MVG-KAN: Multi-View Geo-Wind Guided KAN for PM$_{2.5}$ Forecasting

# MVG-KAN: Multi-View Geo-Wind Guided KAN for PM2.5 Forecasting
Source: [https://arxiv.org/html/2606.24347](https://arxiv.org/html/2606.24347)
Cheng Huang, Muyao Guan, Jairus Yougui Railey, Ning Xu, Honghui Xu, Changjiang Zhang, Zhen Zhang, Shiqing Zhang, and Cong BaiCheng Huang, Muyao Guan, Jairus Yougui Railey, Ning Xu, Honghui Xu, Changjiang Zhang, Zhen Zhang, and Shiqing Zhang are with the College of Artificial Intelligence, Taizhou University, Taizhou 318000, China\.Cong Bai is with the College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China\.Corresponding authors: Zhen Zhang and Shiqing Zhang \(e\-mail: zhangzhen\-911@tzc\.edu\.cn; zhangshiqing@tzc\.edu\.cn\)\.

###### Abstract

Accurate short\-term PM2\.5forecasting is important for public health protection, air\-quality early warning, and urban environmental management\. However, PM2\.5variation is driven by multiple coupled factors, including stable periodic changes induced by human activities and meteorological regularity, station\-specific short\-term concentration evolution, and meteorology\-driven pollutant dispersion among monitoring stations\. Existing spatio\-temporal forecasting methods may capture station relationships to some extent, but distance\-only, correlation\-based, or purely adaptive graphs are often insufficient to comprehensively represent these heterogeneous factors, especially wind\-direction\-dependent pollutant transport\. To address this problem, we propose a Multi\-View Geo\-Wind Guided KAN model for PM2\.5forecasting, namedMVG\-KAN, which models station\-level PM2\.5evolution from three complementary views: local periodic regularity, station\-wise residual temporal dynamics, and meteorological\-environment\-guided spatial dispersion\. Specifically, the periodic\-residual forecasting backbone first separates stable daily and weekly patterns from non\-periodic residual variations\. A Geo\-Wind Graph is constructed by combining geographic distance decay with wind\-direction\- and wind\-speed\-aware transport, providing a lightweight physically motivated directed spatial prior for residual propagation among stations\. In addition, a temporal Kolmogorov\-Arnold network \(TKAN\) residual head is then introduced to learn station\-wise nonlinear autoregressive correction from de\-periodized PM2\.5residuals and historical multi\-pollutant sequences, thereby enhancing the modeling of local residual inertia and pollutant co\-variation\. Experiments on the Beijing PM2\.5dataset show that MVG\-KAN achieves an MAE of 14\.09 and an RMSE of 21\.40\. Ablation studies further validate the complementary effects of TKAN and Geo\-Wind Graph, indicating that multi\-view modeling is effective for station\-level PM2\.5forecasting\.

## IIntroduction

Fine particulate matter with an aerodynamic diameter below2\.5​μ​m2\.5~\\mu\\mathrm\{m\}, namely PM2\.5, is one of the most harmful urban air pollutants\. High PM2\.5concentration is closely associated with respiratory and cardiovascular diseases, reduced atmospheric visibility, and degraded urban environmental quality\[[22](https://arxiv.org/html/2606.24347#bib.bib16),[4](https://arxiv.org/html/2606.24347#bib.bib20),[12](https://arxiv.org/html/2606.24347#bib.bib17),[14](https://arxiv.org/html/2606.24347#bib.bib21)\]\. Accurate PM2\.5forecasting is therefore important for public health protection, air\-quality early warning, exposure risk assessment, and urban environmental management\[[38](https://arxiv.org/html/2606.24347#bib.bib18),[1](https://arxiv.org/html/2606.24347#bib.bib19)\]\. Reliable forecasts over the next several hours or days can support public advisories, temporary emission\-control planning, traffic and industrial regulation, and emergency response during heavy pollution episodes\.

![Refer to caption](https://arxiv.org/html/2606.24347v1/figures/figure1.png)Figure 1:Motivation of MVG\-KAN for PM2\.5forecasting\. PM2\.5evolution at a target stationSiS\_\{i\}is modeled as multi\-view of local periodic patternPi​\(t\)P\_\{i\}\(t\), local residual dynamicsRi​\(t\)R\_\{i\}\(t\), and wind\-driven spatial transportTi​\(t\)T\_\{i\}\(t\)\. The periodic component describes regular daily/weekly variations, the residual component represents local non\-periodic fluctuations and pollution episodes, and the transport component captures directed pollutant propagation from upwind/source stations toSiS\_\{i\}under wind fields\. This decomposition motivates the proposed framework to jointly model temporal regularity, station\-wise residual correction, and physically informed geo\-wind spatial transport, rather than relying only on temporal history or generic distance/correlation/adaptive graphs\.To achieve accurate PM2\.5forecasting, a variety of approaches have been developed over the past decades\. Early studies mainly relied on traditional statistical methods, such as autoregressive integrated moving average \(ARIMA\), regression models, and other time\-series forecasting techniques\[[3](https://arxiv.org/html/2606.24347#bib.bib3)\]\. These methods are computationally efficient and provide relatively good interpretability, making them suitable for capturing simple temporal trends and periodic fluctuations\. However, they usually rely on linear assumptions and handcrafted feature engineering, which limits their ability to model the highly nonlinear and complex interactions among pollutants, meteorological variables, and emission sources\[[40](https://arxiv.org/html/2606.24347#bib.bib23),[15](https://arxiv.org/html/2606.24347#bib.bib22),[24](https://arxiv.org/html/2606.24347#bib.bib4)\]\. As PM2\.5formation and evolution involve intricate nonlinear processes, traditional statistical methods often struggle to maintain forecasting accuracy under rapidly changing environmental conditions\[[1](https://arxiv.org/html/2606.24347#bib.bib19)\]\.

To overcome these limitations, deep learning methods have been increasingly adopted for PM2\.5forecasting\. Models such as recurrent neural networks \(RNNs\)\[[8](https://arxiv.org/html/2606.24347#bib.bib2)\], long short\-term memory networks \(LSTMs\)\[[11](https://arxiv.org/html/2606.24347#bib.bib25)\], gated recurrent units \(GRUs\)\[[6](https://arxiv.org/html/2606.24347#bib.bib26)\], temporal convolutional networks \(TCNs\)\[[2](https://arxiv.org/html/2606.24347#bib.bib27)\], and Transformer\-based architectures have demonstrated strong capabilities in learning nonlinear temporal dependencies directly from historical observations\[[18](https://arxiv.org/html/2606.24347#bib.bib39),[41](https://arxiv.org/html/2606.24347#bib.bib28),[20](https://arxiv.org/html/2606.24347#bib.bib29),[19](https://arxiv.org/html/2606.24347#bib.bib1)\]\. By leveraging large\-scale monitoring data, these methods can automatically extract complex temporal patterns without requiring explicit assumptions about the underlying data distribution\. Nevertheless, most early deep learning approaches primarily focus on temporal modeling at individual stations and often overlook the spatial interactions among monitoring locations\. Since air pollutants can be transported across regions through atmospheric processes, ignoring spatial dependencies may lead to incomplete representations of PM2\.5dynamics\.

To address this issue, spatio\-temporal forecasting models have been proposed to jointly model temporal evolution and spatial interactions\[[17](https://arxiv.org/html/2606.24347#bib.bib24),[35](https://arxiv.org/html/2606.24347#bib.bib5),[23](https://arxiv.org/html/2606.24347#bib.bib6),[39](https://arxiv.org/html/2606.24347#bib.bib43)\]\. These methods integrate temporal neural networks with spatial modules to capture correlations among monitoring stations, thereby improving forecasting performance in regional air\-quality prediction tasks\. Compared with purely temporal models, spatio\-temporal approaches can better characterize pollutant propagation and regional co\-variation patterns\. However, many existing spatio\-temporal methods represent spatial relationships using predefined grids or generic spatial operators, which may not adequately describe the irregular distribution of monitoring stations and the heterogeneous interactions among them\. Consequently, their ability to model realistic inter\-station dependencies remains limited\.

Graph\-based methods provide a natural solution to this challenge by representing monitoring stations as graph nodes and station relationships as graph edges\. Graph neural networks \(GNNs\)\[[13](https://arxiv.org/html/2606.24347#bib.bib31),[27](https://arxiv.org/html/2606.24347#bib.bib32)\]and spatio\-temporal graph neural networks \(STGNNs\)\[[23](https://arxiv.org/html/2606.24347#bib.bib6)\]have achieved remarkable success in air\-quality forecasting because they can flexibly model irregular spatial structures and learn complex dependencies among stations\[[16](https://arxiv.org/html/2606.24347#bib.bib33),[36](https://arxiv.org/html/2606.24347#bib.bib34),[28](https://arxiv.org/html/2606.24347#bib.bib7),[32](https://arxiv.org/html/2606.24347#bib.bib40)\]\. Existing graph construction strategies generally rely on geographic distance, statistical correlation, adaptive graph learning, or combinations thereof\. These approaches effectively capture certain aspects of spatial relationships and have significantly advanced PM2\.5forecasting performance\.

Despite these advances, existing graph\-based PM2\.5forecasting methods still exhibit several limitations\. First, many methods focus primarily on learning spatial dependencies while paying insufficient attention to the intrinsic periodic regularity of PM2\.5evolution\[[30](https://arxiv.org/html/2606.24347#bib.bib38),[37](https://arxiv.org/html/2606.24347#bib.bib41),[25](https://arxiv.org/html/2606.24347#bib.bib42)\]\. In practice, human activities, traffic intensity, industrial production, and meteorological conditions often follow daily and weekly cycles, resulting in stable periodic patterns in PM2\.5concentrations\. Explicitly modeling such periodic regularity can provide strong predictive signals and improve forecasting robustness\.

Second, after periodic patterns are removed, the remaining PM2\.5residuals should not be regarded as random patterns\. These residuals often contain station\-specific temporal dynamics arising from pollutant accumulation, slow dissipation, local emissions, abrupt environmental changes, and interactions among multiple pollutants\. Existing graph\-based methods typically emphasize global spatio\-temporal representations but may underexploit the nonlinear autoregressive characteristics embedded in station\-wise residual sequences\.

Third, pollutant transport is fundamentally influenced by meteorological conditions, especially wind fields\. Pollutants emitted at an upwind station may significantly affect downwind locations, whereas geographically nearby stations may exert limited influence if they are not aligned with the prevailing wind direction\. However, many existing graph construction strategies rely solely on geographic distance, historical correlation, or adaptive learning mechanisms\. Such graphs may capture statistical dependencies but often lack explicit physical guidance regarding meteorology\-driven pollutant dispersion\[[34](https://arxiv.org/html/2606.24347#bib.bib8),[42](https://arxiv.org/html/2606.24347#bib.bib9),[33](https://arxiv.org/html/2606.24347#bib.bib10)\]\.

