relationships between different variables. This dynamic time-lag plot no longer only describes the instantaneous correlations within the same time slice, but rather describes the source variables... At any moment The state of the target variable At any moment The impact.

[0069] For the prediction step and candidate lag order Construct the target node query vector: , Construct the source node key vector: , in, Indicates the lag order Embedded representation, and is a learnable projection matrix. When hour, Retrieve historical Koopman hidden states; when hour, Take the hidden state obtained in the previous prediction step.

[0070] Subsequently, the calculation of the first Individual source variables in lag Next to the Impact rating of each target variable: , in, This represents the attention dimension of the graph. This indicates the strength of prior relationships, such as physical topological distance, spatial proximity, process connectivity, or empirical correlation; if no prior relationship exists, then we can let . This represents the learnable bias for different lag orders. and These are the weighting coefficients.

[0071] Normalize all source nodes and all candidate lag orders to obtain dynamic time-delay adjacency weights: , in, Indicating in the prediction step Below, variables With lag Influencing variables The intensity.

[0072] In one implementation, to enhance the sparsity and interpretability of the graph structure, [the following is done]: Execute Top- filter: , Then the retained edge weights are renormalized.

[0073] Furthermore, the equivalent lag time between pairs of variables can be calculated: , in, Representing variables For variables The average lag time at the current forecast step.

[0074] Through the above processing, the present invention can dynamically generate a time delay graph structure based on the current sample, the current prediction step, and the candidate lag order, avoiding the problem that existing static graphs or instantaneous correlation graphs are difficult to express asynchronous transmission between variables.

[0075] Step Six: Constructing Laplace Constraints for Dynamic Delay Graphs After obtaining the dynamic time-delay adjacency matrix, this invention constructs a dynamic time-delay graph Laplace constraint to constrain the smoothness, consistency and stability of the time-delay propagation process between variables in the latent space.

[0076] For the prediction step and lag order Define the dynamic time-delay adjacency matrix: , Since this graph represents the source node To the target node Based on the time-directed propagation relationship, this invention constructs an extended bipartite graph adjacency matrix: , The corresponding expansion matrix is: , The Laplace matrix of the dynamic time delay graph is: , To ensure that the lagging source node and the current target node remain aligned after Koopman dynamics propagation, this invention constructs an extended graph signal: , in, Indicates the prediction step The fusion of hidden states, This indicates that the source node's lagging state is advanced by the Koopman spectral operator. The state after a time step.

[0077] This yields the Laplace regularization term for the dynamic time-delay graph: , Its equivalent form can be expressed as: , The above formula indicates that if the dynamic time delay diagram considers variables... Lag The following pairs of variables If it has a strong influence, then the variable The current hidden state should be related to the variable The state of the hysteretic latent state remains consistent or approximately consistent after being advanced by Koopman dynamics.

[0078] Furthermore, to avoid drastic fluctuations in the dynamic graph structure due to local noise, this invention can introduce a graph structure time smoothing constraint: , Meanwhile, to avoid an overly dense graph structure, sparse constraints are introduced: , Through the above processing, this invention combines dynamic time delay diagrams with Koopman dynamic propagation processes, enabling the model to not only learn "which variables are related" but also "which variable affects another variable by how long a lag," and improves the stability and interpretability of the time delay propagation structure through Laplace constraints.

[0079] Step 7: Fusion prediction of Koopman spectral prior and dynamic time-delay graph messages After obtaining the Koopman prior hidden states and the dynamic time-delay graph structure, this invention proceeds according to the prediction step. The future hidden states are generated recursively. Each prediction step utilizes both the global dynamic prior provided by the Koopman spectral operator and the local intervariate propagation information provided by the dynamic time delay plot.

[0080] First, obtain the result according to step four. Koopman prior states for each prediction step Furthermore, based on the dynamic time-delay adjacency weights generated in step five, time-delay graph messages from different source variables and different lag orders are aggregated: , in, Indicates the lag order The corresponding learnable message transformation matrix. When When this term represents the immediate relationship between variables within the same prediction step; when In this context, the term indicates the lag propagation relationship across time steps.

[0081] Subsequently, the gating fusion coefficients are constructed: , in, This represents the adaptive fusion weights between the Koopman spectral prior and the graph message.

[0082] Finally, the number was obtained Each node in the prediction step Fusion hidden state: , in, Representation layer normalization, Indicates a feedforward network. This represents the mapping matrix of future external driving variables.

[0083] Through the above processing, this invention forms a fusion prediction mechanism of "Koopman global spectral evolution prior + dynamic time-delay graph local propagation correction". The Koopman spectral operator is used to ensure long-term dynamic consistency, the dynamic time-delay graph is used to characterize the asynchronous transmission relationship between variables, and the gating mechanism is used to adaptively adjust the contributions of both according to different prediction steps and different variables.

[0084] Step 8: Decoding Prediction Results and Mapping to Physical Constraints After obtaining the future hidden state sequence, this invention maps the Koopman hidden space state to the actual predicted value through a predictive decoder.

[0085] For the The variable node in the future The output of each prediction step is represented as: , in, Indicates the predictive decoder, This indicates the prediction result.

[0086] In prediction scenarios with physical upper and lower bounds, bounded mappings can be used to ensure that the prediction results meet business constraints. For example, for non-negative variables such as power, flow rate, and concentration that have upper bounds, the following approach can be used: , in, and They represent the first Each variable node predicts the physical lower and upper bounds of the target.

[0087] In one implementation, the present invention can also output probability prediction results: , in, This represents the predicted mean. This represents the prediction standard deviation. Training can then be performed using Gaussian negative log-likelihood, quantile loss, or interval prediction loss.

[0088] The outputs of all prediction steps constitute the final prediction sequence: , Through the above processing, the present invention transforms the dynamic prediction results in the latent space into actual time-series prediction results that meet the requirements of physical boundaries, business constraints, or probabilistic representations.

[0089] Step Nine: Construction of Joint Loss Function and End-to-End Training This invention employs an end-to-end optimization approach during the training phase, while simultaneously constraining prediction accuracy, Koopman dynamic consistency, spectral stability, dynamic graph smoothness, time-delay propagation consistency, and latent space information preservation capabilities.

[0090] First, construct the prediction error loss. Take the Huber loss as an example: , in, Indicates the number of samples in the batch. Huber's loss function is represented as follows: , Secondly, a Koopman consistency loss is constructed to ensure that the hidden state after the real future state is boosted by the observation function is consistent with the Koopman boosting result: , in, This represents the prediction step size used for the Koopman consistency constraint.

