A self-evolution federated meta-learning method for cross-domain heterogeneous spatio-temporal intelligence

Abstract

The present application relates to the technical field of space-time intelligence and federated learning, in particular to a self-evolution federated meta-learning method for cross-domain heterogeneous space-time intelligence, comprising the following steps: problem definition, basic modeling; constructing a heterogeneous multi-source cross-domain federated meta-learning mechanism; constructing a self-evolution dynamic federated neural architecture search; constructing a federated explainability framework based on causal reasoning; deeply integrating the above models to generate an overall optimization goal suitable for each city; outputting specific prediction values of traffic parameters at future time and providing decision suggestions based on causal relationship for traffic managers; the present application constructs a federated space-time intelligent platform with high performance, high adaptability, high credibility and high privacy protection level through the collaborative innovation and deep integration of three core modules of the heterogeneous multi-source cross-domain federated meta-learning mechanism, the self-evolution dynamic federated neural architecture search and the federated explainability framework based on causal reasoning.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of spatio-temporal intelligent and federated learning, and particularly relates to a self-evolution federated meta-learning method for cross-domain heterogeneous spatio-temporal intelligence. BACKGROUND

[0002] With the acceleration of global urbanization and the continuous growth of the number of motor vehicles, traffic congestion has become a core bottleneck restricting the sustainable development of cities, and intelligent transportation systems have become the key to alleviating this problem. As a core support, traffic flow accurate prediction directly affects the effectiveness of traffic signal optimization, road network planning and path recommendation. In recent years, deep learning technology has achieved remarkable results in spatio-temporal sequence prediction. Graph neural networks, with their excellent spatial dependence capturing ability, have become the mainstream technology for traffic prediction. However, the traditional centralized learning paradigm faces insurmountable challenges in large-scale applications across cities and regions.

[0003] Centralized learning requires the aggregation of traffic data from multiple cities and institutions to a central server, which poses a serious privacy and security risk. Traffic data contains sensitive information such as individual travel patterns, and centralized storage and transmission can easily lead to leaks due to attacks or vulnerabilities. Even after desensitization, the data can still be restored. In addition, cross-domain data sharing is limited due to data sovereignty and commercial competition, forming data silos. Different data sources differ greatly in sampling frequency, spatial granularity, and format, creating a heterogeneous data gap. Traditional models cannot effectively integrate such data, and the accuracy decreases significantly when migrating across cities.

[0004] Different cities differ significantly in road network topology, traffic patterns, and data size, and the unified architecture of centralized models cannot be adapted. For super large cities, shallow models cannot capture complex traffic dependencies. For small and medium-sized cities, deep models are prone to overfitting due to sparse data, resulting in serious generalization and adaptability problems.

[0005] Federated learning provides a solution to the privacy and data silo problems with the characteristic of "data not moving, model moving", but existing federated spatio-temporal learning methods still have limitations. Firstly, there is a lack of multi-source heterogeneous data fusion capability, with a focus on single modal data. The problem of semantic alignment and dynamic weight distribution of heterogeneous data has not been solved, and cross-modal knowledge transfer is difficult. Secondly, the model architecture is not personalized enough, with only parameter-level fine-tuning. The unified architecture cannot adapt to the road network size and computing resources of different cities, and the performance decreases significantly with the increase of clients. Thirdly, the model lacks explainability and causal reasoning ability, acting as a "black box" that can only reveal statistical associations, but cannot distinguish between causal relationships and false associations. Cross-domain causal identification and reasoning in the federated scenario is still a blank.

[0006] In summary, the current traffic spatiotemporal prediction technology faces four core challenges of privacy protection and data sharing, heterogeneous data fusion, model adaptation generalization, explainability and causal reasoning, and urgently needs a new learning method to break through the existing limitations and support the landing of cross-domain heterogeneous spatiotemporal intelligence. SUMMARY

[0007] In order to solve the problems existing in the prior art, the application provides a self-evolutionary federated meta-learning method for cross-domain heterogeneous spatiotemporal intelligence, which solves the problem of lack of multi-source heterogeneous data fusion capability.

[0008] In order to solve the above problems, the technical scheme of the application is as follows: a self-evolutionary federated meta-learning method for cross-domain heterogeneous spatiotemporal intelligence, comprising the following steps:

[0009] S1, problem definition, basic modeling, representing the city traffic network as a weighted directed graph ; for the observed data on the weighted directed graph , define the spatiotemporal sequence with a time step of ; set the federated learning participant set , each participant corresponds to a city or region and has an independent local traffic network and spatiotemporal data ;

[0010] S2, build a heterogeneous multi-source cross-domain federated meta-learning mechanism, process heterogeneous, multi-source raw data into unified feature representation, complete data fusion and knowledge transfer;

[0011] S3, build a self-evolutionary dynamic federated neural architecture search to generate network architectures adapted to different cities;

[0012] S4, build a federated explainability framework based on causal reasoning to provide causal explanation and intervention analysis;

[0013] S5, deeply integrate the heterogeneous multi-source cross-domain federated meta-learning mechanism, self-evolutionary dynamic federated neural architecture search and federated explainability framework based on causal reasoning in S2-S4 to generate an overall optimization target suitable for each city;

[0014] S6, output the specific prediction value of the traffic parameter at the future time and provide decision suggestions based on the causal relationship for the traffic manager.

[0015] Further, in S5, the expression of the overall optimization target is:

[0016] ,

[0017] wherein, denotes the number of cities participating in federated learning, denotes the global shared parameters for encoding cross-city general spatio-temporal patterns, and denote the personalized parameters and network architecture for the th city, respectively, denotes the weight for the th city, denotes the prediction loss for the th city, is a learnable weight parameter for adaptive balancing of different regularizers, is a meta-learning regularizer for promoting fast adaptation capability, is an architecture complexity regularizer for controlling model size, is a causal graph sparsity regularizer for encouraging a simple causal structure.

[0018] Further, in S2, a heterogeneous multi-source cross-domain federated meta-learning mechanism is constructed, including:

[0019] S21, heterogeneous data encoding and modality fusion are performed;

[0020] S22, a cross-domain federated meta-learning framework is constructed, and meta-learning is introduced into federated spatio-temporal prediction, so that the model learns general spatio-temporal representations and adaptation strategies from multiple cities;

[0021] S23, a differential privacy protection mechanism is introduced, and the model parameters are decomposed into global shared and local personalized two parts.