As illustrated in Fig\.[1](https://arxiv.org/html/2606.24347#S1.F1), these observations suggest that accurate PM2\.5forecasting should simultaneously consider three complementary views: periodic regularity, station\-specific residual dynamics, and meteorology\-driven spatial transport\. From this perspective, the evolution of PM2\.5concentration at stationiican be conceptually expressed as

P​M2\.5i​\(t\)=Pi​\(t\)\+Ri​\(t\)\+Ti​\(t\),PM\_\{2\.5\}^\{i\}\(t\)=P\_\{i\}\(t\)\+R\_\{i\}\(t\)\+T\_\{i\}\(t\),\(1\)
wherePi​\(t\)P\_\{i\}\(t\)denotes the local periodic component,Ri​\(t\)R\_\{i\}\(t\)denotes the station\-wise residual temporal component, andTi​\(t\)T\_\{i\}\(t\)denotes the meteorology\-driven transport component\. This formulation is not intended as a strict physical decomposition of atmospheric processes\. Instead, it provides a forecasting\-oriented multi\-view perspective that highlights three major sources of predictability in PM2\.5evolution\.

The periodic component captures stable temporal regularity induced by recurring human activities and environmental cycles\. The residual component captures local nonlinear dynamics that remain predictable after periodicity is removed\. The transport component captures inter\-station influence caused by pollutant dispersion under changing meteorological conditions\. These three views are complementary rather than redundant\. Ignoring any one of them may result in an incomplete understanding of PM2\.5dynamics and consequently limit forecasting performance\.

Motivated by this observation, we proposeMVG\-KAN, a Multi\-View Geo\-Wind Guided KAN framework for PM2\.5forecasting\. The proposed framework explicitly models PM2\.5evolution from the three perspectives described above\. Specifically, we adopt a periodic\-residual forecasting paradigm\[[25](https://arxiv.org/html/2606.24347#bib.bib42)\]as the backbone to separate stable periodic patterns from non\-periodic variations\. Building upon this foundation, we first develop a Geo\-Wind Graph to model meteorology\-driven pollutant dispersion by integrating geographic distance decay, wind direction, and wind speed into a unified directed graph structure\. Unlike conventional distance\-based or correlation\-based graphs, the proposed Geo\-Wind Graph explicitly incorporates physically meaningful transport information, allowing the model to distinguish between upwind and downwind relationships and better characterize pollutant propagation among monitoring stations\.

Furthermore, we introduce a Temporal Kolmogorov–Arnold Network \(TKAN\) residual head to model station\-wise nonlinear residual dynamics\. Instead of directly learning from raw PM2\.5sequences, TKAN operates on de\-periodized residuals together with historical multi\-pollutant observations, enabling it to focus on local residual inertia, abrupt changes, and pollutant co\-variation patterns\.

The periodic\-residual backbone, Geo\-Wind Graph spatial modeling, and TKAN residual temporal modeling jointly operationalize the proposed multi\-view forecasting perspective\. The periodic branch captures stable regularity, the Geo\-Wind Graph captures meteorology\-guided spatial transport, and TKAN captures station\-specific residual evolution\. Together, they provide a more comprehensive representation of PM2\.5dynamics than existing approaches that focus on only one or two aspects of the forecasting problem\.

The main contributions of this paper are summarized as follows:

1. 1\.To address the insufficient decomposition of PM2\.5evolution mechanisms in existing methods, we propose a Multi\-View Geo\-Wind Guided KAN framework \(MVG\-KAN\), which jointly models local periodic regularity, station\-wise residual temporal dynamics, and meteorology\-driven spatial dispersion\.
2. 2\.To overcome the limitation of distance\-only, correlation\-based, or purely adaptive spatial graphs, we develop a Geo\-Wind Graph that integrates geographic distance decay, wind direction, and wind speed for directed pollutant dispersion modeling\.
3. 3\.To better capture non\-periodic local fluctuations after removing stable periodic patterns, we introduce a TKAN residual head that learns nonlinear autoregressive correction from de\-periodized PM2\.5residuals and historical multi\-pollutant sequences\.
4. 4\.Extensive experiments on the Beijing PM2\.5dataset demonstrate the effectiveness of MVG\-KAN\. The proposed model achieves an MAE of14\.0914\.09and an RMSE of21\.4021\.40, yielding the best average MAE and RMSE among the compared deep learning methods\. Further ablation studies verify the complementary roles of TKAN and the Geo\-Wind Graph, where TKAN improves local residual temporal correction and the Geo\-Wind Graph provides physically meaningful spatial enhancement\.

## IIRelated Work

### II\-APM2\.5and Air Quality Forecasting

PM2\.5forecasting has been studied through statistical time\-series modeling, atmospheric transport modeling, remote\-sensing estimation, classical machine learning, and deep learning\[[38](https://arxiv.org/html/2606.24347#bib.bib18),[1](https://arxiv.org/html/2606.24347#bib.bib19)\]\. Statistical and classical models provide interpretable temporal baselines, but their linear assumptions and handcrafted features are often inadequate for nonlinear emission effects, meteorological dispersion, and multi\-station dependencies\[[40](https://arxiv.org/html/2606.24347#bib.bib23),[15](https://arxiv.org/html/2606.24347#bib.bib22)\]\. Chemical transport and atmospheric dispersion systems, such as HYSPLIT, explicitly represent physical processes and trajectory information\[[26](https://arxiv.org/html/2606.24347#bib.bib12)\]\. Remote\-sensing\-based PM2\.5estimation studies further combine aerosol observations, chemical transport modeling, meteorological information, and ground monitoring to characterize spatial pollution fields\[[10](https://arxiv.org/html/2606.24347#bib.bib13),[29](https://arxiv.org/html/2606.24347#bib.bib14),[5](https://arxiv.org/html/2606.24347#bib.bib15)\]\. These studies provide strong environmental motivation, but they are often designed for physical simulation or spatial estimation rather than lightweight station\-level multi\-step forecasting\.

Data\-driven air\-quality models reduce the dependence on explicit physical simulation by learning from observed pollutant, meteorological, and station\-context variables\. Hybrid trajectory\-wavelet models, urban air\-quality inference, DeepAir, distributed fusion networks, GeoMAN, and recent DeepPM2\.5forecasting systems all demonstrate the value of heterogeneous predictors for fine\-grained PM2\.5prediction\[[9](https://arxiv.org/html/2606.24347#bib.bib11),[40](https://arxiv.org/html/2606.24347#bib.bib23),[24](https://arxiv.org/html/2606.24347#bib.bib4),[35](https://arxiv.org/html/2606.24347#bib.bib5),[17](https://arxiv.org/html/2606.24347#bib.bib24),[39](https://arxiv.org/html/2606.24347#bib.bib43)\]\. However, richer inputs alone do not guarantee that a model distinguishes directional pollutant transport from local temporal evolution\. This motivates a station\-level forecasting framework that explicitly separates local periodic regularity, local residual dynamics, and wind\-guided spatial transport\.

### II\-BSpatio\-Temporal Graph Learning for Environmental Prediction

Multi\-station air\-quality forecasting is naturally related to spatio\-temporal graph learning, where monitoring stations are graph nodes and edges encode spatial dependence, transport pathways, or statistical similarity\. Graph convolution, graph attention, diffusion\-style propagation, and spatio\-temporal graph convolution provide common tools for irregular sensor networks\[[13](https://arxiv.org/html/2606.24347#bib.bib31),[27](https://arxiv.org/html/2606.24347#bib.bib32),[16](https://arxiv.org/html/2606.24347#bib.bib33),[36](https://arxiv.org/html/2606.24347#bib.bib34)\]\. In PM2\.5forecasting, graph convolution with recurrent temporal modeling and domain\-knowledge\-enhanced graph construction have been used to capture station relationships and environmental priors\[[23](https://arxiv.org/html/2606.24347#bib.bib6),[28](https://arxiv.org/html/2606.24347#bib.bib7)\]\. Adaptive graph learning, represented by Graph WaveNet, further allows latent dependencies to be inferred when fixed adjacency information is incomplete\[[32](https://arxiv.org/html/2606.24347#bib.bib40)\]\.

Despite these advances, common distance\-based, correlation\-based, and adaptive graphs may not sufficiently reflect pollutant transport\. PM2\.5movement depends not only on proximity or historical similarity, but also on wind direction, wind speed, and upwind/downwind geometry\. A nearby station may be weakly related if it is not aligned with transport direction, whereas an upwind station can be influential even when it is not the nearest neighbor\. Wind\-aware forecasting studies have considered dynamic wind\-field distance, wind\-conditioned graph convolution, and directed dynamic graph structures\[[34](https://arxiv.org/html/2606.24347#bib.bib8),[42](https://arxiv.org/html/2606.24347#bib.bib9),[33](https://arxiv.org/html/2606.24347#bib.bib10)\]\. These works indicate that wind variables should shape spatial propagation rather than serve only as auxiliary node features\.

The present work follows this transport\-aware perspective\. Instead of relying on a generic station graph, it introduces a Geo\-Wind Graph to provide a lightweight directed spatial prior based on geographic proximity and wind\-driven transport tendency\. The graph is intended to guide residual spatial propagation in a physically motivated manner, not to replace chemical transport or atmospheric dispersion models\.

### II\-CDecomposition and Residual Modeling for Time Series Forecasting

Time\-series decomposition is widely used when observations contain both regular temporal structure and irregular fluctuations\. Environmental sequences often exhibit daily or weekly cycles caused by human activity, meteorological variation, and regional background conditions\. In PM2\.5forecasting, trajectory\-based geographic features and wavelet transformation show that separating temporal scales can help model pollution episodes\[[9](https://arxiv.org/html/2606.24347#bib.bib11)\]\. In broader time\-series forecasting, decomposition\-based Transformer models and simple linear temporal mappings also demonstrate that explicit seasonal or residual representations can be competitive\[[30](https://arxiv.org/html/2606.24347#bib.bib38),[37](https://arxiv.org/html/2606.24347#bib.bib41)\]\. Recent periodic\-residual frameworks, such as HyperD, further decouple stable periodic patterns from non\-periodic residual components\[[25](https://arxiv.org/html/2606.24347#bib.bib42)\]\.

For station\-level PM2\.5prediction, however, the residual after removing periodicity should not be treated as unstructured noise\. It may retain pollutant persistence, accumulation, slow dissipation, local emission changes, and co\-variation among PM2\.5, PM10, NO2, CO, SO2, O3, and AQI\. Multi\-source air\-quality models have shown that pollutant and meteorological variables provide complementary evidence for future concentration changes\[[24](https://arxiv.org/html/2606.24347#bib.bib4),[35](https://arxiv.org/html/2606.24347#bib.bib5)\]\. Nevertheless, existing decomposition and residual branches often emphasize global temporal representations or frequency\-domain alignment, while station\-wise nonlinear residual inertia and multi\-pollutant co\-variation remain under\-modeled\.