[0091] Next, construct the spectral stability loss: , The first term is used to limit the spectral radius, and the second term is used to constrain the numerical stability of the spectral basis matrix, avoiding... The pathological condition causes the Koopman propulsion to be unstable.

[0092] Combining the Laplace regularization term, graph smoothing term, and sparsity term of the dynamic time-delay graph obtained in step six, the overall training objective is: , in, These are non-negative weighting coefficients.

[0093] In the implementation, the training process includes the following sub-steps: First, the temporal encoder, Koopman observation function, and decoder are pre-trained using reconstruction loss and prediction loss; then, Koopman consistency loss and spectral stability constraints are added to make the latent space satisfy the linear dynamics propagation characteristics; finally, dynamic time-delay graph Laplace constraints are added to make the delay propagation relationship between variables consistent with Koopman spectral dynamics.

[0094] Through the above processing, the present invention can jointly learn temporal features, Koopman spectral operators, dynamic time delay graphs and predictive decoders in the same optimization framework, thereby improving the accuracy and stability of the model in complex nonlinear, multivariate coupled and long time domain prediction scenarios.

[0095] Step 10: Predictive Reasoning and Explainable Information Output After the model training is completed, the latest historical window is input during the inference phase. and known external conditions in the future The process involves sequentially performing historical encoding, Koopman latent space lifting, spectral operator generation, dynamic time-delay graph construction, graph Laplace constraint fusion, and decoding output to obtain the future prediction sequence. , At the same time, the present invention can output interpretable information related to the prediction process.

[0096] First, output the Koopman dominant spectral modes. Each spectral mode in the prediction step The intensity of contribution can be expressed as: , in, Indicates the first Spectral basis directions Indicates the first Each spectral mode is processed The evolutionary factor after the step. According to The magnitude of the value can be used to determine whether the current forecast is dominated by a trend mode, a cyclical mode, a decay mode, or a growth mode.

[0097] Secondly, output the contribution of time-delay propagation between variables. Variables Lag The following pairs of variables In the prediction step The contribution can be expressed as: , according to The magnitude of the value can be used to obtain the dominant source variable, the dominant lag order, and the main propagation path.

[0098] Furthermore, this invention can output the following interpretation results: Koopman eigenvalues, modal frequencies, modal decay coefficients, dynamic time-delay adjacency matrix, equivalent lag time, variable propagation paths, graph Laplace constraint strength, and prediction uncertainty for the current prediction window. These interpretation results can be used for complex system state analysis, risk localization, variable causal clue analysis, and model credibility assessment.

[0099] Through the above processing, the present invention can not only output predicted values, but also provide the dynamic modes and time-delay propagation structures involved in the prediction process, thereby improving the interpretability and auditability of the model in actual business scenarios.

[0100] This invention also provides a time series prediction system based on a learnable Koopman spectral operator and dynamic time delay graph Laplace constraints. The system includes: a data input and preprocessing module, a historical sequence encoding module, a Koopman latent space lifting module, a learnable Koopman spectral operator module, a dynamic time delay graph construction module, a dynamic time delay graph Laplace constraint module, a spectral fusion prediction module, a prediction decoding output module, a training optimization module, and an interpretable result output module.

[0101] The data input and preprocessing module is used to acquire multi-source, multi-variable time-series data and perform time alignment, missing data completion, anomaly handling, and normalization operations.

[0102] The history sequence encoding module is used to map multivariate sequences within a history window to deep temporal hidden states.

[0103] The Koopman latent space boosting module is used to learn observation functions. Map the nonlinear temporal state to a Koopman latent space suitable for linear dynamics propagation.

[0104] The learnable Koopman spectral operator module is used to generate the spectral basis matrix, spectral mode frequencies, and growth and decay coefficients based on the current prediction context, and to construct the Koopman spectral operator. .

[0105] The dynamic time-delay graph construction module is used to generate dynamic adjacency matrices under different prediction steps and different candidate lag orders. This describes the asynchronous transmission relationship between variables.

[0106] The dynamic time-delay graph Laplace constraint module is used to construct the extended graph Laplace matrix based on the dynamic time-delay adjacency matrix. and through Constrain the consistency and smoothness of delayed propagation.

[0107] The spectral fusion prediction module is used to gating and fusing the Koopman spectral prior prediction state with the dynamic graph message propagation result to form the hidden state for each future prediction step.

[0108] The prediction decoding output module is used to map future hidden states to actual prediction results and apply nonnegativity, capacity limits, power boundaries or other business constraints according to specific application scenarios.

[0109] The training optimization module is used for end-to-end training based on prediction loss, reconstruction loss, Koopman consistency loss, spectral stability loss, dynamic time-delay graph Laplacian loss, graph smoothing loss, and sparsity loss.

[0110] The interpretable results output module is used to output the Koopman dominant spectral modes, dynamic time delay plots, variable propagation paths, equivalent lag times, and predicted contributions.

[0111] Each module can be implemented by software programs, hardware processing units, or a combination of software and hardware, and can be deployed on local servers, edge computing devices, cloud computing platforms, industrial control systems, or power dispatch and analysis platforms.

[0112] Through the above system structure, the present invention can realize a complete closed-loop process from multi-source time-series data input, nonlinear dynamic spectrum modeling, dynamic time-delay graph relationship learning, graph Laplace structure constraint to final prediction output.

[0113] In summary, the technical solution proposed in this invention explicitly characterizes the long-term dynamic evolution of complex multivariate time series using a learnable Koopman spectral operator, adaptively identifies lag propagation relationships between variables through dynamic time-delay graphs, and enhances the consistency and stability of cross-variable and cross-time propagation through dynamic graph Laplace constraints. Compared with prediction methods that rely solely on black-box deep networks, static graph structures, or fixed lag orders, this invention can simultaneously improve prediction accuracy, long-term stability, structural robustness, and prediction interpretability in complex nonlinear time series prediction.

[0114] The present invention also provides an electronic device, comprising: a processor; and a memory for storing processor-executable instructions; The processor is configured to perform the steps of any of the methods described in the above invention.

[0115] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements any one of the methods described above.