[0022] Further, in S21, the heterogeneous data encoding uses a modality encoder to map the heterogeneous data to a unified embedding space. For the th city, the th modality data , the modality encoder realizes dimension alignment:

[0023] ,

[0024] wherein, denotes the length of the time step, denotes the number of spatial nodes, denotes the original feature dimension, denotes the encoder of the th modality, is a learnable parameter of the encoder, denotes the aligned embedding representation;

[0025] The modality fusion introduces a cross-modality attention fusion mechanism, and the expression is:

[0026] ,

[0027] in, Indicates the first The available modal set for each city Indicates attention weights. Indicates the first Global representation of each modality, This is a learnable query weight matrix.

[0028] Furthermore, S22 includes:

[0029] For the Each city constructs a meta-task set from local data. Each meta-task contains a support set and query set Meta-learning rapidly adapts to new learning environments through a two-layer optimization process. Each city obtains task-specific parameters by performing gradient descent on the support set:

[0030] ,

[0031] in, Indicates the first The parameters of each city after adaptation on the support set Indicates globally shared parameters. Indicates the parameter gradient, The inner learning rate parameter is a learnable parameter. For the loss function on the support set;

[0032] Evaluate and compute the meta-gradient on the query set:

[0033] .

[0034] Furthermore, in S23, the expression for the differential privacy protection mechanism is: ,

[0035] in, The learnable outer learning rate parameter. Indicates the first Privacy gradient of each city

[0036] The model parameters are decomposed into globally shared and locally personalized representations as follows: ,

[0037] in, Indicates the first Complete model parameters for each city For globally shared parameters, For the first A personalized parameter offset of a city.

[0038] Further, in S3, a self-evolution dynamic federated neural architecture search is constructed, including:

[0039] S31, constructing a hierarchical search space;

[0040] S32, constructing a weight sharing super network containing all candidate operations;

[0041] S33, privacy-protected architecture evaluation and personalized generation, using a ranking-based evaluation mechanism to evaluate privacy-protected candidate architectures, and using a counting method to aggregate global preferences on the server side;

[0042] S34, continuous evolution and elastic consolidation, introducing an architecture performance monitoring mechanism, and using an elastic weight consolidation strategy to protect important parameters.

[0043] Further, in S31, the hierarchical search space expression is:

[0044] ,

[0045] wherein, represents the network depth, represents the inter-layer connection mode, represents the module type of the layer, represents the operation hyperparameter.

[0046] Further, in S4, a federated explainability framework based on causal reasoning is constructed, including:

[0047] S41, using a structural causal model to formalize the causal relationship of the transportation system, each participant uses a non-stationary causal discovery algorithm based on local data to identify stable causal edges and dynamic causal edges, and constructs a local causal graph;

[0048] S42, privacy-protected causal graph aggregation, using an encryption mechanism to aggregate local causal graphs, completing homomorphic encryption aggregation, dividing causal effects into global sharing and local specificity, and completing causal effect decomposition;

[0049] S43, designing a causal-aware graph neural network, limiting message passing only along causal edges;

[0050] S44, counterfactual reasoning and intervention analysis, based on the learned causal graph and causal effect, calculating the causal effect of the intervention measure on the target variable, while designing a causal explanation generation module to decompose the prediction result into the causal contribution of each factor and convert it into a natural language explanation, providing intuitive basis for traffic management decision-making.

[0051] Further, in S41, the structural causal model uses a set of spatiotemporal variables For the core modeling object, the generation mechanism of each variable is represented as:

[0052] ,

[0053] wherein, is the th spatiotemporal variable, is the causal parent node set of is the exogenous noise, is the causal function, is the learnable causal function parameter; The causal graph is wherein,

[0054] is a set of spatiotemporal variables containing nodes, represents a set of directed edges. Compared with the prior art, the present application has the following beneficial effects: the present application, through the synergistic innovation and deep integration of the three core modules of the heterogeneous multi-source cross-domain federated meta-learning mechanism, self-evolution dynamic federated neural architecture search and federated explainability framework based on causal reasoning, builds a federated spatiotemporal intelligent platform with high performance, high adaptability, high credibility and high privacy protection level.

[0055] The heterogeneous multi-source cross-domain federated meta-learning mechanism is introduced, which uses the paradigm of "learning how to learn" to extract general spatiotemporal representations and rapid adaptation strategies from multiple city tasks through double-layer optimization, realizes effective alignment and information complementation of heterogeneous data in a unified embedding space through modal perception encoder and cross-modal attention fusion, and greatly improves the performance of the model in the real world.

[0056] The core evaluation indicators of the model on key prediction tasks such as traffic speed, flow and occupancy are greatly better than those of existing mainstream federated learning baselines. In addition, the introduction of the meta-learning mechanism enables the model to quickly fine-tune and adapt to new cities with scarce data or sudden shifts in data distribution, greatly improving the practicality and robustness of the model in dynamic and open real-world scenarios.

[0057] ​A self-evolution dynamic federated neural architecture search is constructed, the algorithm designs a hierarchical search space, and uses a weight-shared super network technology, so that each client can automatically search and generate the optimal personalized graph neural network architecture according to the local traffic network scale, topological complexity and computing resources at an acceptable cost; The fundamental personalization at the level of neural network architecture is realized; The self-evolution dynamic federated neural architecture search can automatically configure a deep and wide complex network for a large city with a complex road network to capture long-distance dependence and fine patterns; At the same time, it can also generate a lightweight and efficient network structure for small and medium-sized cities, avoiding overfitting and resource waste while ensuring performance; In addition, through the weight-shared super network and the ranking-based privacy protection evaluation mechanism, the huge computing and communication overhead brought by traditional neural architecture search is greatly reduced.