The TKAN residual temporal head is introduced to address this local residual modeling gap\. Motivated by Kolmogorov–Arnold networks\[[21](https://arxiv.org/html/2606.24347#bib.bib37)\], it focuses on station\-wise residual correction from de\-periodized PM2\.5and recent multi\-pollutant histories\. It is complementary to the Geo\-Wind Graph: TKAN targets local temporal residual dynamics, whereas the graph module targets wind\-guided inter\-station transport\.

![Refer to caption](https://arxiv.org/html/2606.24347v1/figures/framework_0616.png)Figure 2:Overall framework of the geo\-wind guided KAN residual temporal learning model\. \(a\) Overall framework\. The input pollutant and meteorological sequences are decomposed into periodic and residual patterns\. The periodic branch models regular temporal variations, while the residual branch is refined by the proposed Geo\-Wind Graph and TKAN residual head to represent wind\-driven spatial transportTi​\(t\)T\_\{i\}\(t\)and local residual dynamicsRi​\(t\)R\_\{i\}\(t\), respectively\. The final PM2\.5forecast is reconstructed by combining the periodic component with the refined residual component\. \(b\) Geo\-Wind Graph module\. A dynamic directed transport graph is constructed by combining geographic proximity with wind\-direction\- and wind\-speed\-aware modulation, providing a physically motivated spatial prior for pollutant propagation\. \(c\) TKAN residual head\. A station\-wise temporal KAN maps de\-periodized residual and pollutant histories to nonlinear residual corrections, strengthening local temporal residual modeling after periodic effects are removed\.

## IIIMethod

### III\-AProblem Formulation

Let𝐗∈ℝT×N×C\\mathbf\{X\}\\in\\mathbb\{R\}^\{T\\times N\\times C\}denote a multi\-station and multi\-variate historical observation window, whereTTis the input sequence length,NNis the number of PM2\.5monitoring stations, andCCis the number of value features\. In the current implementation,T=48T=48,N=39N=39, andC=12C=12for the final full\-feature configuration\. The value features are AQI, CO, NO2, O3, PM10, SO2, dew\-point temperature, ground pressure, temperature, wind direction, wind speed, and PM2\.5\. In addition to these value features, the model receives hour\-of\-day and day\-of\-week indicators for periodic indexing\. The target is to predict the future PM2\.5sequence

𝐘∈ℝH×N,\\mathbf\{Y\}\\in\\mathbb\{R\}^\{H\\times N\},\(2\)whereHHis the prediction horizon\. The implemented setting usesH=48H=48\. For a mini\-batch, we write the historical value tensor as𝐗∈ℝB×T×N×C\\mathbf\{X\}\\in\\mathbb\{R\}^\{B\\times T\\times N\\times C\}and the PM2\.5target as𝐘∈ℝB×H×N\\mathbf\{Y\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}\. The forecasting objective is short\-term station\-level PM2\.5prediction; the output channel is PM2\.5only\.

### III\-BOverall Framework

The proposed framework is built on a periodic\-residual forecasting backbone\[[25](https://arxiv.org/html/2606.24347#bib.bib42)\]\. The backbone contains a periodic representation branch and a residual representation branch\. The periodic branch uses temporal indices to obtain an input\-period periodic pattern𝐒i​n∈ℝB×T×N\\mathbf\{S\}^\{in\}\\in\\mathbb\{R\}^\{B\\times T\\times N\}and a future periodic pattern𝐒o​u​t∈ℝB×H×N\\mathbf\{S\}^\{out\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}\. Let𝐗p∈ℝB×T×N\\mathbf\{X\}^\{p\}\\in\\mathbb\{R\}^\{B\\times T\\times N\}denote the PM2\.5channel selected from𝐗\\mathbf\{X\}\. The input residual is

𝐑i​n=𝐗p−𝐒i​n\.\\mathbf\{R\}^\{in\}=\\mathbf\{X\}^\{p\}\-\\mathbf\{S\}^\{in\}\.\(3\)The residual branch models the non\-periodic part of the sequence\. In our framework, two PM2\.5\-oriented modules are added to this branch, as shown in Fig\.[2](https://arxiv.org/html/2606.24347#S2.F2)\.\(a\)\. First, a Geo\-Wind Graph constructs a dynamic directed adjacency tensor𝐀G​W∈ℝB×T×N×N\\mathbf\{A\}\_\{GW\}\\in\\mathbb\{R\}^\{B\\times T\\times N\\times N\}from station coordinates and wind variables\. This module models wind\-driven spatial transport of non\-periodic PM2\.5residual disturbances across stations\. Second, a TKAN residual head is applied after the residual output to provide station\-wise residual correction from de\-periodized PM2\.5and historical pollutant variables\.

Let𝐑^∈ℝB×H×N\\widehat\{\\mathbf\{R\}\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}denote the final residual prediction after Geo\-Wind Graph enhancement and TKAN residual correction\. Specifically, the Geo\-Wind Graph provides a dynamic spatial correction for the transport\-related residual component𝐓i​\(t\)\\mathbf\{T\}\_\{i\}\(t\), while the TKAN residual head captures station\-wise nonlinear temporal residual dynamics𝐑i​\(t\)\\mathbf\{R\}\_\{i\}\(t\)\. The final PM2\.5prediction𝐘^\\widehat\{\\mathbf\{Y\}\}is obtained by adding the future periodic component𝐒o​u​t\\mathbf\{S\}^\{out\}and the final residual prediction:

𝐘^=𝐒o​u​t\+𝐑^\.\\widehat\{\\mathbf\{Y\}\}=\\mathbf\{S\}^\{out\}\+\\widehat\{\\mathbf\{R\}\}\.\(4\)Here,𝐒o​u​t∈ℝB×H×N\\mathbf\{S\}^\{out\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}represents the predicted daily\-weekly periodic component, and𝐑^\\widehat\{\\mathbf\{R\}\}represents the non\-periodic residual component refined by spatial transport modeling and local temporal correction\. Therefore, the final prediction decomposes PM2\.5into an explicit periodic part and a learned residual part\.

### III\-CGeo\-Wind Guided Graph Construction

PM2\.5dependence among monitoring stations is affected by both geographic proximity and wind\-driven transport\. As visualized in Fig\.[3](https://arxiv.org/html/2606.24347#S3.F3)and Fig\.[2](https://arxiv.org/html/2606.24347#S2.F2)\.\(b\), the Beijing monitoring stations form an irregular spatial network in which plausible transport edges should depend on station locations and downwind alignment rather than distance alone\. The graph of PM2\.5should therefore not be treated as a simple static distance graph or a purely data\-driven correlation graph, but as a physically informed transport graph that explicitly models wind\-driven pollutant propagation between monitoring stations\. The implemented Geo\-Wind Graph constructs a directed adjacency tensor in which𝐀G​W​\[b,t,i,j\]\\mathbf\{A\}\_\{GW\}\[b,t,i,j\]represents the strength of pollutant propagation from source stationiito target stationjjat timettin batch samplebb\.

#### III\-C1Geographic Proximity

For each stationii, let\(λi,ϕi\)\(\\lambda\_\{i\},\\phi\_\{i\}\)denote longitude and latitude in radians\. We precompute the haversine distance between stationsiiandjjas

ai​j=sin2⁡\(ϕj−ϕi2\)\+cos⁡ϕi​cos⁡ϕj​sin2⁡\(λj−λi2\),a\_\{ij\}=\\sin^\{2\}\\left\(\\frac\{\\phi\_\{j\}\-\\phi\_\{i\}\}\{2\}\\right\)\+\\cos\\phi\_\{i\}\\cos\\phi\_\{j\}\\sin^\{2\}\\left\(\\frac\{\\lambda\_\{j\}\-\\lambda\_\{i\}\}\{2\}\\right\),\(5\)di​j=2​Re​atan2⁡\(ai​j,1−ai​j\),d\_\{ij\}=2R\_\{e\}\\operatorname\{atan2\}\\left\(\\sqrt\{a\_\{ij\}\},\\sqrt\{1\-a\_\{ij\}\}\\right\),\(6\)whereRe=6371R\_\{e\}=6371km\. It also precomputes the bearingβi​j\\beta\_\{ij\}from stationiito stationjj:

βi​j=atan2\(\\displaystyle\\beta\_\{ij\}=\\operatorname\{atan2\}\\big\(sin⁡\(λj−λi\)​cos⁡ϕj,\\displaystyle\\sin\(\\lambda\_\{j\}\-\\lambda\_\{i\}\)\\cos\\phi\_\{j\},\(7\)cosϕisinϕj−sinϕicosϕjcos\(λj−λi\)\),\\displaystyle\\cos\\phi\_\{i\}\\sin\\phi\_\{j\}\-\\sin\\phi\_\{i\}\\cos\\phi\_\{j\}\\cos\(\\lambda\_\{j\}\-\\lambda\_\{i\}\)\\big\),wrapped to\[0,2​π\)\[0,2\\pi\)\. The geographic graph is initialized by an exponential distance kernel,

A~i​jg​e​o=\{exp⁡\(−di​j/τd\),i≠j,0,i=j,\\widetilde\{A\}^\{geo\}\_\{ij\}=\\begin\{cases\}\\exp\(\-d\_\{ij\}/\\tau\_\{d\}\),&i\\neq j,\\\\ 0,&i=j,\\end\{cases\}\(8\)where the final configuration usesτd=50\.0\\tau\_\{d\}=50\.0\. A top\-kkoperation keeps the largestkkoutgoing neighbors for each source station, followed by row normalization:

Ai​jg​e​o=Mi​j​A~i​jg​e​o∑ℓ=1NMi​ℓ​A~i​ℓg​e​o\+ϵ,A^\{geo\}\_\{ij\}=\\frac\{M\_\{ij\}\\widetilde\{A\}^\{geo\}\_\{ij\}\}\{\\sum\_\{\\ell=1\}^\{N\}M\_\{i\\ell\}\\widetilde\{A\}^\{geo\}\_\{i\\ell\}\+\\epsilon\},\(9\)whereMi​j∈\{0,1\}M\_\{ij\}\\in\\\{0,1\\\}is the top\-kkmask,k=8k=8in the final configuration, andϵ\\epsilonis a small constant for numerical stability\.

#### III\-C2Wind Graph

The wind graph uses wind direction and wind speed from each source station, as shown in Fig\.[2](https://arxiv.org/html/2606.24347#S2.F2)\.\(b\)\. For degree\-valued input, the downwind transport directionθb​t​id​o​w​n\\theta^\{down\}\_\{bti\}is

θb​t​id​o​w​n=deg2rad⁡\(\(θb​t​iw​i​n​d\+180\)mod360\),\\theta^\{down\}\_\{bti\}=\\operatorname\{deg2rad\}\\left\(\(\\theta^\{wind\}\_\{bti\}\+180\)\\bmod 360\\right\),\(10\)whereθb​t​iw​i​n​d\\theta^\{wind\}\_\{bti\}is the recorded wind direction\.