[0116] The following are further examples of corresponding implementation schemes, in conjunction with the accompanying drawings and practical application scenarios: Example 1: A Multivariate Time Series Prediction Method Based on Learnable Koopman Spectral Operator and Dynamic Delay Graph Laplace Constraints This embodiment provides a time series forecasting method based on a learnable Koopman spectral operator and a dynamic time-delay graph Laplace constraint. This method is applicable to wind power forecasting, photovoltaic power forecasting, source-load net load forecasting, traffic flow forecasting, industrial process forecasting, robot state forecasting, equipment health status forecasting, and other multivariate time series forecasting tasks.

[0117] In this embodiment, the system to be predicted includes There are 1 variable nodes, and each variable node is at time 1 / 2. The observation feature dimension is Then the multivariate time-series input is represented as: , in, Indicates the first Each variable node at time... The observation characteristics. The historical observation window length is... The future predicted step size is The historical input sequence is then: , The target sequence to be predicted in the future is: , When there are known external driving variables in the future, such as weather forecasts, control commands, time calendar information, load planning information, or robot control inputs, they are represented as: , in, , .

[0118] The method in this embodiment includes the following steps.

[0119] Step S1: Input and preprocessing of multi-source, multi-variable time series data First, acquire multi-source, multi-variable time series data of the system to be predicted. The time series data includes one or more of the following: historical sequence of the target variable, external driving variables, environmental variables, control variables, time context variables, and system state variables.

[0120] In the scenario of new energy power prediction, the target variable may include wind power, photovoltaic power, or net source-load; the external driving variables may include wind speed, wind direction, temperature, humidity, air pressure, irradiance, cloud cover, numerical weather forecast data, unit operating status, and station capacity information. In the scenario of robot state prediction, the target variable may include joint angle, joint velocity, end effector pose, or chassis pose; the external driving variables may include control torque, speed command, contact status, or sensor observation data.

[0121] Since there may be inconsistencies in sampling frequency, timestamp offsets, missing values, and outliers between different data sources, this step first performs time alignment, resampling, missing value completion, outlier correction, and normalization on the original data.

[0122] For the The first variable node The features, after normalization, can be represented as: , in, and Let represent the mean and standard deviation of the corresponding variable feature in the training set, respectively. To prevent extremely small positive numbers with a denominator of zero.

[0123] When the original data contains missing values, a missing value mask is introduced: , in, The position with a value of 1 indicates that the feature is at time 1. If valid observations exist, a value of 0 indicates that the feature is missing. The completed input can be represented as: , in, This represents the estimate obtained by interpolation, neighbor variable estimation, historical mean, Kalman filtering, or neural network completion model. This indicates element-wise multiplication.

[0124] After the above processing, training samples are constructed: , Through this step, the original multi-source time series data are organized into a unified multivariate state sequence, providing an input basis for subsequent Koopman latent space lifting, dynamic time delay graph construction, and prediction output.

[0125] Step S2: Historical sequence embedding and temporal feature encoding After obtaining the standardized historical sequence, this step maps the historical observation features of each variable node to a unified hidden feature space.

[0126] For the Each variable node at time... Given the input features, construct the embedding vector: , in, For the observation feature mapping matrix, For external driving variable mapping matrix, For the first Learnable node embeddings of variable nodes Embed for time location or calendar context. This is the embedded feature vector.

[0127] Perform time-series encoding on the embedded sequences within all historical windows: , in, Indicates a timing encoder. This indicates its learnable parameters. It represents the historical implicit state sequence.

[0128] The temporal encoder can employ a Transformer encoder, a temporal convolutional network, a gated recurrent network, a long short-term memory network, a multi-scale convolutional network, an attention network, or a combination thereof. Taking the Transformer encoder as an example, for any time step and variable node, the query, key, and value vectors are first calculated: , Attention output is: , After multiple layers of attention, feedforward network, residual connection and normalization processing, the historical time-series hidden states are obtained. .

[0129] To facilitate the subsequent construction of lag relationships between variables, this embodiment further sets up a candidate lag order set: , in, This represents the maximum candidate lag order. For any candidate lag order... Construct time-delay candidate hidden states: , Through this step, the model not only obtains deep temporal features of historical sequences, but also establishes multiple candidate lag states, providing candidate inputs for subsequent dynamic time-lag graph learning.

[0130] Step S3: Learn the Koopman observation function and latent space state lifting. This step constructs a learnable Koopman observation function, mapping the historical time-series hidden states to the Koopman latent space. This latent space is used to approximate the evolution of the original nonlinear time-series system by satisfying a linear operator progression relation.

[0131] For the Each variable node at time... The historical hidden states are defined, and a learnable Koopman observation function is defined: , in, This indicates a learnable Koopman observation function. Indicates the length of the history used for local state promotion. Indicates the current time context or runtime context. Indicates the first The state of each variable node in the Koopman latent space.

[0132] The Koopman hidden state representation of all variable nodes is as follows: , In one alternative implementation, to ensure that the original state information is preserved in the Koopman latent space, this embodiment sets up a reconstruction decoder: , And set the reconstruction loss: , The Koopman observation function It can be implemented using multilayer perceptrons, convolutional networks, Transformer encoders, graph neural networks, or hybrid neural networks. Its function is not to directly output predicted values, but rather to learn a latent space state representation suitable for linear advancement by spectral operators.

[0133] Through this step, the original time-series state of complex nonlinearity and multivariate coupling is elevated to the Koopman latent space, providing a state foundation for the subsequent construction of learnable Koopman spectral operators.

[0134] Step S4: Learnable Koopman spectral operator generation After obtaining the Koopman hidden state, this step constructs a learnable Koopman spectral operator to characterize the long-term evolution mode, oscillation frequency, growth and decay characteristics, and stable propagation law of the system within the prediction window.

[0135] First, a prediction context is generated based on historical Koopman hidden states and known future external driving variables: , in, This represents a pooling or context aggregation function. This indicates the overall context state of the current prediction window.

[0136] Then, the Koopman spectral parameters are obtained through the spectral parameter generation network: , in, Indicates the spectral parameter generation network, Represents the spectral basis matrix. Indicates the number of spectral modes. and Used to generate the first The amplitude and frequency of each spectral mode.

[0137] To ensure long-term recursive stability, for the th The amplitude coefficients of each spectral mode are bounded: , in, The upper bound of the spectral radius, Let be the Sigmoid function. When strictly stable prediction is required, we can let be When a slight growth mode is allowed in the system, it can be set to... ,in It is a small positive number.

[0138] The frequency parameter can be expressed as: , in, This indicates the maximum permissible angular frequency.