[0058] A federated explainability framework based on causal reasoning is constructed, the framework adopts a structural causal model, learns the stable causal relationship between variables in the traffic system through a federated causal discovery algorithm, and constructs a causal-aware graph neural network accordingly, ensuring that information only propagates along the true causal path; Provide reliable causal explanation and intervention analysis: the model not only can accurately predict "what", but also can clearly explain "why" a particular traffic condition occurs, for example, pointing out that congestion is caused by an accident at a specific upstream intersection or the aggregation effect of a specific POI, and can perform counterfactual reasoning to evaluate the expected effect of taking certain control measures, which upgrades the model from a prediction tool to a decision support system, providing clear, reliable and actionable insights for traffic managers; In addition, by discarding non-causal false correlations, causal-aware models exhibit stronger stability when facing changes in data distribution or rare events. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 It is the overall architecture schematic diagram of the present application;

[0060] Figure 2 It is the data privacy schematic diagram of the present application;

[0061] Figure 3 It is the performance comparison of 12 time steps on Know Air of the present application;

[0062] Figure 4 It is the model performance comparison schematic of the present application Figure 1 ;

[0063] Figure 5 It is the model performance comparison schematic of the present application Figure 2 ;

[0064] Figure 6 It is the ablation experiment result visualization effect schematic diagram of METR_LA data set of the present application;

[0065] Figure 7 Fig. 2 is a schematic diagram of the visualization effect of the ablation experiment results of the KnowAir data set of the present application;

[0066] Figure 8 Fig. 3 is a schematic diagram of the visualization effect of the ablation experiment results of the PEMS BAY data set of the present application;

[0067] Figure 9 Fig. 4 is a multi-client heat map of the PEMS08 (occupancy) data set of the present application;

[0068] Figure 10 Fig. 5 is a multi-client heat map of the METR LA (speed) data set of the present application;

[0069] Figure 11 Fig. 6 is a multi-client heat map of the PEMS04 (speed) data set of the present application;

[0070] Figure 12 Fig. 7 is a multi-client heat map of the PEMS04 (flow) data set of the present application;

[0071] Figure 13 Fig. 8 is a multi-client heat map of the PEMS07 (speed) data set of the present application;

[0072] Figure 14 Fig. 9 is a multi-client heat map of the PEMS08 (speed) data set of the present application. DETAILED DESCRIPTION

[0073] The present application provides a self-evolution federated meta-learning method for cross-domain heterogeneous spatio-temporal intelligence, which is characterized by the synergistic optimization of three technical modules: a heterogeneous multi-source cross-domain federated meta-learning mechanism, a self-evolution dynamic federated neural architecture search, and a federated explainability framework based on causal reasoning, thereby constructing a federated spatio-temporal intelligent platform that balances prediction accuracy, personalized adaptation, explainability, and privacy protection, as shown in Fig. 1. Figure 1 As shown in Fig. 1, the system adopts a federated learning paradigm of "data not moving, model moving", and each city client performs training locally and only exchanges encrypted model update information with the server, effectively protecting data privacy.

[0074] The specific steps are as follows:

[0075] S1, problem definition and basic modeling, abstracting the complex traffic system into a graph structure to facilitate the processing of spatial dependence by graph neural networks, and providing a basis for cross-city heterogeneous data fusion with a unified data representation, the specific steps are as follows:

[0076] Representing the urban traffic network as a weighted directed graph , wherein is a set of spatio-temporal variables containing nodes, each node representing a traffic monitoring point (such as a road segment or an area). is the set of edges, representing the spatial connection relationship between nodes; is the adjacency matrix, represents the node and the spatial correlation strength between them.

[0077] For the observed data on the weighted directed graph , define the time step as spatiotemporal sequence , where, represents the dimensional feature observation value (such as flow, speed, occupancy, etc.) of all nodes at time .

[0078] In the federated learning scenario, define a set of participating parties (clients) , each client corresponding to a city or region, with local traffic network and local spatiotemporal data , the network size, topology structure and data distribution of different clients may differ significantly. S2, build a heterogeneous multi-source cross-domain federated meta-learning mechanism, process heterogeneous and multi-source raw data into unified feature representation, complete data fusion and knowledge transfer, as follows:

[0079] S21, encode and modal fusion of heterogeneous data, provide a high-quality, unified feature representation for the data processing front end, and provide high-quality, unified feature representation for subsequent meta-learning, solve the alignment problem of multi-source heterogeneous data, automatically focus on important modalities, and improve feature expression ability. In the PEMS07 dataset experiment in Table 3, this mechanism helps the model reduce the RMSE of speed prediction from 5.83 to 4.52 under a 4-client setting, with the following specific steps:

[0080]

[0081] Heterogeneous data encoding uses modal encoder to map heterogeneous data to a unified embedding space. For the th city's th modal data , the modal encoder realizes dimension alignment: ,

[0082] where, denotes the time step length, denotes the number of spatial nodes, denotes the original feature dimension, denotes the encoder of the th modal, ​These are the learnable parameters of the encoder. This represents the aligned embedding representation;

[0083] Modality fusion introduces a cross-modality attention fusion mechanism, expressed as:

[0084] ,

[0085] in, Indicates the first The available modal set for each city Indicates attention weights. Indicates the first Global representation of each modality, This is a learnable query weight matrix.

[0086] S22. Construct a cross-domain federated meta-learning framework to achieve knowledge transfer based on data fusion, enabling the model to gain cross-city generalization ability. The specific steps are as follows:

[0087] For the Each city constructs a meta-task set from local data. Each meta-task contains a support set and query set Meta-learning rapidly adapts to new learning capabilities through a two-layer optimization process. The inner layer optimization involves: ... Each city obtains task-specific parameters by performing gradient descent on the support set:

[0088] ,

[0089] in, Indicates the first The parameters of each city after adaptation on the support set Indicates globally shared parameters. Indicates the parameter gradient, The inner learning rate parameter is a learnable parameter. For the loss function on the support set;

[0090] Outer optimization evaluates and computes the meta-gradient on the query set: .

[0091] The two-layer optimization simulates the process of "learning to learn". The inner layer optimization quickly adapts to new tasks, while the outer layer optimization evaluates the adaptability of the initial parameters. As shown in Table 4, on the KnowAir dataset, the MAE of the 12-step prediction is 27.92, which is significantly better than the traditional federated learning method.

[0092] S23. A differential privacy protection mechanism is introduced, and the model parameters are decomposed into two parts: globally shared and locally personalized. Personalized modeling is achieved while protecting privacy. As shown in Table 5, this method achieves optimal performance with a communication cost of 152.35GB, achieving a good balance between efficiency and privacy, as detailed below:

[0093] The expression for differential privacy protection is: ,

[0094] in, The learnable outer learning rate parameter. Indicates the first Privacy gradient of each city

[0095] The model parameters are decomposed into globally shared and locally personalized representations as follows: ,

[0096] in, Indicates the first Complete model parameters for each city For globally shared parameters, For the first Individual parameter offsets for each city.