The directional alignment between the downwind direction at source stationiiand the bearing fromiitojjis

Db​t​i​j=\[ReLU⁡\(cos⁡\(βi​j−θb​t​id​o​w​n\)\)\]pd,D\_\{btij\}=\\left\[\\operatorname\{ReLU\}\\left\(\\cos\(\\beta\_\{ij\}\-\\theta^\{down\}\_\{bti\}\)\\right\)\\right\]^\{p\_\{d\}\},\(11\)whereDb​t​i​jD\_\{btij\}measures the likelihood that pollutants emitted or accumulated at stationiican be transported toward stationjjunder the current wind conditions\. It acts as a directional transport gate that favors downwind connections and suppresses upwind or crosswind interactions,pd=2\.0p\_\{d\}=2\.0in the final configuration\. A wind\-speed gate is computed as

Gb​t​i=σ​\(vb​t​i−v0sv\+ϵ\),G\_\{bti\}=\\sigma\\left\(\\frac\{v\_\{bti\}\-v\_\{0\}\}\{s\_\{v\}\+\\epsilon\}\\right\),\(12\)wherevb​t​iv\_\{bti\}is the wind speed,v0=1\.0v\_\{0\}=1\.0, andsv=0\.5s\_\{v\}=0\.5in the final configuration\. Theσ​\(\)\\sigma\(\)gate converts wind speed into a smooth transport\-strength coefficient in the range\(0,1\)\(0,1\)\. When the wind speed is lower thanv0v\_\{0\}, the gate value is suppressed, indicating limited pollutant transport\. As wind speed increases beyondv0v\_\{0\}, the gate gradually approaches 1, allowing stronger wind\-driven spatial interactions\. For travel\-time decay, the code first converts the wind speed into an effective transport velocity and enforces a lower bound to avoid unrealistically large travel times under near\-calm conditions:

v~b​t​i=max⁡\(vb​t​i,vm​i​n\)×3\.6,\\widetilde\{v\}\_\{bti\}=\\max\(v\_\{bti\},v\_\{min\}\)\\times 3\.6,\(13\)wherevm​i​n=0\.2v\_\{min\}=0\.2\. The factor3\.63\.6converts wind speed from m/s to km/h, making it consistent with the distance unit used in the graph construction\.

The estimated travel time required for pollutants to move from stationiito stationjjis then approximated as

Tb​t​i​jt​r​a​v​e​l=di​jv~b​t​i\+ϵ,T^\{travel\}\_\{btij\}=\\frac\{d\_\{ij\}\}\{\\widetilde\{v\}\_\{bti\}\+\\epsilon\},\(14\)wheredi​jd\_\{ij\}denotes the geographic distance between the two stations\. Larger distances or weaker winds result in longer travel times\.

To model the gradual attenuation of transport influence over time, an exponential decay function is applied:

Qb​t​i​j=exp⁡\(−Tb​t​i​jt​r​a​v​e​lτt\+ϵ\),Q\_\{btij\}=\\exp\\left\(\-\\frac\{T^\{travel\}\_\{btij\}\}\{\\tau\_\{t\}\+\\epsilon\}\\right\),\(15\)whereτt=6\.0\\tau\_\{t\}=6\.0controls the characteristic transport timescale\. The decay factorQb​t​i​j∈\(0,1\]Q\_\{btij\}\\in\(0,1\]assigns larger weights to station pairs that can be connected more rapidly by the current wind field, while suppressing interactions requiring excessively long transport times\. Consequently, nearby stations under strong winds retain strong connections, whereas distant stations or stations connected through weak winds receive smaller weights\.

The wind\-directed graph is then constructed by combining geographic proximity, directional consistency, wind\-speed strength, and travel\-time feasibility:

A~b​t​i​jw​i​n​d=Ai​jg​e​o​Db​t​i​j​Gb​t​i​Qb​t​i​j,\\widetilde\{A\}^\{wind\}\_\{btij\}=A^\{geo\}\_\{ij\}\\,D\_\{btij\}\\,G\_\{bti\}\\,Q\_\{btij\},\(16\)whereAi​jg​e​oA^\{geo\}\_\{ij\}provides the distance\-based spatial prior,Db​t​i​jD\_\{btij\}favors station pairs aligned with the downwind direction,Gb​t​iG\_\{bti\}controls whether the wind at the source station is strong enough to support transport, andQb​t​i​jQ\_\{btij\}suppresses connections whose estimated travel time is too long\.

The raw wind\-directed graph is further sparsified and normalized using the same top\-kkrow normalization as in \([9](https://arxiv.org/html/2606.24347#S3.E9)\)\. This ensures that each station only aggregates information from the most physically plausible upstream or downstream neighbors\.

Since both the downwind directionθb​t​id​o​w​n\\theta^\{down\}\_\{bti\}and wind speedvb​t​iv\_\{bti\}depend on the source stationiiand time steptt, the resulting wind graph is dynamic and directed\. In general, the transport influence from stationiito stationjjdiffers from that from stationjjto stationii, i\.e\.,

Ab​t​i​jw​i​n​d≠Ab​t​j​iw​i​n​d\.A^\{wind\}\_\{btij\}\\neq A^\{wind\}\_\{btji\}\.This asymmetry is consistent with wind\-driven pollutant transport, because pollutants are more likely to propagate along the downwind direction rather than equally in both directions\.

#### III\-C3Graph Fusion

The final confirmed Geo\-Wind configuration uses two graphs,𝐀g​e​o\\mathbf\{A\}^\{geo\}and𝐀w​i​n​d\\mathbf\{A\}^\{wind\}\. They are fused by learnable softmax weights:

αm=exp⁡\(lm\)∑qexp⁡\(lq\),𝐀~G​W=∑m∈\{g​e​o,w​i​n​d\}αm​𝐀m,\\alpha\_\{m\}=\\frac\{\\exp\(l\_\{m\}\)\}\{\\sum\_\{q\}\\exp\(l\_\{q\}\)\},\\qquad\\widetilde\{\\mathbf\{A\}\}\_\{GW\}=\\sum\_\{m\\in\\\{geo,wind\\\}\}\\alpha\_\{m\}\\mathbf\{A\}^\{m\},\(17\)wherelml\_\{m\}are learnable graph logits\. The fused graph is again sparsified by top\-kkrow normalization, passed through dropout, and normalized once more\. The final graph is𝐀G​W∈ℝB×T×N×N\\mathbf\{A\}\_\{GW\}\\in\\mathbb\{R\}^\{B\\times T\\times N\\times N\}\.

#### III\-C4Graph Convolution Based on Geo\-Wind Graph

The Geo\-Wind graph is inserted into the residual branch to provide wind\-aware spatial correction before temporal residual forecasting\. Given the historical PM2\.5residual𝐑i​n∈ℝB×T×N\\mathbf\{R\}^\{in\}\\in\\mathbb\{R\}^\{B\\times T\\times N\}, which is calculated by equation\.[3](https://arxiv.org/html/2606.24347#S3.E3), we first normalize and project it into a hidden feature space:

𝐇\(0\)=fenc​\(Norm⁡\(𝐑i​n\)\),\\mathbf\{H\}^\{\(0\)\}=f\_\{\\mathrm\{enc\}\}\\left\(\\operatorname\{Norm\}\\left\(\\mathbf\{R\}^\{in\}\\right\)\\right\),\(18\)where𝐇\(0\)∈ℝB×T×N×C\\mathbf\{H\}^\{\(0\)\}\\in\\mathbb\{R\}^\{B\\times T\\times N\\times C\},CCdenotes the hidden dimension, andfencf\_\{\\mathrm\{enc\}\}dotes an MLP encoder\. In the final configuration, only the PM2\.5residual is used as the node feature for graph convolution, while meteorological variables are used to construct the dynamic Geo\-Wind adjacency𝐀G​W\\mathbf\{A\}\_\{GW\}\.

For the directed adjacency𝐀G​W\\mathbf\{A\}\_\{GW\}, the entryAG​W,b​t​i​jA\_\{GW,btij\}denotes the transport weight from source stationiito target stationjjat batchbband time steptt\. A largerAG​W,b​t​i​jA\_\{GW,btij\}means that the residual information at stationiiis more likely to influence stationjjunder the current wind field\. Thekk\-th order forward diffusion is defined as

𝐇f,b​t​j\(k\)=∑i=1NAG​W,b​t​i​j​𝐇f,b​t​i\(k−1\),\\mathbf\{H\}^\{\(k\)\}\_\{f,btj\}=\\sum\_\{i=1\}^\{N\}A\_\{GW,btij\}\\mathbf\{H\}^\{\(k\-1\)\}\_\{f,bti\},\(19\)where𝐇f,b​t​j\(k\)\\mathbf\{H\}^\{\(k\)\}\_\{f,btj\}is the forward aggregated representation of target stationjj\. This operation updates stationjjby collecting residual features from source stations according to their wind\-directed transport weights\. Therefore, stations that are geographically close, aligned with the downwind direction, and reachable within a reasonable travel time contribute more strongly to the representation of stationjj\.

Since the Geo\-Wind graph is directed, the transport influence fromiitojjis generally different from that fromjjtoii\. To model reverse directional dependencies, backward diffusion is computed using the transposed adjacency𝐀G​W⊤\\mathbf\{A\}\_\{GW\}^\{\\top\}:

𝐇b,b​t​i\(k\)=∑j=1NAG​W,b​t​i​j​𝐇b,b​t​j\(k−1\)\.\\mathbf\{H\}^\{\(k\)\}\_\{b,bti\}=\\sum\_\{j=1\}^\{N\}A\_\{GW,btij\}\\mathbf\{H\}^\{\(k\-1\)\}\_\{b,btj\}\.\(20\)Thus, forward diffusion captures information propagated along the wind\-directed graph, while backward diffusion𝐇b,b​t​i\(k\)\\mathbf\{H\}^\{\(k\)\}\_\{b,bti\}captures information propagated along the reverse direction\.