[0139] For oscillation modes with complex eigenvalues, this embodiment uses a two-dimensional real matrix block representation: , in, Indicates the time step. A block diagonal spectral matrix is ​​composed of multiple spectral blocks: , like For odd number or latent space dimensions and If they are not completely consistent, a one-dimensional real eigenvalue block or a zero-filled block can be added to the block diagonal matrix.

[0140] To enhance numerical stability, the spectral basis matrix is ​​adjusted. Orthogonalization or near-orthogonalization is performed, for example, through QR decomposition, Cayley transform, or orthogonal regularization, to obtain the spectral basis matrix that satisfies the approximate conditions: , The final construction of the learnable Koopman spectral operator: , In another implementation, a general invertible spectral basis matrix can also be used. And construct: , Through the above method, the Koopman spectral operator Instead of a fixed matrix, it is a learnable spectral operator that is adaptively generated based on the current historical state and future external conditions.

[0141] For the future For each prediction step, the Koopman spectrum prior hidden state can be represented as: , in, This represents the mapping function from external driving variables to the Koopman latent space.

[0142] Through this step, the model can explicitly express the trend mode, periodic mode, decay mode, and growth mode of the system in the latent space, avoiding complete reliance on a black-box network to directly fit future sequences.

[0143] Step S5: Construction of Dynamic Delay Graph This step constructs a dynamic time-delay graph to describe the time-varying coupling relationships between variables with delayed responses. Unlike a static adjacency matrix, the graph structure in this embodiment also depends on the prediction step. Source variables Target variable and lag order .

[0144] For the prediction step and candidate lag order Define the source node state: , Among them, when When a state falls within the prediction interval, the hidden state obtained from the previous prediction is used; when a state falls within the historical interval, the historical Koopman hidden state is used.

[0145] Construct the target node query vector: , Construct the source node key vector: , in, Indicates the lag order Learnable embeddings and It is a learnable projection matrix.

[0146] Calculate the first Individual source variables in lag Next to the Impact rating of each target variable: , in, This represents the attention dimension of the graph. Representing variables With variables The strength of the prior association between them Indicates the lag order bias. and is the weighting parameter. The prior association strength can come from physical topology, spatial distance, process connection relationships, robot link topology, historical correlation coefficients, or manual configuration; when no prior relationship exists, it can be set as follows: .

[0147] Normalize all source variables and candidate lag orders to obtain dynamic time-delay adjacency weights: , in, Indicating in the prediction step Below, source variables With lag For target variable The intensity of the impact.

[0148] To improve the sparsity and interpretability of the graph structure, each target variable can be... Top- Execution of Incoming Edges filter: , Then the retained edge weights are renormalized.

[0149] Furthermore, variables For variables The equivalent lag time can be calculated as follows: , Through this step, this embodiment can dynamically identify the time-delay coupling relationship between variables, avoiding insufficient delay propagation modeling caused by using only static graphs or instantaneous correlation graphs.

[0150] Step S6: Constructing Laplace Constraints for Dynamic Delay Graphs After obtaining the dynamic time-delay adjacency weights, this step constructs a dynamic time-delay graph Laplace constraint to enhance the smoothness, consistency, and stability of the lag propagation process between variables.

[0151] For the prediction step and lag order Define the dynamic time-delay adjacency matrix: , Since this graph describes the source node To the target node To understand the cross-time propagation relationships, this embodiment constructs an extended bipartite graph adjacency matrix: , The corresponding degree matrix is: , The Laplace matrix of the dynamic time delay plot is: , In an alternative implementation, a normalized graphical Laplace matrix can also be used: , To ensure that the source node's lagging state remains consistent with the target node's current state after being advanced by the Koopman spectral operator, an extended graph signal is constructed: , in, Indicates the prediction step The fusion of hidden states, This indicates that the source variable's lagged hidden state is advanced by the Koopman spectral operator. The state after taking a step.

[0152] The Laplace regularization term for a dynamic time-delay graph is expressed as: , Its equivalent form is: , This constraint states that when the dynamic time-delay diagram considers variables... Lag The following pairs of variables When the variable has a strong influence In the current prediction step, the Koopman hidden state should be related to the variable. The state of the hysteretic hidden state remains close after being advanced by the Koopman spectral operator.

[0153] To avoid drastic fluctuations in the dynamic graph structure due to noise or outliers, this embodiment also sets a dynamic graph time smoothing constraint: , To avoid overly dense time-delay plots, sparse constraints are also set: , Through this step, this embodiment not only learns the correlation between variables, but also further constrains the stability and interpretability of this correlation during the propagation process "across variables, across time, and across lag orders".

[0154] Step S7: Fusion of Koopman Spectral Priors and Dynamic Delay Graph Messages This step follows the prediction step. The future hidden states are generated recursively. Each prediction step utilizes both the global dynamic prior given by the Koopman spectral operator and the local variable propagation information given by the dynamic time delay diagram.

[0155] First, according to step S4, the first... Koopman spectral prior states for each prediction step: , Then, graph messages from different source variables and different lag orders are aggregated based on dynamic time-delay adjacency weights: , in, Indicates the lag order The corresponding message transformation matrix.

[0156] Subsequently, the gating fusion coefficients are constructed: , in, This represents the fusion weight between the Koopman spectral prior and the dynamic time-delay graph message.

[0157] Initial fusion hidden state obtained: , in, Representation layer normalization, Indicates a feedforward network. This represents the mapping matrix of external driving variables.

[0158] In a further implementation, this embodiment may also perform a closed-loop Laplace correction on the initially fused hidden state. Specifically, let: , Solve the following optimization problem: , Its closed-form approximate solution can be expressed as: , in, This indicates the strength of the Laplace correction. This correction causes the current node's hidden state to move closer to the Koopman advance state of the high-weight source node in the dynamic time-delay graph, thereby enhancing the graph structure consistency of the predicted hidden state.

[0159] Through this step, this embodiment forms a fusion prediction mechanism of "Koopman spectrum prior propagation - dynamic time-delay graph message propagation - graph Laplace correction", which enables the model to have both long-term dynamic stability and local variable coupling adaptability.

[0160] Step S8: Decoding the prediction results and mapping them to physical constraints In obtaining the future hidden state Then, this step outputs the actual prediction result through the prediction decoder.

[0161] For the The variable node in the future Output of each prediction step: , in, Indicates the predictive decoder, Indicates the target of the prediction.

[0162] For prediction objects with physical upper and lower bounds, bounded mappings can be used: , in, and They represent the first Each variable node predicts the physical lower and upper bounds of the target.