[0097] S3. Construct a self-evolving dynamic federated neural architecture search to generate a network architecture adapted to different cities. The specific steps are as follows:

[0098] S31. Construct a hierarchical search space to provide customized architecture options for cities of different sizes, avoiding the limitations of a uniform architecture. The expression for the hierarchical search space is: ,

[0099] in, Indicates network depth. Indicates the inter-layer connection mode. Indicates the first Layer module type, This indicates the operation hyperparameters.

[0100] S32. Construct a weight-shared supernetwork containing all candidate operations. , No. Random sampling architecture for each city And optimize: ,

[0101] in, For super network shared weights, The learning rate parameter is a learnable parameter. For training data.

[0102] S33, privacy-protected architecture evaluation and individualization generation, when evaluating candidate architectures, directly uploading verification loss can leak data distribution information. To protect privacy, an evaluation mechanism based on ranking is proposed. Each participant only uploads relative ranking instead of absolute loss value. The server side uses Borda counting method to aggregate global preferences:

[0103] ,

[0104] wherein, is the th candidate architecture, is the total number of candidate architectures, is the local ranking, is the city weight. Although the global ranking provides general guidance, the optimal architecture of different cities may still be different. To achieve architecture individualization, the city feature vector is extracted, and a mapping is established through the neural architecture predictor: ,

[0105] wherein, is the architecture predictor, is the learnable parameter, is the predicted optimal architecture. New participants do not need to search from scratch. Directly generate adaptive architecture according to their own characteristics. As shown in Table 6, the MAE of METR_LA increases from 3.42 to 3.58 after removing this component, proving its necessity.

[0106] S34, continuous evolution and elastic consolidation, traffic networks and data distribution evolve over time, and fixed architectures are difficult to maintain optimal performance in the long term. To cope with dynamic environments, an architecture performance monitoring mechanism is introduced. When the performance drops by more than a preset threshold, the architecture is re-searched. To avoid catastrophic forgetting, an elastic weight consolidation strategy is used to protect important parameters:

[0107] ,

[0108] wherein, is the loss on the current data, is the new data distribution, is the Fisher information matrix of the parameter , is the parameter value before evolution, is the learnable consolidation strength parameter. This mechanism ensures that the architecture evolves to adapt to new data distribution while retaining effective representation ability for old data.

[0109] S4, build a federated explainability framework based on causal reasoning, provide causal explanation and intervention analysis, the specific steps are as follows:

[0110] S41. The causal relationships of the traffic system are formalized using a structural causal model. Each participant uses a non-stationary causal discovery algorithm based on local data to identify stable and dynamic causal edges and construct a local causal graph.

[0111] Structural causal models use a set of spatiotemporal variables For the core modeling object, the generation mechanism of each variable is represented as follows:

[0112] ,

[0113] in, For the first A spatiotemporal variable, for The set of causal parent nodes, For exogenous noise, It is a causal function. These are learnable causal function parameters;

[0114] Cause-and-effect diagram ,in, It is a set of spatiotemporal variables containing N nodes. It represents the set of directed edges.

[0115] S42. Privacy-preserving causal graph aggregation and heterogeneity handling: Directly sharing causal graphs may leak data distribution information. To protect privacy, a secure multi-party computation protocol is designed, representing the local causal graph as a binary adjacency matrix. Aggregation via homomorphic encryption:

[0116] ,

[0117] in, This is a homomorphic encryption operation. For decryption operation, For learnable threshold parameters, As an indicator function, this mechanism preserves causal edges that consistently appear among most participants. Causal relationships may exhibit heterogeneity across different cities. To capture this causal heterogeneity, the causal effect is decomposed as follows: ,

[0118] in, Indicates in the city From variables arrive The strength of the causal effect For globally shared causal effects, Shifts in causal effects specific to the city.

[0119] S43. Design a causal perceptual graph neural network to restrict message passing to only along causal edges;

[0120] Integrating the learned causal structure into the prediction model can significantly improve the generalization ability and interpretability. Traditional graph neural networks construct graph structures based on physical adjacency or statistical correlation, but these structures may contain false associations. The present application designs a causally aware graph neural network, whose message passing is only along the causal edges:

[0121] ,

[0122] wherein, is the hidden representation of the node at the layer, is the self-connection weight matrix, is the causal message passing weight matrix, is the causal parent node set, is the activation function. This design ensures that information propagation follows real causal dependencies and avoids the propagation of false associations.

[0123] S44, counterfactual reasoning and intervention analysis. The core advantage of the causal model is to support counterfactual reasoning. In traffic management, decision makers need to evaluate the expected effects of different intervention measures. Based on the learned causal graph and causal effect, the causal effect of intervention on the target variable can be calculated:

[0124] ,

[0125] wherein, indicates that the variable is artificially set to a specific value, is the expected value, is the causal effect coefficient. This formula enables managers to quantitatively evaluate the expected effect of intervention measures. To further improve interpretability, a causal explanation generation module is designed to automatically decompose the prediction result into the causal contribution of each factor and convert it into a natural language explanation. This causal explanation provides decision makers with intuitive and credible prediction basis, significantly improving the interpretability and credibility of the model.

[0126] S5, integrate the three modules under the overall optimization objective. The expression of the overall optimization objective is:

[0127] ,

[0128] wherein, denotes the number of cities participating in federated learning, denotes the global shared parameters used to encode cross-city general spatiotemporal patterns, and denote the personalized parameters and network architecture of the city, the weight of the the prediction loss of the the weight of the the prediction loss of the a learnable weight parameter for adaptive balancing of different regularizers, a meta-learning regularizer for promoting fast adaptation capability, an architecture complexity regularizer for controlling model size, a causal graph sparsity regularizer for encouraging a simple causal structure.

[0129] S6, three core modules are implemented in the logic of "data fusion-architecture adaptation-causal explanation", the output of the previous module is the input of the next module, forming a complete technical link, finally serving the overall optimization goal, the expression of the overall optimization goal is:

[0130] ,

[0131] wherein, represents the number of cities participating in federated learning, represents the global shared parameter for encoding cross-city general spatiotemporal patterns, and respectively represent the personalized parameters and network architecture of the the weight of the the prediction loss of the the weight of the the prediction loss of the a learnable weight parameter for adaptive balancing of different regularizers, a meta-learning regularizer for promoting fast adaptation capability, an architecture complexity regularizer for controlling model size, a causal graph sparsity regularizer for encouraging a simple causal structure.