With graph orderK=2K=2, the graph convolution considers both one\-hop and two\-hop spatial diffusion\. The original residual feature, two forward diffusion supports, and two backward diffusion supports are concatenated as

𝐙b​t=Concat⁡\(𝐇b​t\(0\),𝐇f,b​t\(1\),𝐇f,b​t\(2\),𝐇b,b​t\(1\),𝐇b,b​t\(2\)\)\.\\mathbf\{Z\}\_\{bt\}=\\operatorname\{Concat\}\\left\(\\mathbf\{H\}^\{\(0\)\}\_\{bt\},\\mathbf\{H\}^\{\(1\)\}\_\{f,bt\},\\mathbf\{H\}^\{\(2\)\}\_\{f,bt\},\\mathbf\{H\}^\{\(1\)\}\_\{b,bt\},\\mathbf\{H\}^\{\(2\)\}\_\{b,bt\}\\right\)\.\(21\)The first\-order supports capture direct wind\-driven interactions between stations, while the second\-order supports capture indirect interactions through intermediate stations\. The concatenated representation is then passed through a linear projection, GELU activation, dropout, and a residual connection to obtain the graph\-enhanced hidden𝐇g​r​a​p​h\\mathbf\{H\}^\{graph\}representation:

𝐇g​r​a​p​h=𝐇\(0\)\+Dropout⁡\(GELU⁡\(Linear⁡\(𝐙\)\)\)\.\\mathbf\{H\}^\{graph\}=\\mathbf\{H\}^\{\(0\)\}\+\\operatorname\{Dropout\}\\left\(\\operatorname\{GELU\}\\left\(\\operatorname\{Linear\}\\left\(\\mathbf\{Z\}\\right\)\\right\)\\right\)\.\(22\)
Finally, the graph\-enhanced hidden representation is decoded back to the residual space:

Δ​𝐑g​r​a​p​h=fdec​\(𝐇g​r​a​p​h\),\\Delta\\mathbf\{R\}^\{graph\}=f\_\{\\mathrm\{dec\}\}\\left\(\\mathbf\{H\}^\{graph\}\\right\),\(23\)whereΔ​𝐑g​r​a​p​h∈ℝB×T×N\\Delta\\mathbf\{R\}^\{graph\}\\in\\mathbb\{R\}^\{B\\times T\\times N\}andfdecf\_\{\\mathrm\{dec\}\}dotes an MLP decoder\. This decoded graph output is used as a correction to the historical residual input:

𝐑~i​n=𝐑i​n\+γg​Δ​𝐑g​r​a​p​h,\\widetilde\{\\mathbf\{R\}\}^\{in\}=\\mathbf\{R\}^\{in\}\+\\gamma\_\{g\}\\Delta\\mathbf\{R\}^\{graph\},\(24\)whereγg\\gamma\_\{g\}is a learnable gate initialized to0\.00\.0in the final configuration\. This initialization makes the model start from the original residual branch and gradually learn whether and how strongly the wind\-aware graph correction should be used\.

The graph\-enhanced residual sequence𝐑~i​n\\widetilde\{\\mathbf\{R\}\}^\{in\}is then fed into the residual temporal forecasting module, named Spatio\-Temporal Frequency Encoder, inherited from the baseline branch\. This module models temporal dependencies in the enhanced historical residual sequence and produces the preliminary future residual prediction𝐑o​u​t∈ℝB×H×N\\mathbf\{R\}^\{out\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}\. The output𝐑o​u​t\\mathbf\{R\}^\{out\}is further combined with the station\-wise TKAN residual correction introduced in the next subsection to obtain the final residual prediction𝐑^\\widehat\{\\mathbf\{R\}\}\. Therefore, the proposed graph convolution does not replace the temporal residual predictor; instead, it provides a dynamic wind\-aware spatial correction before future residual forecasting, while TKAN further refines the predicted residuals through local nonlinear temporal correction\.

![Refer to caption](https://arxiv.org/html/2606.24347v1/x1.png)Figure 3:Geo\-Wind Graph illustration\. Blue nodes denote the 39 PM2\.5monitoring stations, and orange arrows show representative directed transport edges constructed from geographic proximity and downwind alignment\. The upper\-left arrow indicates the downwind transport direction used for visualization\.

### III\-DTKAN Residual Temporal Module

The Geo\-Wind Graph models wind\-aware spatial propagation among monitoring stations, but it does not explicitly capture the nonlinear temporal evolution of the residual sequence at each individual station\. After removing the daily and weekly periodic components, the remaining PM2\.5residual may still contain short\-term persistence, pollutant accumulation, slow dissipation, abrupt fluctuations, and co\-variation with other pollutants\. Therefore, we introduce a TKAN residual temporal module as a station\-wise nonlinear correction head for future residual prediction, as shown in Fig\.[2](https://arxiv.org/html/2606.24347#S2.F2)\.\(c\)\.

Let𝐗t​k​a​n∈ℝB×T×N×Cp\\mathbf\{X\}^\{tkan\}\\in\\mathbb\{R\}^\{B\\times T\\times N\\times C\_\{p\}\}denote the input tensor of the TKAN module, whereCpC\_\{p\}is the number of selected pollutant channels\. In the final configuration, we use seven pollutant\-related variables, including PM2\.5, AQI, PM10, NO2, CO, SO2, and O3, soCp=7C\_\{p\}=7\. To avoid duplicating the periodic modeling branch, the PM2\.5channel is replaced by its de\-periodized residual:

rb​t​ip=xb​t​ip−Sb​t​ii​n,r^\{p\}\_\{bti\}=x^\{p\}\_\{bti\}\-S^\{in\}\_\{bti\},\(25\)wherexb​t​ipx^\{p\}\_\{bti\}denotes the raw PM2\.5observation at stationiiand time steptt, andSb​t​ii​nS^\{in\}\_\{bti\}is the learned periodic component\. This replacement forces TKAN to focus on residual temporal dynamics rather than relearning daily or weekly periodic patterns\.

For each stationii, the local historical sequence is processed independently\. Specifically, the temporal window and pollutant channels are flattened into a station\-wise vector:

𝐳b​i=Flatten⁡\(𝐗b,:,i,:t​k​a​n\)∈ℝT​Cp\.\\mathbf\{z\}\_\{bi\}=\\operatorname\{Flatten\}\\left\(\\mathbf\{X\}^\{tkan\}\_\{b,:,i,:\}\\right\)\\in\\mathbb\{R\}^\{TC\_\{p\}\}\.\(26\)In implementation, all station\-wise vectors are reshaped from\[B,N,T​Cp\]\[B,N,TC\_\{p\}\]to\[B​N,T​Cp\]\[BN,TC\_\{p\}\]so that the same TKAN head is shared across stations\. This design enables parameter sharing while preserving station\-wise independent temporal modeling\.

Following the KAN formulation\[[21](https://arxiv.org/html/2606.24347#bib.bib37)\], the two\-layer TKAN head is formulated as

𝐡b​i=KAN1⁡\(𝐳b​i;g,o\),\\mathbf\{h\}\_\{bi\}=\\operatorname\{KAN\}\_\{1\}\\left\(\\mathbf\{z\}\_\{bi\};g,o\\right\),\(27\)Δ​𝐘^b​i=KAN2⁡\(Dropout⁡\(𝐡b​i\);g,o\),\\Delta\\widehat\{\\mathbf\{Y\}\}\_\{bi\}=\\operatorname\{KAN\}\_\{2\}\\left\(\\operatorname\{Dropout\}\\left\(\\mathbf\{h\}\_\{bi\}\\right\);g,o\\right\),\(28\)whereggandoodenote the spline grid size and spline order, respectively\. In the final configuration,g=3g=3,o=3o=3, and the hidden size is 32\. The outputΔ​𝐘^b​i∈ℝH\\Delta\\widehat\{\\mathbf\{Y\}\}\_\{bi\}\\in\\mathbb\{R\}^\{H\}represents the TKAN\-predicted residual correction for stationiiover the future horizon\. After reshaping, we obtainΔ​𝐘^∈ℝB×H×N\\Delta\\widehat\{\\mathbf\{Y\}\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}\.

The TKAN output is not treated as an independent forecast\. Instead, it is used as a gated correction to the residual prediction produced by the main residual forecasting branch:

𝐑^=𝐑o​u​t\+γt​Δ​𝐘^,\\widehat\{\\mathbf\{R\}\}=\\mathbf\{R\}^\{out\}\+\\gamma\_\{t\}\\Delta\\widehat\{\\mathbf\{Y\}\},\(29\)where𝐑o​u​t∈ℝB×H×N\\mathbf\{R\}^\{out\}\\in\\mathbb\{R\}^\{B\\times H\\times N\}denotes the residual output after Geo\-Wind graph module and before TKAN correction, andγt\\gamma\_\{t\}is a learnable gate initialized to0\.050\.05\. This small initialization allows the model to start from the main residual predictor and gradually learn the contribution of TKAN during training\. The corrected residual prediction𝐑^\\widehat\{\\mathbf\{R\}\}is then added to the future periodic component𝐒o​u​t\\mathbf\{S\}^\{out\}to obtain the final PM2\.5prediction, as defined in Equation\. \([4](https://arxiv.org/html/2606.24347#S3.E4)\)\.

Overall, TKAN serves as a station\-wise nonlinear autoregressive residual correction module\. It does not use graph information and does not perform spatial propagation\. Therefore, it is complementary to the Geo\-Wind Graph: the Geo\-Wind Graph captures dynamic spatial transport under meteorological conditions, while TKAN captures local nonlinear temporal residual dynamics\.

### III\-ETraining Objective

The model is trained using a horizon\-aware masked mean absolute error \(MAE\) loss\. Letℋ=\{1,3,6,12,24,48\}\\mathcal\{H\}=\\\{1,3,6,12,24,48\\\}denote the set of evaluation horizons, and letYb​h​iY\_\{bhi\}andY^b​h​i\\widehat\{Y\}\_\{bhi\}represent the ground\-truth and predicted PM2\.5values for samplebb, horizonhh, and stationii, respectively\. Since missing values are encoded as0\.00\.0in the dataset, a binary maskMb​h​iM\_\{bhi\}is defined as

Mb​h​i=\{1,Yb​h​i≠0,0,Yb​h​i=0\.M\_\{bhi\}=\\begin\{cases\}1,&Y\_\{bhi\}\\neq 0,\\\\ 0,&Y\_\{bhi\}=0\.\\end\{cases\}\(30\)The prediction loss is computed only on valid observations and selected evaluation horizons:

ℒp​r​e​d=∑b,h∈ℋ,iMb​h​i​\|Y^b​h​i−Yb​h​i\|∑b,h∈ℋ,iMb​h​i\+ϵ,\\mathcal\{L\}\_\{pred\}=\\frac\{\\sum\_\{b,h\\in\\mathcal\{H\},i\}M\_\{bhi\}\\left\|\\widehat\{Y\}\_\{bhi\}\-Y\_\{bhi\}\\right\|\}\{\\sum\_\{b,h\\in\\mathcal\{H\},i\}M\_\{bhi\}\+\\epsilon\},\(31\)whereϵ\\epsilonis a small constant to avoid division by zero\. This design ensures that invalid records do not contribute to optimization and that the training objective remains consistent with the reported forecasting horizons\. The backbone also returns an auxiliary frequency alignment loss inherited from the periodic\-residual framework\[[25](https://arxiv.org/html/2606.24347#bib.bib42)\]\. Specifically, the prediction is decomposed by FFT into low\-frequency and high\-frequency components, and the periodic output and residual output are aligned with these components by mean squared error\. Letℒl​o​w\\mathcal\{L\}\_\{low\}andℒh​i​g​h\\mathcal\{L\}\_\{high\}denote these two inherited losses\. The training objective is

ℒ=ℒp​r​e​d\+α​\(ℒl​o​w\+ℒh​i​g​h\),\\mathcal\{L\}=\\mathcal\{L\}\_\{pred\}\+\\alpha\\left\(\\mathcal\{L\}\_\{low\}\+\\mathcal\{L\}\_\{high\}\\right\),\(32\)where the final configuration usesα=2\.0\\alpha=2\.0\. This auxiliary loss is part of the inherited backbone and is not introduced as a contribution of this paper\.