[0163] In wind power forecasting, we can assume: , in, This represents the rated capacity of the wind farm or wind turbine. In photovoltaic power forecasting, we can set: , It can also be combined with sunrise / sunset conditions or irradiance thresholds to constrain the predicted power to be close to zero in nighttime scenarios.

[0164] The outputs of all prediction steps constitute the final prediction sequence: , In one alternative implementation, this embodiment may also output probability prediction results: , in, This represents the predicted mean. This represents the standard deviation of the forecast. This method can be used to generate forecast intervals, confidence intervals, or risk assessment results.

[0165] Step S9: Joint Loss Function Construction and Model Training This step involves end-to-end training of the model. The training objectives simultaneously constrain prediction accuracy, latent space reconstruction capability, Koopman dynamic consistency, spectral stability, dynamic time-delay graph smoothness, and graph Laplacian propagation consistency.

[0166] Loss prediction can be achieved using mean squared error, mean absolute error, or Huber loss. Taking Huber loss as an example: , in, Indicates the number of samples in the batch. Huber's loss function is represented as follows: , The Koopman consistency loss is used to ensure that the hidden state after the observation function boosts the true future state is consistent with the Koopman boosted state: , in, This represents the prediction step size used for the Koopman consistency constraint.

[0167] The spectral stability loss can be expressed as: , The first term is used to limit the spectral radius, and the second term is used to constrain the near orthogonality of the spectral basis matrix, preventing numerical instability in the Koopman spectral operator during multi-step recursion.

[0168] Combining the aforementioned reconstruction loss, dynamic time-delay graph Laplace constraint, graph smoothing constraint, and sparsity constraint, the total loss function is expressed as: , in, These are non-negative weighting coefficients. This represents the loss due to physical constraints.

[0169] In the scenario of new energy power prediction, physical constraint losses can include capacity overrun penalties and ramping constraint penalties: , in, and These represent the lower and upper limits of power, respectively. Indicates the maximum permissible gradient. This represents the penalty coefficient.

[0170] Training can employ Adam, AdamW, or stochastic gradient descent optimizers. The training process may include a pre-training phase and a joint training phase. The pre-training phase primarily optimizes... , and This enables the model to possess basic predictive capabilities and Koopman latent space representation capabilities; during the joint training phase, [the following is added]... , , and This allows dynamic graph structure, time-delay propagation relationship, and spectral stability to participate in the optimization.

[0171] Through this step, the model can jointly learn temporal features, the Koopman spectral operator, dynamic time delay graphs, graph Laplace constraints, and predictive decoders within the same training framework.

[0172] Step S10: Predictive Inference and Explainable Result Output After the model training is complete, enter the latest history window. and known external conditions in the future The process involves sequentially performing historical encoding, Koopman latent space lifting, spectral operator generation, dynamic time-delay graph construction, graph Laplacian fusion correction, and prediction decoding to obtain the future prediction results. , Meanwhile, this embodiment outputs interpretable information during the prediction process.

[0173] First, output the Koopman dominant spectral modes. Each spectral mode in the prediction step The intensity of contribution can be expressed as: , in, Indicates the first Spectral basis directions Indicates the first Each spectral mode is processed The evolutionary factor after the step. According to The magnitude of the value can be used to determine whether the current forecast is dominated by a trend mode, a cyclical mode, a decay mode, or a growth mode.

[0174] Secondly, output the contribution of time-delay propagation between variables. Variables Lag The following pairs of variables In the prediction step The contribution can be expressed as: , according to The size of the variable can be used to output the dominant source variable, the dominant lag order, and the main propagation path.

[0175] Furthermore, this embodiment can also output the dynamic time-delay adjacency matrix, equivalent lag time, Koopman eigenvalues, modal frequencies, modal decay coefficients, graph Laplace constraint strength, and prediction uncertainty. The above interpretation results can be used for power dispatch risk analysis, industrial process state diagnosis, robot motion prediction reliability assessment, or complex system variable correlation analysis.

[0176] Example 2: Specific Implementation Methods for Wind Power Forecasting This embodiment uses short-term power prediction of wind farms as an example to explain the above method in detail.

[0177] Assuming the wind farm sampling interval is 15 minutes, and the historical window length is taken as... This indicates that data from the past 24 hours is used; the prediction step size is... This represents a forecast of wind power output for the next 4 hours. Variable nodes include at least a subset of the following: wind power output, wind speed, wind direction, temperature, humidity, air pressure, available turbine capacity, numerical weather forecast wind speed, numerical weather forecast wind direction, and time context variables. If the wind turbine is taken as the node, then the... The node corresponds to the first Typhoon machine or the first Each zone has node characteristics including the historical power, wind speed, wind direction, and operating status of the wind turbine or zone.

[0178] In this embodiment, the input sequence is represented as: , The future weather forecast sequence is represented as follows: , The output is a future wind power forecast sequence: , In its implementation, the temporal encoder employs a two-layer Transformer encoder to hide dimensions. Koopman latent space dimension ; Number of spectral modes Maximum candidate lag order This indicates that a maximum lag of 3 hours is considered; Top-Dynamic Lag Plot Take 5; Upper bound of spectral radius .

[0179] During the forecasting process, the model first generates Koopman hidden states based on the power and meteorological sequences of the past 24 hours; then, it generates Koopman spectral operators based on current wind conditions, future NWP weather forecasts, and historical power inertia; and finally, it generates dynamic time-lag plots for each 15-minute forecast step. For example, when the model forecasts the 8th time step in the future, the dynamic time-lag plot can identify that "upstream wind speed variables have a strong impact on wind power variables after a lag of 4 time steps," and it can also identify that "wind direction changes alter the power output trend after a lag of several time steps."

[0180] Wind power output uses capacity constraint mapping: , in, Indicates the rated capacity of the wind farm. Indicates the power node in the prediction step The Koopman fusion hidden state.

[0181] After training, the model outputs wind power predictions for the next 4 hours, along with the following explanatory information: the dominant Koopman mode in the current prediction window, the equivalent lag time from wind speed to power, the dynamic influence strength from wind direction to power, the propagation paths between meteorological variables, and the strength of the graphical Laplace constraint. For uphill or sudden drop prediction scenarios, the above explanatory information can help determine whether the predicted changes are mainly caused by sudden changes in wind speed, wind direction changes, changes in generator status, or historical power inertia.