[0132] Experimental verification and performance analysis

[0133] Dataset, we conducted experiments on the following datasets: METR_LA, PEMSBA Y, PEMS04, PEMS07, PEMS08 and KnowAir.

[0134] ​METR LA contains traffic speed data collected by 207 loop detectors on the county highway network in A from March to June in 2012. The PEMS dataset is from the performance measurement system in C, which contains multi-dimensional traffic information such as traffic speed, flow and occupancy. Among them, the PEMS BAY dataset covers the traffic speed data of 325 sensors in B from January to May 2017, which are distributed in the main highway network of B and record the traffic running state for about 6 months; the PEMS04 dataset collects data from 307 sensors in B from January to February 2018; the PEMS07 contains traffic flow data from 883 sites in A and D from May to August 2017; and the PEMS08 covers traffic speed information from 170 detectors in E from July to August 2016. KnowAir is a multi-modal dataset containing air quality and traffic flow information, which collects data from 184 monitoring stations in a major city of a country from January 2015 to December 2018. The unique feature of this dataset is that it contains PM2.5 concentration, weather conditions and traffic flow data, which can be used to study the correlation between air quality and traffic conditions. In the graph modeling of traffic data, each node represents a sensor or monitoring station, and the edges between nodes are constructed according to the road network connection relationship or geographical distance. The detailed statistical information of the dataset is shown in Table 1.

[0135]

[0136] Table 1

[0137] Setting

[0138] The network architecture is set to contain 2 hidden layers, each with a feature dimension of 64. The dimension of the feature embedding space is set to 2. In terms of optimizer parameters, we set the learning rate to 0.003 and each batch contains 64 samples. The training process includes 200 global training and 2 local training, and the data set division strategy is: the training set accounts for 60%, the validation set and the test set each accounts for 20%. In order to evaluate the performance of the model comprehensively, we use three commonly used regression task evaluation indicators: root mean square error, mean absolute error and mean absolute percentage error. The smaller the value of these indicators, the higher the prediction accuracy of the model. All the calculation experiments are completed on a computer equipped with Nvidia GeForce RTX3090 GPU. For details, see Table 2.

[0139]

[0140] Table 2

[0141] Model comparison experiment

[0142] The comparative experiments are carried out on six real-world datasets, METR LA, PEMS BAY, PEMS04, PEMS07, PEMS08 and KnowAir, which have heterogeneous spatial topologies, data modalities and statistical characteristic distributions. All baseline models are strictly reproduced according to the optimal hyperparameter settings reported in their original papers. To ensure the fairness and repeatability of the experiments, all models are implemented in the PyTorch framework, with Kaiming strategy for parameter initialization, and a fixed random seed is set for each round of experiment. For each experimental configuration (combination of different datasets and heterogeneous scenarios), we perform 10 independent repeated experiments for each model, with resampled initialization parameters for each run. The prediction accuracy, generalization performance and robustness of the models under different random conditions are comprehensively evaluated by statistical mean and standard deviation.

[0143] As shown in Table 3, a self-evolving federated meta-learning method (FedSTMeta) for cross-domain heterogeneous spatio-temporal intelligence is presented and compared with six benchmark models on five real-world traffic datasets. Experimental results show that our method achieves the best performance in most configurations. In the speed prediction of the PEMS07 dataset (4 clients), the self-evolving federated meta-learning method reduces the root mean square error by 22.5%, the mean absolute error by 38.3%, and the mean absolute percentage error by 44.0% compared to the suboptimal method FASTGNN. On the PEMSBay dataset (8 clients), the mean absolute percentage error is optimized from 4.48% to 3.34% compared to MFVSTGNN. In the larger-scale PEMS08 experiment (14 clients), the self-evolving federated meta-learning method exhibits more significant advantages, reducing the root mean square error by 5.0%, the mean absolute error by 6.6%, and the mean absolute percentage error by 7.4% in speed prediction compared to FedGTP. In the difficult occupancy prediction, the mean absolute error is 0.0054, significantly better than all baseline methods. The advantages of this method are due to the synergistic effect of three core innovations. The heterogeneous multi-source cross-domain federated meta-learning mechanism enables the model to learn universal spatio-temporal representations from multiple source cities and quickly adapt to new scenarios. In the occupancy prediction of PEMS08, the mean absolute error is reduced by 8.5% compared to the suboptimal method, fully verifying the effectiveness of meta-learning in low-resource scenarios. The self-evolving dynamic federated neural architecture search algorithm dynamically generates personalized graph neural network structures based on the network size and topological complexity of each city, enabling the entire model to maintain stable and superior performance on different scale datasets. In small-scale PEMS07, the mean absolute percentage error is reduced by 44.0%, and in large-scale PEMSBay, it still maintains a reduction of 33.0%, while the baseline methods with unified architecture show significant fluctuations. The federated explainability framework based on causal reasoning avoids spurious correlations by identifying real causal relationships in the traffic system. In the long-term task of PEMS08 traffic prediction, the error growth rate of our method is significantly lower than FedGTP, with the mean absolute percentage error increasing from 9.02% to about 10%, while FedGTP rapidly climbs from 9.09%. At the same time, it provides interpretable intervention suggestions for traffic management decisions. In addition, as the number of clients increases from 4 to 14, the performance of most baseline methods significantly degrades, while the entire model quickly adapts and remains stable in high-heterogeneous scenarios through meta-learning and architecture personalization. In the 14-client configuration of PEMS08, the mean absolute percentage error of speed prediction is only 1.25%, fully demonstrating its scalability and robustness in large-scale heterogeneous federated scenarios. It should be noted that some baseline methods have data missing in certain indicators, which is mainly due to the inability to reproduce the model and the absence of reporting of the corresponding evaluation indicators in the original literature.In summary, a self-evolving federated meta-learning for cross-domain heterogeneous spatio-temporal intelligence achieves breakthrough progress in spatio-temporal prediction under the premise of protecting data privacy through the synergistic effect of three major innovative mechanisms, providing a practical technical solution for the actual deployment of intelligent transportation systems.

[0144]

[0145] Table 3 Performance comparison of federated learning methods

[0146] To comprehensively evaluate the effectiveness of the FedSTMeta framework in multi-source spatio-temporal prediction scenarios, we selected nine representative benchmark models covering three types of training paradigms for comparative analysis:

[0147] a) Separated method: This type of method independently models on each client's local data, with training processes isolated from each other. Due to the lack of consideration of spatial correlation characteristics of spatio-temporal data, classical time series modeling techniques are usually used.