## IVExperiments

### IV\-ADataset and Experimental Settings

The experiments are conducted on the Beijing PM2\.5dataset used in DeepPM2\.5\[[39](https://arxiv.org/html/2606.24347#bib.bib43)\]\. The dataset contains air\-quality and meteorological observations from 39 monitoring stations in and around Beijing, China\. According to the DeepPM2\.5dataset description, the observations cover the period from January 1, 2018 00:00 to December 31, 2020 00:00 with an hourly resolution\. The 12 physical variables are PM2\.5, PM10, SO2, NO2, CO, O3, AQI, surface pressure, air temperature, dew\-point temperature, wind direction, and average wind speed\. In the current codebase, these variables correspond to the stored fields AQI, CO, NO2, O3, PM10, SO2, dew\-point temperature, ground pressure, temperature, wind direction, wind speed, and PM2\.5; the code additionally stores temporal fields including hour, day of week, and month\. The implemented model uses the 12 value features together with hour\-of\-day and day\-of\-week indicators, and predicts PM2\.5only\.

The train, validation, and test subsets follow a 60%/20%/20% chronological split\. The current processed tensor recorded in the project configuration contains 25,593 hourly time steps, 39 stations, and 15 raw fields\. The input sequence length is 48 hours and the prediction horizon is 48 hours\. The current implementation uses channel\-wise z\-score normalization with rescaling during evaluation\. The main evaluation horizons are 1, 3, 6, 12, 24, and 48 hours\. Table[I](https://arxiv.org/html/2606.24347#S4.T1)summarizes the dataset, variables, split, prediction setting, and evaluation metrics used in the experiments\.

TABLE I:Dataset and Experimental Setting Summary
### IV\-BBaselines

The main comparison uses the baseline results publicly reported in DeepPM2\.5\[[39](https://arxiv.org/html/2606.24347#bib.bib43)\]under the same Beijing dataset and experimental protocol\. These cited rows include temporal sequence models, transformer\-based models, spatio\-temporal graph models, and the DeepPM2\.5model: LSTM\[[11](https://arxiv.org/html/2606.24347#bib.bib25)\], GRU\[[6](https://arxiv.org/html/2606.24347#bib.bib26)\], Informer\[[41](https://arxiv.org/html/2606.24347#bib.bib28)\], iTransformer\[[20](https://arxiv.org/html/2606.24347#bib.bib29)\], Graph\-WaveNet\[[32](https://arxiv.org/html/2606.24347#bib.bib40)\], MTGNN\[[31](https://arxiv.org/html/2606.24347#bib.bib30)\], DSTGN\[[7](https://arxiv.org/html/2606.24347#bib.bib35)\], MSTGAN\[[43](https://arxiv.org/html/2606.24347#bib.bib36)\], and DeepPM2\.5\[[39](https://arxiv.org/html/2606.24347#bib.bib43)\]\. We additionally include HyperD\[[25](https://arxiv.org/html/2606.24347#bib.bib42)\]as the inherited periodic\-residual backbone reproduced in our codebase\. Since the reported DeepPM2\.5table provides MAE and RMSE only, the main comparison is restricted to these two metrics\.

### IV\-CComparisons with Deep Learning Methods

The main comparison is presented from two perspectives\. We first use representative prediction cases to examine whether the model follows realistic temporal changes at individual stations\. We then report average and horizon\-wise quantitative results to compare MVG\-KAN with representative baselines\.

#### IV\-C1Qualitative Analysis

![Refer to caption](https://arxiv.org/html/2606.24347v1/x2.png)Figure 4:Selected PM2\.5forecasting examples at three monitoring stations and six evaluation horizons\. The curves compare the ground truth \(GT\), HyperD, DeepPM2\.5, and MVG\-KAN under aligned test samples\.Fig\.[4](https://arxiv.org/html/2606.24347#S4.F4)shows three station\-level examples selected from the aligned test predictions\. At Station 29, the ground truth remains high in the early horizons and then decreases toward the 48H horizon\. HyperD tends to dissipate the concentration too rapidly, while DeepPM2\.5preserves a higher long\-horizon level but underestimates the early high\-concentration segment\. MVG\-KAN follows the early peak and subsequent decline more closely\. This behavior is consistent with the role of TKAN, which uses de\-periodized PM2\.5residuals and multi\-pollutant histories to model local residual persistence and gradual dissipation\. At Station 33, the concentration drops sharply after the first few horizons\. MVG\-KAN captures this rapid local transition better than HyperD, whose prediction remains too high after the drop\. This suggests that modeling the residual after removing daily and weekly periodicity helps the model respond to non\-periodic local changes rather than relying mainly on inherited periodic patterns\. At Station 31, MVG\-KAN follows the later decreasing trend, but it underestimates the abrupt 12H spike\. This case is useful because it shows both the benefit and the limitation of the proposed design: TKAN improves station\-wise residual correction and the Geo\-Wind Graph provides transport\-aware spatial context, but extremely abrupt local peaks may still be difficult when they are weakly supported by recent multi\-pollutant and wind\-guided signals\.

#### IV\-C2Quantitative Analysis

Table[II](https://arxiv.org/html/2606.24347#S4.T2)reports the average MAE/RMSE comparison\. Rows except HyperD and MVG\-KAN are cited from DeepPM2\.5\[[39](https://arxiv.org/html/2606.24347#bib.bib43)\]; HyperD and MVG\-KAN use saved test metrics from the current project\. Lower values indicate better forecasting performance\. The horizon\-wise behavior of representative methods, including DeepPM2\.5and MVG\-KAN, is further visualized in Fig\.[5](https://arxiv.org/html/2606.24347#S4.F5)\.

TABLE II:Performance comparison on the Beijing PM2\.5dataset\. Rows except HyperD and MVG\-KAN are cited from DeepPM2\.5under the same dataset and experimental protocol; HyperD and MVG\-KAN are evaluated in our codebase\.Table[II](https://arxiv.org/html/2606.24347#S4.T2)shows that MVG\-KAN achieves the lowest average errors in the comparison, with an Avg MAE of 14\.09 and an Avg RMSE of 21\.40\. Compared with DeepPM2\.5, the Avg MAE is reduced from 14\.45 to 14\.09, and the Avg RMSE is reduced from 22\.18 to 21\.41\. Compared with the inherited HyperD backbone, MVG\-KAN also reduces Avg MAE from 14\.79 to 14\.09 and Avg RMSE from 22\.46 to 21\.41, indicating that the proposed PM2\.5\-oriented modules improve the underlying periodic\-residual foundation\. Fig\.[5](https://arxiv.org/html/2606.24347#S4.F5)further compares MVG\-KAN with representative Transformer\-, graph\-, inherited HyperD\-, and PM2\.5\-oriented baselines\. The improvement over DeepPM2\.5is most evident at short and medium forecasting horizons from 1H to 12H, where MVG\-KAN consistently yields lower MAE and RMSE\. At longer horizons, the two methods become closer, and DeepPM2\.5remains slightly lower for 48H RMSE\. Overall, these results support the complementary benefits of the transport\-aware Geo\-Wind Graph and the TKAN residual temporal head: the former provides physically motivated spatial guidance for residual propagation, while the latter strengthens station\-wise local residual dynamics modeling through nonlinear multi\-pollutant temporal correction\.

![Refer to caption](https://arxiv.org/html/2606.24347v1/x3.png)Figure 5:Horizon\-wise forecasting performance comparison among representative methods on the Beijing PM2\.5dataset\. The selected methods include Informer, Graph\-WaveNet, DSTGN, HyperD, DeepPM2\.5, and the proposed MVG\-KAN\. MVG\-KAN achieves lower errors across most forecasting horizons\.

### IV\-DAblation Study

Table[III](https://arxiv.org/html/2606.24347#S4.T3)reports the controlled ablation results from the current project\. The baseline row uses PM2\.5as the only value input and disables the Geo\-Wind Graph, TKAN, and residual AR setting\. The other rows use multi\-pollutant and meteorological inputs with residual AR enabled, and isolate the effects of the Geo\-Wind Graph and TKAN\. The ablation results are our own test results\.

TABLE III:Ablation Study of the Proposed ComponentsAdding only the Geo\-Wind Graph reduces MAE from 14\.79 to 14\.69 and RMSE from 22\.45 to 22\.21, indicating that geography\- and wind\-guided spatial propagation provides a useful but moderate enhancement\. Adding TKAN without the Geo\-Wind Graph reduces MAE to 14\.28 and RMSE to 21\.70, showing that station\-wise KAN residual autoregression and multi\-pollutant co\-variation provide a stronger local temporal correction path\. The full MVG\-KAN model further improves MAE to 14\.09 and RMSE to 21\.4062\. This result supports the complementary roles of the two proposed components: the Geo\-Wind Graph supplies transport\-aware inter\-station residual propagation, while TKAN strengthens intra\-station residual dynamics after the periodic component has been removed\.

### IV\-EStation\-Wise and Distributional Diagnostics

![Refer to caption](https://arxiv.org/html/2606.24347v1/x4.png)Figure 6:Station\-wise MAE comparison computed from aligned test predictions at the selected evaluation horizons\. The figure compares HyperD, DeepPM2\.5, and MVG\-KAN across the 39 monitoring stations\.Fig\.[6](https://arxiv.org/html/2606.24347#S4.F6)compares station\-wise MAE across the 39 monitoring stations\. MVG\-KAN obtains lower errors than HyperD and DeepPM2\.5at most stations, while all methods show larger errors at several high\-variance stations\. This station\-wise pattern is consistent with the main comparison: replacing raw local temporal prediction with de\-periodized TKAN residual correction improves most stations, but the remaining hard stations indicate that some local pollution episodes are still difficult to capture\. The Geo\-Wind Graph is expected to help when station interactions are supported by geographic proximity and wind direction, but its effect can vary spatially because the Beijing monitoring network only sparsely samples possible transport pathways\.

Two observations are particularly relevant to the proposed decomposition view\. First, the reduction of station\-wise error is not concentrated at a single station\. MVG\-KAN tends to track the lower envelope of the compared methods over many low\- and medium\-error stations, suggesting that TKAN provides a broadly useful local residual correction after daily and weekly periodicity has been removed\. This agrees with the ablation result, where TKAN contributes the larger numerical improvement\. Second, the stations with the largest errors are often difficult for all methods simultaneously\. Such cases are likely associated with abrupt local events, sparse upstream evidence, or pollutant changes that are weakly represented by recent multi\-pollutant and wind variables\. In these cases, the Geo\-Wind Graph can provide a transport\-aware spatial prior, but it cannot fully compensate for unobserved emissions or unresolved meteorological processes\. Therefore, Fig\.[6](https://arxiv.org/html/2606.24347#S4.F6)supports a balanced interpretation: the proposed modules improve the robustness of station\-level forecasting across the network, while the remaining station\-wise peaks identify where additional meteorological, emission, or remote\-sensing information may be needed\.