[0182] Example 3: Specific Implementation Methods for Robot State Prediction This embodiment can also be applied to robot state prediction. Assume the robot includes... Each node is a joint or sensor node, and the state characteristics of each node include one or more of the following: joint angle, joint velocity, joint torque, IMU observation, end effector state, and control input.

[0183] The robot's multivariable state input is represented as follows: , Among them, the The node can correspond to the first node. The first joint, the first The first link or the first One sensor. Future control commands are represented as: , In robotic systems, mechanical transmission coupling, control delays, and contact response lags typically exist between different joints. The dynamic time-delay graph of this invention can be used to learn asynchronous coupling relationships such as "hip joint control changes affecting knee joint state after several time steps" and "chassis acceleration changes affecting robotic arm end-effector pose after a certain delay." The Koopman spectral operator can be used to describe the oscillatory modes, damped modes, and stable motion modes of the robot's overall motion state.

[0184] The predictive decoder outputs the future joint states: , It can also add constraints on joint angle range, velocity range, and torque range: , Through the above methods, the present invention can be used for robot motion prediction, trajectory planning assistance, state estimation, abnormal motion early warning, and control system safety assessment.

[0185] Example 4: Time Series Prediction System This embodiment also provides a time series prediction system based on a learnable Koopman spectral operator and dynamic time delay graph Laplace constraints. The system includes a data input and preprocessing module, a historical sequence encoding module, a Koopman latent space lifting module, a learnable Koopman spectral operator module, a dynamic time delay graph construction module, a dynamic time delay graph Laplace constraint module, a spectral fusion prediction module, a prediction decoding output module, a training optimization module, and an interpretable result output module.

[0186] The data input and preprocessing module is used to acquire multi-source, multi-variable time series data and perform time alignment, missing value completion, outlier correction, normalization, and training sample construction.

[0187] The history sequence encoding module is used to map multivariate sequences within a history window to deep temporal hidden states. This module can employ Transformer, temporal convolutional networks, recurrent neural networks, or hybrid encoding networks.

[0188] The Koopman latent space boosting module is used to learn observation functions. Mapping the temporal hidden state to the Koopman hidden space state .

[0189] The learnable Koopman spectral operator module is used to generate spectral basis matrices, spectral mode magnitudes, spectral mode frequencies, and Koopman spectral operators based on the current prediction context. .

[0190] The dynamic time-delay graph construction module is used to generate a dynamic time-delay adjacency matrix based on the target node state, the source node's lag state, node embedding, lag order embedding, and prior relation strength. .

[0191] The dynamic time-delay graph Laplace constraint module is used to construct the extended graph Laplace matrix based on the dynamic time-delay adjacency matrix. Constrained hysteresis propagation consistency is corrected by graph Laplace regularization or closed Laplace correction.

[0192] The spectral fusion prediction module is used to fuse the Koopman spectral prior state and dynamic time-delay graph messages to generate the fused hidden state for each future prediction step.

[0193] The prediction decoding output module is used to map future hidden states to actual prediction results and apply power capacity, joint range, non-negativity, ramp rate or other business constraints according to specific application scenarios.

[0194] The training optimization module is used to perform end-to-end optimization of system parameters based on prediction loss, reconstruction loss, Koopman consistency loss, spectral stability loss, dynamic time-delay graph Laplacian loss, graph smoothing loss, and sparsity loss.

[0195] The interpretable results output module is used to output the Koopman dominant spectral modes, dynamic time delay plots, variable propagation paths, equivalent lag times, graphical Laplace constraint strength, and prediction uncertainty.

[0196] The above modules can be implemented by software programs, hardware processing units, or a combination of software and hardware, and can be deployed in power dispatch servers, industrial edge computing devices, robot control computing platforms, cloud prediction platforms, or embedded intelligent terminals.

[0197] Example 5: Electronic Devices and Storage Media This embodiment also provides an electronic device, including a processor, a memory, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in any of the above embodiments.

[0198] This embodiment also provides a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, it implements the steps of the method described in any of the above embodiments.

[0199] Through the specific embodiments described above, this invention can simultaneously achieve nonlinear dynamic spectrum modeling, dynamic time-delay coupling relationship learning, graph Laplace propagation consistency constraints, and physical constraint prediction output in multivariate time series forecasting. Compared with prediction methods that rely solely on black-box deep networks, static adjacency matrices, or fixed lag orders, this invention can improve the accuracy, stability, robustness, and interpretability of medium- and long-term forecasts for complex non-stationary systems.

[0200] It is important to emphasize that this patent application, compared with existing technologies, achieves key innovations in the following aspects, which are also the key areas this patent application aims to protect: First, it protects a holistic technical process for reconstructing multivariate time series prediction into a fusion of "Koopman spectral dynamics advancement and dynamic time delay graph constraints"; second, it protects a mechanism for elevating nonlinear time series states to the Koopman latent space through learnable observation functions; third, it protects a mechanism for adaptively generating Koopman spectral operators based on input context and characterizing long-term dynamic laws through spectral mode frequencies and growth / decay coefficients; fourth, it protects a dynamic time delay graph construction mechanism oriented towards asynchronous influences between variables, used to represent the propagation relationship of different variables at different lag orders; fifth, it protects a method for advancing the state of lagging source nodes through Koopman operators and then... The following are the key points: 1) A mechanism for constraining the current state of the target node using graph Laplace consistency; 2) A gated fusion prediction mechanism for Koopman spectral priors and dynamic time-delay graph messages; 3) An end-to-end training mechanism for joint prediction error, Koopman consistency, spectral stability, dynamic graph smoothness, and graph sparsity; 4) An interpretable prediction mechanism for outputting the dominant Koopman mode, variable propagation path, equivalent lag time, and dynamic graph contribution; 5) A time series prediction system composed of the above modules, and its application in new energy power prediction, traffic prediction, industrial process prediction, equipment status prediction, and other complex multivariate time series prediction tasks.

[0201] The present invention and its embodiments have been described above. This description is not restrictive, and the accompanying drawings are only one embodiment of the present invention. The actual solution is not limited to this. In conclusion, if those skilled in the art are inspired by this description and design similar solutions and embodiments without departing from the spirit of the present invention, they should all fall within the protection scope of the present invention.