[0148] b) Centralized method: This type of method aggregates global node data for unified training, with mainstream algorithms mostly based on graph neural network architecture.

[0149] c) Federated training method: This type of method is the baseline paradigm we focus on comparing, which meets the privacy protection requirements. Unlike the multi-client scenario comparison in Table 3, in this experiment, all monitoring sites are unified as participating nodes of federated learning, focusing on evaluating the performance of each method under different prediction time domains to verify the effectiveness of the model in long-term time series prediction tasks. In the experiment on the KnowAir dataset, we follow the standard settings of federated learning and conduct collaborative training with 184 monitoring sites as federated clients. Since air quality monitoring sites are distributed in different cities across the country, they have significant geographical heterogeneity and data distribution differences, which are essentially consistent with the setting of dividing clients by city in Table 3, both of which reflect the application value of federated learning in multi-source heterogeneous data scenarios. This experiment mainly focuses on the time series prediction ability of the model, and by comparing the performance of 6-step and 12-step prediction time domains, it comprehensively evaluates the stability of each method in short-term and medium-term prediction tasks.

[0150] From Table 4, it can be seen that the average absolute error of FedSTMeta is 20.65 in 6-step prediction and 27.92 in 12-step prediction, which is improved by 0.43% and 4.19% compared with the optimal federated baseline FedCroST and FedSTG, respectively, and the performance advantage is more significant as the prediction step increases. The comparison with the three types of training paradigm fully verifies the effectiveness of the framework: compared with the separation training method ARIMA (6-step average absolute error = 28.45), FedSTMeta reduces the error by 27.44% through cross-site collaborative learning, proving that federated collaboration can effectively tap the spatiotemporal correlation patterns between sites; compared with the centralized method STGCN (6-step average absolute error = 20.82), FedSTMeta still has an advantage under the premise of protecting data privacy, indicating that personalized architecture design can make up for the disadvantage of federated learning in data availability; compared with the traditional federated method FedGRU and FedSTN, the error is reduced by 12.05% and 8.79%, respectively, fully verifying the synergistic effect of the three core innovative modules. Specifically, the meta-learning mechanism extracts a general spatiotemporal representation and a fast adaptation strategy from 184 heterogeneous sites, effectively handling geographical distribution differences, and the modal perception encoder and cross-modal attention fusion integrate multi-source data such as air quality, weather, and geography; the self-evolving neural architecture search algorithm automatically generates the optimal network structure for different size sites, with deep networks used in large cities to capture complex patterns and shallow networks used in small and medium-sized cities to avoid overfitting, reducing the 12-step prediction error by 8.89% and 10.06% compared with the fixed architecture CNFGNN and SFL; the federated explainability framework identifies stable causal edges (such as the influence of the upwind direction site and the role of weather conditions on pollutant dispersion) through distributed causal structure learning, and constructs a causal perception graph neural network to ensure that information propagation only occurs along true causal dependencies, reducing the average absolute error by 4.52% and 4.19% compared with FedCroST and FedSTG in 12-step long-term prediction when short-term statistical correlations gradually fail. Experiments show that FedSTMeta exhibits stable advantages in short-term and medium-term prediction, especially in challenging scenarios such as handling heterogeneous multi-source data, adapting to different size networks, and long-term prediction, providing a solution for federated spatiotemporal prediction that combines performance, privacy, and explainability. The visualization effect is shown in Figure 3

[0151]

[0152] Table 4 Performance of KnowAir dataset

[0153] Cost experiment

[0154] ​We evaluate the efficiency of FedSTMeta by comparing it with four federated spatio-temporal baseline models, as shown in Table 5, which reports the computational cost, training communication cost, and performance of each model on the KnowAir dataset. Figure 4 、 5 As shown in Figure 3, the 3D visualization intuitively presents the efficiency-performance trade-off of different methods from the perspectives of computational cost, communication cost, and prediction accuracy. FedGRU, which adopts a simple GRU architecture, performs best in terms of computational and communication efficiency, with a computational cost of only 0.15 GFLOPs and a training communication cost of 74.58 GB, making it the lowest in the 3D space. This is due to its avoidance of complex graph structure modeling and spatial dependency learning. However, FedGRU's mean absolute error is as high as 31.38, indicating poor performance in prediction. This reflects the limitations of lightweight architectures in capturing complex spatio-temporal dependencies. CNFGNN, while maintaining the same computational complexity, has a mean absolute error of 30.64 and a training communication cost of 141.78 GB, nearly twice that of FedGRU. This is essentially the communication overhead resulting from the server-side cross-node dependency extraction and spatial dynamic modeling in the separate spatio-temporal modeling strategy. SFL, as a segmentation federated learning method, requires the transmission of intermediate layer activation values between clients and servers, resulting in a computational cost of 0.36 and a communication cost of 163.55 GB, the highest in the 3D space. Its mean absolute error is 31.04, making it a method with both low efficiency and poor performance. FedCroST strikes a good balance between efficiency and cross-domain knowledge transfer, with a computational cost of 0.18, a training communication cost of 149.88 GB, and a mean absolute error of 29.28. Although the introduction of the domain adaptation mechanism increases the additional overhead, it improves performance. Most importantly, our FedSTMeta model is located in the optimal position in the 3D space, with a mean absolute error of only 27.92, significantly better than all baseline methods, while maintaining reasonable computational and communication costs. Specifically, FedSTMeta improves the mean absolute error by 4.6% compared to the suboptimal FedCroST, while increasing the computational cost by only 16.7% and the training communication cost by 1.6%. This fully demonstrates the superiority of our proposed meta-learning mechanism, adaptive architecture search, and causal reasoning mechanism in balancing efficiency and performance. Figure 4 、 5The green semi-transparent plane in the figure marks the high-performance region (average absolute error < 28.5), which is a threshold set based on the average performance level of all baseline methods to intuitively distinguish models that meet practical accuracy requirements from methods that only provide theoretical value. FedSTMeta, as the only method located in this region, is clearly marked with a red star, demonstrating its ability to achieve outstanding performance while maintaining reasonable efficiency. These experimental results fully demonstrate that FedSTMeta achieves the optimal Pareto balance between model complexity, communication efficiency, and prediction accuracy, providing an efficient and accurate solution for federated spatio-temporal prediction.