![Refer to caption](https://arxiv.org/html/2606.24347v1/x5.png)Figure 7:Distributional and spatial\-spectrum diagnostics of PM2\.5forecasts\. \(a\) Frequency distributions compare GT and model\-predicted PM2\.5concentrations at the selected evaluation horizons\. \(b\) RAPSD compares the radially averaged spatial power spectral density after interpolating station observations onto a regular geographic grid\.Fig\.[7](https://arxiv.org/html/2606.24347#S4.F7)provides two additional descriptive diagnostics\. In the frequency distribution, MVG\-KAN follows the overall PM2\.5concentration distribution of the ground truth and better preserves the middle\-to\-high concentration range than the compared prediction files\. This is relevant to TKAN because residual correction is designed to retain short\-term pollutant accumulation and slow dissipation after periodicity is removed\. In the RAPSD plot, MVG\-KAN stays close to the ground\-truth curve over much of the low\- and middle\-wavenumber range, suggesting that the broad spatial variability of the station field is reasonably preserved after interpolation\. This observation is consistent with the role of the Geo\-Wind Graph as a lightweight spatial prior\. However, RAPSD here should be interpreted only as a diagnostic visualization because it is computed after interpolating irregular station observations onto a regular grid, not as direct evidence that the model physically simulates atmospheric transport\.

## VDiscussion

The experimental results support the proposed forecasting view that station\-level PM2\.5should be modeled through local periodic patterns, local residual dynamics, and wind\-driven spatial transport\. The qualitative examples in Fig\.[4](https://arxiv.org/html/2606.24347#S4.F4)show that MVG\-KAN can better follow several local concentration transitions after periodic effects are removed\. The quantitative comparison in Table[II](https://arxiv.org/html/2606.24347#S4.T2)and Fig\.[5](https://arxiv.org/html/2606.24347#S4.F5)shows lower average MAE and RMSE than the cited baselines across most forecasting horizons, and the ablation study in Table[III](https://arxiv.org/html/2606.24347#S4.T3)further separates the roles of the two proposed modules\. TKAN provides the stronger local residual temporal correction, while the Geo\-Wind Graph contributes a complementary transport\-aware spatial prior\. The station\-wise, distributional, and RAPSD diagnostics in Figs\.[6](https://arxiv.org/html/2606.24347#S4.F6)and[7](https://arxiv.org/html/2606.24347#S4.F7)also suggest that the improvements are not limited to a single averaged metric, although the gains vary across stations and horizons\.

Several limitations remain\. First, the Geo\-Wind Graph is a lightweight physically motivated graph rather than a full atmospheric transport simulator\. It depends on the quality and spatial representativeness of the available wind variables, and the 39\-station network may not fully capture transport pathways through unmonitored areas\. Second, the current graph does not explicitly model emissions, boundary\-layer height, vertical mixing, deposition, or chemical formation, which may be important during severe or chemically complex pollution episodes\. Third, TKAN is intentionally station\-wise; it improves local residual correction but does not itself model spatial propagation\. Finally, the main comparison uses several baseline rows reported by DeepPM2\.5rather than fully reproduced under one codebase, so the comparison should be interpreted with this practical constraint\.

Future work will investigate more refined meteorological fields, trajectory\-based transport features, boundary\-layer information, emission inventories, remote sensing observations, and chemical\-transport\-model outputs\. More extensive reproduced baselines, repeated runs, uncertainty analysis, and cross\-city or cross\-season evaluation would also help clarify the robustness of MVG\-KAN across different pollution regimes and forecasting horizons\.

## VIConclusion

Short\-term PM2\.5forecasting is challenging because observed concentrations are affected by temporal periodicity, non\-periodic pollution episodes, wind\-driven transport, and multi\-pollutant co\-variation\. This paper presented a geo\-wind guided KAN residual temporal learning framework for station\-level PM2\.5forecasting\. The framework uses a periodic\-residual prediction backbone as its foundation and introduces two PM2\.5\-oriented components: a Geo\-Wind Graph and a TKAN residual temporal module\.

The Geo\-Wind Graph combines geographic proximity and wind\-driven directionality to enhance physically meaningful spatial dependency modeling among monitoring stations\. It uses station locations and wind variables to guide directed residual information flow, so that spatial interaction is more consistent with potential pollutant transport\. TKAN uses the de\-periodized PM2\.5residual and historical multi\-pollutant sequences for station\-wise nonlinear residual correction\. It does not perform spatial propagation; instead, it complements the graph module by modeling local residual inertia, abrupt temporal changes, and pollutant co\-variation\.

Experiments on the Beijing PM2\.5dataset show that MVG\-KAN achieves MAE 14\.09 and RMSE 21\.40\. The ablation study indicates that TKAN provides the major performance gain, reducing MAE and RMSE more clearly than the graph\-only variant\. The Geo\-Wind Graph provides complementary spatial enhancement, with moderate improvements in MAE and RMSE in the graph\-only setting\. These results suggest that station\-wise KAN residual temporal correction and physically guided spatial propagation play different but compatible roles in PM2\.5forecasting\.

Overall, the results indicate that separating local residual temporal correction from wind\-guided spatial propagation is a practical direction for PM2\.5forecasting\.