Claims

1. A time series prediction method based on learnable Koopman spectral operators and dynamic time delay graph Laplace constraints, characterized in that: Includes the following steps: Step 1: Inputting multi-source, multi-variable time-series data and constructing a unified state sequence; Step 2: Historical sequence embedding and multi-scale time-delay candidate state construction; Step 3 involves learning the Koopman observation function and latent space state lifting; Step four allows you to learn about the construction of Koopman spectral operators and the advancement of stable dynamics; Step 5: Constructing a dynamic time delay graph for asynchronous effects between variables; Step 6: Constructing Laplace constraints for dynamic time delay graphs; Step 7: Fusion prediction of Koopman spectral prior and dynamic time-delay graph messages; Step 8: Decoding the prediction results and mapping them to physical constraints; Step 9: Construction of joint loss function and end-to-end training; Step 10: Predictive reasoning and output of interpretable information.

2. The time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in claim 1, characterized in that: Step one specifically refers to acquiring multi-source, multi-variable time series data of the object to be predicted and organizing it into a unified time series state representation; the object to be predicted is a time series system with multi-variable coupling relationships; the specific operation method is as follows: assuming the system contains... There are 1 variable nodes, and each variable node is at time 1 / 2. have If we consider the observed features, then the original multivariate input is represented as: ； in, Indicates the first Each variable node at time... Observational characteristics; The observed features include one or more of the following: the target variable's own historical values, external driving variables, environmental variables, time context variables, equipment status variables, and spatial location variables; if there are known external driving variables in the future, they are denoted as: ； in, Indicates the prediction step size. Indicates the future number Known external conditions at each predicted time point; To address the inconsistencies in temporal resolution, missing values, outliers, and dimensional differences in the original multi-source data, the first step is to perform time alignment, missing value completion, outlier removal, and smoothing normalization. For the second step... The first variable node The following normalization method is used for each feature: ； in, and Let represent the mean and standard deviation of the corresponding feature in the training set, respectively. To prevent extremely small positive numbers with a denominator of zero.

3. The time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in claim 1, characterized in that: The specific execution of step two is as follows: First, an embedding transformation is performed on the input features of each variable node at each time step: ； in, and For learnable linear transformation matrices, Indicates the first Node embedding of a variable node, This indicates either time location embedding or calendar context embedding. This represents the temporal characteristics of the embedded nodes; Subsequently, a temporal encoder is used to extract long-short-term dependencies within the historical window; the temporal encoder employs a Transformer encoder, a temporal convolutional network, a gated recurrent network, a multi-scale convolutional network, or a combination thereof; represented by a general encoder as follows: ； in, Represents an embedded sequence. The parameter is The timing encoder, Represents a sequence of historical implicit states; Constructing a multi-scale time delay candidate set : ； in, Represents the maximum candidate lag order; for any candidate lag order... Construct time-delay candidate states : ； in, Indicates the first Each variable node is lagging. The historical implicit state at each time step.

4. The time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in claim 1, characterized in that: The specific execution of step three is as follows: For the first Each variable node at time... The historical encoded state is used to define a learnable Koopman observation function: ； in, This indicates a learnable Koopman observation function. This indicates the short history length used to construct the local observation function. Indicates time context or operating condition context. Indicates the first The state of each variable node in the Koopman latent space; The Koopman hidden state of all nodes is represented as follows: ； Configure the reconstruction decoder: ； in, Represents the latent space reconstruction function; The corresponding reconstruction loss is expressed as: ； Through the above processing, the state of complex nonlinear and multivariate coupled time series is elevated to the Koopman latent space, providing a state foundation for subsequent modeling of learnable spectral operators.

5. The time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph as described in claim 1, characterized in that: Step four is executed as follows: After obtaining the Koopman latent space state, a learnable Koopman spectral operator is constructed to characterize the global dynamic evolution modes, periodic frequencies, growth and decay characteristics, and long-term propagation laws of the system within the prediction window; the construction method is as follows: For the prediction starting point First, the context state is generated based on the historical hidden state and the known external conditions for the future: ； in, This represents a pooling or context aggregation function. This indicates the overall state context of the current prediction window; Subsequently, the Koopman spectral parameters are obtained through a spectral parameter generation network: ； in, Denotes the Koopman spectral basis transformation matrix. Indicates the first The amplitude or growth / decrease coefficient of each spectral mode. Indicates the first The oscillation frequency of each spectral mode Indicates the number of spectral modes. This represents a learnable spectral parameter generation network; Each spectral mode is represented by a two-dimensional real block matrix: ； in, Indicates the time step; A block diagonal spectral matrix is ​​composed of multiple spectral blocks: ； This leads to the learnable Koopman spectral operator within the prediction window: ； in, This represents the local Koopman linear advance operator within the current prediction window.

6. The time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph as described in claim 1, characterized in that: Step five involves constructing a dynamic time-delay plot to describe the lag propagation relationships between different variables over time. This dynamic time-delay plot no longer only describes the instantaneous correlations within the same time slice, but rather describes the source variables... At any moment The state of the target variable At any moment The impact of this is specifically addressed in the following construction method: For the prediction step and candidate lag order Construct the target node query vector: ； Construct the source node key vector: ； in, Indicates the lag order Embedded representation, and For a learnable projection matrix; when hour, Retrieve historical Koopman hidden states; when hour, Take the hidden states that have been obtained in the previous prediction step; Subsequently, the calculation of the first Individual source variables in lag Next to the Impact rating of each target variable: ； in, This represents the attention dimension of the graph. Let represent the strength of the prior relation; if no prior relation exists, then let . ; This represents the learnable bias for different lag orders. and These are the weighting coefficients; Normalize all source nodes and all candidate lag orders to obtain dynamic time-delay adjacency weights: ； in, Indicating in the prediction step variable With lag Influencing variables The strength; right Execute Top- filter: ； Then, the retained edge weights are renormalized, and the equivalent lag time between variable pairs is calculated: ； in, Representing variables For variables The average lag time at the current forecast step.