[0155]

[0156] Table 5 Cost comparison of KnowAir dataset

[0157] Ablation experiments

[0158] To verify the effectiveness of the model, we conducted ablation experiments on different components to verify the effectiveness of the key modules we proposed, without considering the impact of multiple clients. We designed the following variants, and the experimental results are shown in Table 6:

[0159] Remove federated learning mechanism: To evaluate the impact of the heterogeneous multi-source cross-domain federated meta-learning mechanism on overall performance, we removed this module and replaced it with standard FedAvg. The results show that the performance decreased significantly on all datasets, for example, the average absolute error increased from 3.42 to 3.82 (an increase of 11.7%) on METR_LA, and the average absolute error increased to 30.84 (an increase of 10.5%) on KnowAir. This experiment provides evidence that this mechanism plays a key role in cross-domain knowledge transfer and multi-modal fusion, which is crucial for improving the model's generalization ability in heterogeneous urban traffic scenarios.

[0160] Remove meta-learning: To verify the contribution of the meta-learning mechanism, we kept the multi-modal fusion mechanism but removed the meta-learning framework and trained it using traditional supervised learning. The experimental results show that each indicator has decreased to varying degrees, especially on the PEMSBAY dataset, where the average absolute error increased from 1.89 to 2.12 (an increase of 12.2%). This proves the effectiveness of the meta-learning mechanism in learning universal spatio-temporal representations from multiple cities and achieving rapid scenario adaptation, and the necessity of surpassing traditional supervised learning methods is verified.

[0161] Remove information fusion: Removing multi-modal fusion and only using single traffic flow data, the model performance shows a stable decline. For example, on the KnowAir dataset, the mean absolute error increases from 27.92 to 29.02 (increasing by 3.9%), and the root mean square error increases from 37.88 to 39.03. This shows that multi-modal fusion has obvious advantages in providing rich context information and capturing complex traffic patterns by integrating heterogeneous data sources such as traffic flow, weather, and POI, but its impact is relatively moderate compared to meta-learning and federated frameworks.

[0162] Single modeling: Model variants using only single spatio-temporal modeling methods perform the worst on all datasets, especially on the METR_LA, where the mean absolute error increases to 4.02 (increasing by 17.5%). In addition, on the KnowAir dataset, the mean absolute error of this variant reaches 31.12, and the root mean square error reaches 42.34, which is much higher than the complete model. This shows that although a unified global spatio-temporal modeling architecture can capture basic traffic patterns, it has obvious limitations in dealing with spatial heterogeneity and complex spatio-temporal dependencies in different cities.

[0163] Remove self-evolving neural architecture search: Removing the self-evolving dynamic neural architecture search algorithm and using a fixed graph neural network architecture, the model shows uneven performance decline on different scale cities. On the PEMSBAY dataset, the mean absolute error increases from 1.89 to 2.11 (increasing by 11.6%). This shows that this algorithm can optimize the architecture level according to the characteristics of the city's traffic network, but its contribution is slightly lower than the federated meta-learning framework on some datasets, confirming the importance of dynamic architecture search in large and complex urban scenarios.

[0164] Remove federated explainability: Removing only the federated explainability mechanism based on causal reasoning, the model performs relatively well on most datasets, but shows a significant decline in extreme value prediction. On the METR_LA, the mean absolute error increases from 3.42 to 3.48 (increasing by 1.8%), while on the KnowAir, the mean absolute error only increases to 28.75. This shows that the causal reasoning mechanism has certain advantages in identifying real causal dependencies and filtering false correlations. However, compared to other core modules, this mechanism has relatively limited overall performance improvement for regular prediction tasks, and its value is more reflected in providing explainability and dealing with abnormal scenarios.

[0165] From the experimental results in Table 6, it can be seen that the FedSTMeta complete model achieves the best performance on all data sets and evaluation indicators, verifying the effectiveness of our architecture design that integrates heterogeneous multi-source cross-domain federated meta-learning, self-evolving neural architecture search, and causal reasoning mechanism. Especially on the KnowAir data set with higher data complexity, the complete model outperforms each variant, demonstrating the necessity of multi-module collaborative integration. The experimental results fully prove the important contribution of each core component to improving the performance of federated spatio-temporal prediction, and the visualization effects are shown in Figs. Figure 6 、 7 、8.

[0166]

[0167] Table 6 Ablation experiment of FedSTMeta

[0168] Multi-client performance comparison experiment

[0169] To verify the effectiveness of FedSTMeta in dealing with cross-task differences, as Figures 9-14As shown, the prediction variance heatmaps of each client on four traffic datasets are displayed, covering three tasks of speed prediction (METR_LA, PEMS04, PEMS07, PEMS08), flow prediction (PEMS04 flow) and occupancy prediction (PEMS08 occupancy), with the horizontal axis representing node ID, the vertical axis representing client, and the color intensity representing normalized prediction variance (dark red for high variance and light yellow for low variance). The heatmaps reveal the complexity of the federated learning scenario from three dimensions: first, different task types show significant differences, with speed prediction showing a large number of dark red high variance areas due to the influence of instantaneous events, flow prediction showing a more smooth sparse high variance distribution as a cumulative indicator, and occupancy prediction mainly with light yellow low variance due to slow changes and physical constraints, which verifies the necessity of the heterogeneous multi-source cross-domain federated meta-learning module to extract general spatio-temporal knowledge through multi-task meta-learning and enable the model to quickly adapt to new tasks; second, the prediction difficulty of different clients in the same task is significantly different, such as PEMS04 where client 2 has dark red blocks at nodes 0 and 7 while clients 1 and 3 have moderate variance, and PEMS08 occupancy where clients 2 and 4 show local high variance while clients 1 and 3 remain low variance, which proves the value of the self-evolving dynamic federated neural architecture search to configure lightweight architectures for low variance clients and deep networks for high variance clients; finally, the complexity of datasets of different scale cities is significantly different, with METR_LA large-scale road network showing about 40% high variance nodes reflecting the complexity of super-large cities, PEMS04 and PEMS07 medium-scale road networks showing 25-30% high variance nodes, and PEMS08 small-scale road network showing sparse extreme high variance, which verifies the cross-domain migration ability of the heterogeneous multi-source cross-domain federated meta-learning mechanism to enable new cities to quickly adapt using existing city knowledge. In addition, the high variance nodes in the heatmaps provide a direct motivation for federated causal reasoning and explainability mechanisms, with federated explainability identifying real causal factors (congestion propagation, POI density, weather, etc.) of difficult-to-predict nodes through distributed causal structure learning and supporting counterfactual reasoning to provide actionable intervention suggestions for managers, thus providing sufficient design motivation and experimental evidence for the three core innovations of FedSTMeta from the task, client, and dataset levels.