## References

- \[1\]\(2018\)Air pollution forecasts: an overview\.International Journal of Environmental Research and Public Health15\(4\),pp\. 780\.External Links:[Document](https://dx.doi.org/10.3390/ijerph15040780)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4),[§I](https://arxiv.org/html/2606.24347#S1.p2.2),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[2\]S\. Bai, J\. Z\. Kolter, and V\. Koltun\(2018\)An empirical evaluation of generic convolutional and recurrent networks for sequence modeling\.arXiv preprint arXiv:1803\.01271\.External Links:[Link](https://arxiv.org/abs/1803.01271)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2)\.
- \[3\]P\. J\. Brockwell and R\. A\. Davis\(2002\)Introduction to time series and forecasting\.Springer\.Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p2.2)\.
- \[4\]R\. D\. Brook, S\. Rajagopalan, C\. A\. Pope, J\. R\. Brook, A\. Bhatnagar, A\. V\. Diez\-Roux, F\. Holguin, Y\. Hong, R\. V\. Luepker, M\. A\. Mittleman, A\. Peters, D\. Siscovick, S\. C\. Smith, L\. Whitsel, and J\. D\. Kaufman\(2010\)Particulate matter air pollution and cardiovascular disease: an update to the scientific statement from the american heart association\.Circulation121\(21\),pp\. 2331–2378\.External Links:[Document](https://dx.doi.org/10.1161/CIR.0b013e3181dbece1)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4)\.
- \[5\]D\. Chen, H\. Guo, X\. Gu, T\. Cheng, J\. Yang, Y\. Zhan, and X\. Wei\(2023\)A spatial neighborhood deep neural network model for pm2\.5 estimation across china\.IEEE Transactions on Geoscience and Remote Sensing61,pp\. 1–15\.External Links:[Document](https://dx.doi.org/10.1109/TGRS.2023.3317905)Cited by:[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[6\]K\. Cho, B\. van Merrienboer, C\. Gulcehre, D\. Bahdanau, F\. Bougares, H\. Schwenk, and Y\. Bengio\(2014\)Learning phrase representations using RNN encoder–decoder for statistical machine translation\.InProceedings of the 2014 Conference on Empirical Methods in Natural Language Processing,pp\. 1724–1734\.External Links:[Document](https://dx.doi.org/10.3115/v1/D14-1179)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[7\]Z\. Diao, X\. Wang, D\. Zhang, Y\. Liu, K\. Xie, and S\. He\(2019\)Dynamic spatial\-temporal graph convolutional neural networks for traffic forecasting\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.33,pp\. 890–897\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v33i01.3301890)Cited by:[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[8\]J\. L\. Elman\(1990\)Finding structure in time\.Cognitive science14\(2\),pp\. 179–211\.Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2)\.
- \[9\]X\. Feng, Q\. Li, Y\. Zhu, J\. Hou, L\. Jin, and J\. Wang\(2015\)Artificial neural networks forecasting of pm2\.5 pollution using air mass trajectory based geographic model and wavelet transformation\.Atmospheric Environment107,pp\. 118–128\.External Links:[Document](https://dx.doi.org/10.1016/j.atmosenv.2015.02.030)Cited by:[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p1.1)\.
- \[10\]G\. Geng, Q\. Zhang, R\. V\. Martin, A\. van Donkelaar, H\. Huo, H\. Che, J\. Lin, and K\. He\(2015\)Estimating long\-term pm2\.5 concentrations in china using satellite\-based aerosol optical depth and a chemical transport model\.Remote Sensing of Environment166,pp\. 262–270\.External Links:[Document](https://dx.doi.org/10.1016/j.rse.2015.05.016)Cited by:[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[11\]S\. Hochreiter and J\. Schmidhuber\(1997\)Long short\-term memory\.Neural Computation9\(8\),pp\. 1735–1780\.External Links:[Document](https://dx.doi.org/10.1162/neco.1997.9.8.1735)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[12\]N\. P\. Hyslop\(2009\)Impaired visibility: the air pollution people see\.Atmospheric Environment43\(1\),pp\. 182–195\.External Links:[Document](https://dx.doi.org/10.1016/j.atmosenv.2008.09.067)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4)\.
- \[13\]T\. N\. Kipf and M\. Welling\(2017\)Semi\-supervised classification with graph convolutional networks\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=SJU4ayYgl)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[14\]J\. Lelieveld, J\. S\. Evans, M\. Fnais, D\. Giannadaki, and A\. Pozzer\(2015\)The contribution of outdoor air pollution sources to premature mortality on a global scale\.Nature525\(7569\),pp\. 367–371\.External Links:[Document](https://dx.doi.org/10.1038/nature15371)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4)\.
- \[15\]X\. Li, L\. Peng, X\. Yao, S\. Cui, Y\. Hu, C\. You, and T\. Chi\(2017\)Long short\-term memory neural network for air pollutant concentration predictions: method development and evaluation\.Environmental Pollution231,pp\. 997–1004\.External Links:[Document](https://dx.doi.org/10.1016/j.envpol.2017.08.114)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p2.2),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[16\]Y\. Li, R\. Yu, C\. Shahabi, and Y\. Liu\(2018\)Diffusion convolutional recurrent neural network: data\-driven traffic forecasting\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=SJiHXGWAZ)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[17\]Y\. Liang, S\. Ke, J\. Zhang, X\. Yi, and Y\. Zheng\(2018\)GeoMAN: multi\-level attention networks for geo\-sensory time series prediction\.InProceedings of the Twenty\-Seventh International Joint Conference on Artificial Intelligence,pp\. 3428–3434\.External Links:[Document](https://dx.doi.org/10.24963/ijcai.2018/476)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p4.1),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2)\.
- \[18\]B\. Lim, S\. O\. Arik, N\. Loeff, and T\. Pfister\(2021\)Temporal fusion transformers for interpretable multi\-horizon time series forecasting\.International Journal of Forecasting37\(4\),pp\. 1748–1764\.External Links:[Document](https://dx.doi.org/10.1016/j.ijforecast.2021.03.012)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2)\.
- \[19\]J\. Lin, Y\. Cui, C\. Fang, B\. Pan, C\. Pan, G\. Jiang, S\. Zhang, S\. Ma, and Q\. Tian\(2026\)Multi\-modal cross\-attention\-guided network for audio\-visual quality evaluation via visual saliency and mel\-spectrum features\.IEEE Transactions on Circuits and Systems for Video Technology36\(5\),pp\. 6783–6798\.External Links:[Document](https://dx.doi.org/10.1109/TCSVT.2026.3652641)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2)\.
- \[20\]Y\. Liu, T\. Hu, H\. Zhang, H\. Wu, S\. Wang, L\. Ma, and M\. Long\(2024\)iTransformer: inverted transformers are effective for time series forecasting\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=JePfAI8fah)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[21\]Z\. Liu, Y\. Wang, S\. Vaidya, F\. Ruehle, J\. Halverson, M\. Soljacic, T\. Y\. Hou, and M\. Tegmark\(2024\)KAN: kolmogorov–arnold networks\.arXiv preprint arXiv:2404\.19756\.External Links:[Link](https://arxiv.org/abs/2404.19756)Cited by:[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p3.1),[§III\-D](https://arxiv.org/html/2606.24347#S3.SS4.p4.8)\.
- \[22\]C\. A\. Pope and D\. W\. Dockery\(2006\)Health effects of fine particulate air pollution: lines that connect\.Journal of the Air & Waste Management Association56\(6\),pp\. 709–742\.External Links:[Document](https://dx.doi.org/10.1080/10473289.2006.10464485)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4)\.
- \[23\]Y\. Qi, Q\. Li, H\. Karimian, and D\. Liu\(2019\)A hybrid model for spatiotemporal forecasting of pm2\.5 based on graph convolutional neural network and long short\-term memory\.Science of the Total Environment664,pp\. 1–10\.External Links:[Document](https://dx.doi.org/10.1016/j.scitotenv.2019.01.333)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p4.1),[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[24\]Z\. Qi, T\. Wang, G\. Song, W\. Hu, X\. Li, and Z\. Zhang\(2018\)Deep air learning: interpolation, prediction, and feature analysis of fine\-grained air quality\.IEEE Transactions on Knowledge and Data Engineering30\(12\),pp\. 2285–2297\.External Links:[Document](https://dx.doi.org/10.1109/TKDE.2018.2823740)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p2.2),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p2.6)\.
- \[25\]M\. Shao, Z\. Zhang, Y\. Wang, Y\. Dai, X\. Shen, and X\. Wang\(2026\)HyperD: hybrid periodicity decoupling framework for traffic forecasting\.Proceedings of the AAAI Conference on Artificial Intelligence40\(18\),pp\. 15689–15697\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v40i18.38599)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p13.2),[§I](https://arxiv.org/html/2606.24347#S1.p6.3),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p1.1),[§III\-B](https://arxiv.org/html/2606.24347#S3.SS2.p1.5),[§III\-E](https://arxiv.org/html/2606.24347#S3.SS5.p1.12),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[26\]A\. F\. Stein, R\. R\. Draxler, G\. D\. Rolph, B\. J\. B\. Stunder, M\. D\. Cohen, and F\. Ngan\(2015\)NOAA’s hysplit atmospheric transport and dispersion modeling system\.Bulletin of the American Meteorological Society96\(12\),pp\. 2059–2077\.External Links:[Document](https://dx.doi.org/10.1175/BAMS-D-14-00110.1)Cited by:[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[27\]P\. Velickovic, G\. Cucurull, A\. Casanova, A\. Romero, P\. Lio, and Y\. Bengio\(2018\)Graph attention networks\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=rJXMpikCZ)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[28\]S\. Wang, Y\. Li, J\. Zhang, Q\. Meng, L\. Meng, and F\. Gao\(2020\)PM2\.5\-gnn: a domain knowledge enhanced graph neural network for pm2\.5 forecasting\.InProceedings of the 28th International Conference on Advances in Geographic Information Systems,pp\. 163–166\.External Links:[Document](https://dx.doi.org/10.1145/3397536.3422208)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[29\]J\. Wei, W\. Huang, Z\. Li, W\. Xue, Y\. Peng, L\. Sun, and M\. Cribb\(2019\)Estimating 1\-km\-resolution pm2\.5 concentrations across china using the space\-time random forest approach\.Remote Sensing of Environment231,pp\. 111221\.External Links:[Document](https://dx.doi.org/10.1016/j.rse.2019.111221)Cited by:[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[30\]H\. Wu, J\. Xu, J\. Wang, and M\. Long\(2021\)Autoformer: decomposition transformers with auto\-correlation for long\-term series forecasting\.InAdvances in Neural Information Processing Systems,Vol\.34,pp\. 22419–22430\.External Links:[Link](https://proceedings.neurips.cc/paper/2021/hash/bcc0d400288793e8bdcd7c19a8ac0c2b-Abstract.html)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p6.3),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p1.1)\.
- \[31\]Z\. Wu, S\. Pan, G\. Long, J\. Jiang, X\. Chang, and C\. Zhang\(2020\)Connecting the dots: multivariate time series forecasting with graph neural networks\.InProceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining,pp\. 753–763\.External Links:[Document](https://dx.doi.org/10.1145/3394486.3403118)Cited by:[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[32\]Z\. Wu, S\. Pan, G\. Long, J\. Jiang, and C\. Zhang\(2019\)Graph wavenet for deep spatial\-temporal graph modeling\.InProceedings of the Twenty\-Eighth International Joint Conference on Artificial Intelligence,pp\. 1907–1913\.External Links:[Document](https://dx.doi.org/10.24963/ijcai.2019/264)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[33\]X\. Xiao, Z\. Jin, S\. Wang, J\. Xu, Z\. Peng, R\. Wang, W\. Shao, and Y\. Hui\(2022\)A dual\-path dynamic directed graph convolutional network for air quality prediction\.Science of the Total Environment827,pp\. 154298\.External Links:[Document](https://dx.doi.org/10.1016/j.scitotenv.2022.154298)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p8.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p2.1)\.
- \[34\]M\. Yang, H\. Fan, and K\. Zhao\(2019\)PM2\.5 prediction with a novel multi\-step\-ahead forecasting model based on dynamic wind field distance\.International Journal of Environmental Research and Public Health16\(22\),pp\. 4482\.External Links:[Document](https://dx.doi.org/10.3390/ijerph16224482)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p8.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p2.1)\.
- \[35\]X\. Yi, J\. Zhang, Z\. Wang, T\. Li, and Y\. Zheng\(2018\)Deep distributed fusion network for air quality prediction\.InProceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,pp\. 965–973\.External Links:[Document](https://dx.doi.org/10.1145/3219819.3219822)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p4.1),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p2.6)\.
- \[36\]B\. Yu, H\. Yin, and Z\. Zhu\(2018\)Spatio\-temporal graph convolutional networks: a deep learning framework for traffic forecasting\.InProceedings of the Twenty\-Seventh International Joint Conference on Artificial Intelligence,pp\. 3634–3640\.External Links:[Document](https://dx.doi.org/10.24963/ijcai.2018/505)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p5.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p1.1)\.
- \[37\]A\. Zeng, M\. Chen, L\. Zhang, and Q\. Xu\(2023\)Are transformers effective for time series forecasting?\.Proceedings of the AAAI Conference on Artificial Intelligence37\(9\),pp\. 11121–11128\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v37i9.26317)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p6.3),[§II\-C](https://arxiv.org/html/2606.24347#S2.SS3.p1.1)\.
- \[38\]Y\. Zhang, M\. Bocquet, V\. Mallet, C\. Seigneur, and A\. Baklanov\(2012\)Real\-time air quality forecasting, part i: history, techniques, and current status\.Atmospheric Environment60,pp\. 632–655\.External Links:[Document](https://dx.doi.org/10.1016/j.atmosenv.2012.06.031)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p1.4),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2)\.
- \[39\]H\. Zheng, C\. Huang, Z\. Zhang, and S\. Zhang\(2026\)DeepPM2\.5: PM2\.5 prediction under dynamic and heterogeneous conditions with contrastive learning and spatio\-temporal graph convolution\.IEEE Transactions on Geoscience and Remote Sensing\.External Links:[Document](https://dx.doi.org/10.1109/TGRS.2026.3656010)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p4.1),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2),[§IV\-A](https://arxiv.org/html/2606.24347#S4.SS1.p1.14),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4),[§IV\-C2](https://arxiv.org/html/2606.24347#S4.SS3.SSS2.p1.2)\.
- \[40\]Y\. Zheng, F\. Liu, and H\. Hsieh\(2013\)U\-Air: when urban air quality inference meets big data\.InProceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,pp\. 1436–1444\.External Links:[Document](https://dx.doi.org/10.1145/2487575.2488188)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p2.2),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p1.2),[§II\-A](https://arxiv.org/html/2606.24347#S2.SS1.p2.2)\.
- \[41\]H\. Zhou, S\. Zhang, J\. Peng, S\. Zhang, J\. Li, H\. Xiong, and W\. Zhang\(2021\)Informer: beyond efficient transformer for long sequence time\-series forecasting\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.35,pp\. 11106–11115\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v35i12.17325)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p3.2),[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.
- \[42\]H\. Zhou, F\. Zhang, Z\. Du, and R\. Liu\(2021\)Forecasting pm2\.5 using hybrid graph convolution\-based model considering dynamic wind\-field to offer the benefit of spatial interpretability\.Environmental Pollution273,pp\. 116473\.External Links:[Document](https://dx.doi.org/10.1016/j.envpol.2021.116473)Cited by:[§I](https://arxiv.org/html/2606.24347#S1.p8.1),[§II\-B](https://arxiv.org/html/2606.24347#S2.SS2.p2.1)\.
- \[43\]X\. Zhou, J\. Wang, J\. Wang, and Q\. Guan\(2024\)Predicting air quality using a multi\-scale spatiotemporal graph attention network\.Information Sciences680,pp\. 121072\.External Links:[Document](https://dx.doi.org/10.1016/j.ins.2024.121072)Cited by:[§IV\-B](https://arxiv.org/html/2606.24347#S4.SS2.p1.4)\.