7. The time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in claim 1, characterized in that: The specific execution of step six is ​​as follows: After obtaining the dynamic time-delay adjacency matrix, construct the dynamic time-delay graph Laplace constraint to constrain the smoothness, consistency and stability of the time-delay propagation process between variables in the latent space; For the prediction step and lag order Define the dynamic time-delay adjacency matrix: ； Because the dynamic time delay graph represents the source node To the target node Based on the time-directed propagation relationship, an extended bipartite graph adjacency matrix is ​​further constructed: ； The corresponding expansion matrix is: ； The Laplace matrix of the dynamic time delay graph is: ； Constructing extended graph signals: ； in, Indicates the prediction step The fusion of hidden states, This indicates that the source node's lagging state is advanced by the Koopman spectral operator. The state after one time step; This yields the Laplace regularization term for the dynamic time-delay graph: ； Its equivalent form is expressed as: 。 8. The time series prediction method based on the learnable Koopman spectral operator and dynamic time delay graph Laplace constraint as described in claim 1, characterized in that: In step seven, after obtaining the Koopman prior hidden states and the dynamic time delay graph structure, the prediction step is performed. The future hidden states are generated recursively; each prediction step utilizes both the global dynamic priors provided by the Koopman spectral operator and the local intervariate propagation information provided by the dynamic time delay graph. First, obtain the result according to step four. Koopman prior states for each prediction step Furthermore, based on the dynamic time-delay adjacency weights generated in step five, time-delay graph messages from different source variables and different lag orders are aggregated: ； in, Indicates the lag order The corresponding learnable message transformation matrix; when When this term represents the immediate relationship between variables within the same prediction step; when In this case, the term represents the lag propagation relationship across time steps; Subsequently, the gating fusion coefficients are constructed: ； in, This represents the adaptive fusion weights between the Koopman spectral prior and the graph message; Finally, the number was obtained Each node in the prediction step Fusion hidden state: ； in, Representation layer normalization, Indicates a feedforward network. This represents the mapping matrix of future external driving variables.

9. The time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph as described in claim 1, characterized in that: In step eight, after obtaining the future hidden state sequence, the Koopman hidden space state is mapped to the actual predicted value through the prediction decoder. For the The variable node in the future The output of each prediction step is represented as: ； in, Indicates the predictive decoder, This indicates the prediction result.

10. The time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph as described in claim 1, characterized in that: In step nine, an end-to-end optimization approach is adopted, simultaneously constraining prediction accuracy, Koopman dynamic consistency, spectral stability, dynamic graph smoothness, time-delay propagation consistency, and latent space information preservation capability. First, construct the prediction error loss, specifically the Huber loss: ； in, Indicates the number of samples in the batch. Huber's loss function is represented as follows: ； Secondly, a Koopman consistency loss is constructed to ensure that the hidden state after the real future state is boosted by the observation function is consistent with the Koopman boosting result: ； in, This indicates the prediction step size used for the Koopman consistency constraint; Next, construct the spectral stability loss: ； The first term is used to limit the spectral radius, and the second term is used to constrain the numerical stability of the spectral basis matrix, avoiding... Pathological conditions lead to unstable Koopman propulsion; Combining the Laplace regularization term, graph smoothing term, and sparsity term of the dynamic time-delay graph obtained in step six, the overall training objective is: ； in, These are non-negative weighting coefficients; The specific training process includes the following sub-steps: First, the temporal encoder, Koopman observation function, and decoder are pre-trained using reconstruction loss and prediction loss; then, Koopman consistency loss and spectral stability constraints are added to make the latent space satisfy the linear dynamics propagation characteristics; finally, dynamic time-delay graph Laplace constraints are added to make the delay propagation relationship between variables consistent with Koopman spectral dynamics.

11. The time series prediction method based on the learnable Koopman spectral operator and the Laplace constraint of the dynamic time delay graph as described in claim 1, characterized in that: In step ten, after the model training is complete, the latest history window is input during the inference phase. and known external conditions in the future The process involves sequentially performing historical encoding, Koopman latent space lifting, spectral operator generation, dynamic time-delay graph construction, graph Laplace constraint fusion, and decoding output to obtain the future prediction sequence. ； First, output the Koopman dominant spectral modes; the second... Each spectral mode in the prediction step The contribution intensity is expressed as: ； in, Indicates the first Spectral basis directions Indicates the first Each spectral mode is processed Evolutionary factors after the step; according to The magnitude of the value can determine whether the current forecast is dominated by a trend mode, a cyclical mode, a decay mode, or a growth mode. Secondly, output the contribution of time delay propagation between variables; variables Lag The following pairs of variables In the prediction step The contribution is expressed as: ； according to The magnitude of the value can be used to obtain the dominant source variable, the dominant lag order, and the main propagation path.

12. A time series prediction system based on a learnable Koopman spectral operator and a dynamic time delay graph Laplace constraint, characterized in that: The system includes: a data input and preprocessing module, a historical sequence encoding module, a Koopman latent space boosting module, a learnable Koopman spectral operator module, a dynamic time delay graph construction module, a dynamic time delay graph Laplace constraint module, a spectral fusion prediction module, a prediction decoding output module, a training optimization module, and an interpretable result output module. The data input and preprocessing module is used to acquire multi-source, multi-variable time series data and perform time alignment, missing data completion, anomaly handling, and normalization operations. The history sequence encoding module is used to map multivariate sequences within a history window to deep temporal hidden states; The Koopman latent space boosting module is used to learn observation functions. Map the nonlinear temporal state to a Koopman latent space suitable for linear dynamics propagation; The learnable Koopman spectral operator module is used to generate the spectral basis matrix, spectral mode frequencies, and growth and decay coefficients based on the current prediction context, and to construct the Koopman spectral operator. ; The dynamic time-delay graph construction module is used to generate dynamic adjacency matrices under different prediction steps and different candidate lag orders. This describes the asynchronous transmission relationship between variables; The dynamic time-delay graph Laplace constraint module is used to construct the extended graph Laplace matrix based on the dynamic time-delay adjacency matrix. and through Constrain the consistency and smoothness of delayed propagation; The spectral fusion prediction module is used to gating and fusing the Koopman spectral prior prediction state with the dynamic graph message propagation result to form the hidden state of each future prediction step. The prediction decoding output module is used to map future hidden states to actual prediction results and apply nonnegativity, capacity limits, power boundaries or other business constraints according to specific application scenarios. The training optimization module is used for end-to-end training based on prediction loss, reconstruction loss, Koopman consistency loss, spectral stability loss, dynamic time-delay graph Laplacian loss, graph smoothing loss, and sparsity loss. The interpretable results output module is used to output the Koopman dominant spectral modes, dynamic time delay plots, variable propagation paths, equivalent lag times, and predicted contributions. Each module can be implemented by software programs, hardware processing units, or a combination of software and hardware, and can be deployed on local servers, edge computing devices, cloud computing platforms, industrial control systems, or power dispatch and analysis platforms.

13. An electronic device, characterized in that, include: processor; Memory used to store processor-executable instructions; The processor is configured to perform the steps of the method according to any one of claims 1 to 11.

14. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method described in any one of claims 1 to 11.

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