[0170] The above detailed description is merely intended to illustrate the technical solutions of the present application and not to limit the same. Although the present application has been described in detail with reference to the examples, those skilled in the art should understand that the technical solutions of the present application can be modified or equivalently replaced without departing from the scope of the technical solutions of the present application, and all modifications and equivalent replacements should be included in the scope of the claims of the present application.

Claims

1. A self-evolving federated meta-learning method for cross-domain heterogeneous spatio-temporal intelligence, characterized in that, The method comprises the following steps: S1, problem definition, basic modeling, representing the urban transportation network as a weighted directed graph ; for observed data on a weighted directed graph , define a spatio-temporal sequence with time step ; Setting a set of federated learning participants Each participant corresponds to a city or region and has an independent local transportation network And space-time data ; S2, constructing a heterogeneous multi-source cross-domain federated meta-learning mechanism, processing heterogeneous and multi-source raw data into a unified feature representation, completing data fusion and knowledge transfer; S3, constructing a self-evolution dynamic federated neural architecture search, generating network architectures adapted to different cities; S4, constructing a federated explainability framework based on causal reasoning, providing causal explanation and intervention analysis; S5, deeply integrating the heterogeneous multi-source cross-domain federated meta-learning mechanism, the self-evolution dynamic federated neural architecture search, and the federated explainability framework based on causal reasoning in S2-S4 to generate an overall optimization target applicable to each city; S6, outputting specific prediction values of traffic parameters at future time points and providing decision suggestions based on causal relationships for traffic managers.

2. The method of claim 1, wherein: In S5, the expression of the overall optimization target is: ， wherein, denotes the number of cities participating in federated learning, denotes global shared parameters for encoding cross-city general spatio-temporal patterns, and denote the personalized parameters and network architecture of the th city, respectively, denotes the weight of the th city, denotes the prediction loss of the th city, is a learnable weight parameter for adaptive balancing of different regularizers, is a meta-learning regularizer for promoting fast adaptation capability, is an architecture complexity regularizer for controlling model size, is a causal graph sparsity regularizer for encouraging a simple causal structure.

3. The method of claim 2, wherein: In S2, the heterogeneous multi-source cross-domain federated meta-learning mechanism comprises: S21, encoding and modality fusion of heterogeneous data; S22, constructing a cross-domain federated meta-learning framework, introducing meta-learning into federated spatiotemporal prediction, and enabling the model to learn universal spatiotemporal representation and adaptation strategies from multiple cities; S23, introducing a differential privacy protection mechanism and decomposing model parameters into global shared and local personalized parts.

4. The method of claim 3, wherein: In S21, the heterogeneous data encoding employs a modal encoder to map the heterogeneous data to a unified embedding space, for the i-th city's j-th modal data , the modal encoder implements dimension alignment: :​ ， wherein, denotes the time step length, denotes the number of spatial nodes, denotes the original feature dimension, denotes the encoder for the th modality, are the learnable parameters of the encoder, denotes the aligned embedding representation; The modality fusion introduces a cross-modal attention fusion mechanism, and the expression is: ， wherein, denotes a set of modalities available for the th city, denotes an attention weight, denotes a global representation of the th modality, is a learnable query weight matrix.

5. The method of claim 4, wherein: S22 comprises: For the first city, the meta-task set is constructed from local data Each meta-task contains a support set and a query set , meta-learning quickly adapts to the ability through double-layer optimization learning, the first city obtains task-specific parameters by gradient descent on the support set: ， wherein, denotes the adapted parameters of the th city on the support set, denotes the global shared parameters, denotes the gradient of the parameters , is a learnable inner learning rate parameter, is the loss function on the support set; Evaluating and calculating meta-gradient on the query set: 。 6. The method of claim 5, wherein: In S23, the expression of the differential privacy protection mechanism is: , wherein, is a learnable outer learning rate parameter, denotes the privacy-preserved metagradient for the i-th city, The model parameters are decomposed into global shared and local personalized representations as: , in, Indicates the first Complete model parameters for each city For globally shared parameters, For the first Individual parameter offsets for each city.

7. The method of claim 6, wherein: In S3, the self-evolution dynamic federated neural architecture search comprises: S31, constructing a hierarchical search space; S32, constructing a weight-sharing super network containing all candidate operations; S33, privacy-protected architecture evaluation and personalized generation, using a ranking-based evaluation mechanism to evaluate privacy-protected candidate architectures, and using a counting method to aggregate global preferences on the server side; S34, continuous evolution and elastic consolidation, introducing an architecture performance monitoring mechanism and using an elastic weight consolidation strategy to protect important parameters.

8. The method of claim 7, wherein: In S31, the expression of the hierarchical search space is: ， wherein, represents the network depth, represents the inter-layer connection mode, represents the module type of the layer, represents the operation hyperparameters.

9. The method of claim 8, wherein: In S4, the federated explainability framework based on causal reasoning comprises: S41, using a structural causal model to formalize the causal relationship of the traffic system, using a non-stationary causal discovery algorithm based on local data to identify stable causal edges and dynamic causal edges, and constructing a local causal graph; S42, privacy-protected causal graph aggregation, using an encryption mechanism to aggregate local causal graphs, completing homomorphic encryption aggregation, dividing causal effects into global shared and local specific, and completing causal effect decomposition; S43, designing a causal-aware graph neural network to limit message passing along causal edges only; S44, counterfactual reasoning and intervention analysis, calculating the causal effect of intervention measures on target variables based on the learned causal graph and causal effect, and designing a causal explanation generation module to decompose prediction results into causal contributions of each factor and convert them into natural language explanations, providing intuitive basis for traffic management decisions.

10. The method of claim 9, wherein: In S41, the structural causal model uses a set of spatiotemporal variables. For the core modeling object, the generation mechanism of each variable is represented as follows: ， in, For the first A spatiotemporal variable, for The set of causal parent nodes, For exogenous noise, It is a causal function. These are learnable causal function parameters; The causal graph is wherein, is a set of spatiotemporal variables comprising N nodes, represents a set of directed edges.

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