Title: Mesh Interpolation Graph Network for Dynamic and Spatially Irregular Global Weather Forecasting
Venue: NeurIPS 2025
Accepted: yes
Keywords: Weather forecasting

Graph neural network

Earth sciences
Abstract: Graph neural networks have shown promising results in weather forecasting, which is critical for human activity such as agriculture planning and extreme weather preparation. However, most studies focus on finite and local areas for training, overlooking the influence of broader areas and limiting their ability to generalize effectively. Thus, in this work, we study global weather forecasting that is irregularly distributed and dynamically varying in practice, requiring the model to generalize to unobserved locations.
To address such challenges, we propose a general Mesh Interpolation Graph Network (MIGN) that models the irregular weather station forecasting, consisting of two key designs: (1) learning spatially irregular data with regular mesh interpolation network to align the data; (2) leveraging parametric spherical harmonics location embedding to further enhance spatial generalization ability. Extensive experiments on an up-to-date observation dataset show that MIGN significantly outperforms existing data-driven models. Besides, we show that MIGN has spatial generalization ability, and is capable of generalizing to previously unseen stations.

Mesh Interpolation Graph Network for Dynamic and Spatially Irregular Global Weather Forecasting

Abstract

Graph neural networks have shown promising results in weather forecasting, which is critical for human activity such as agriculture planning and extreme weather preparation. However, most studies focus on finite and local areas for training, overlooking the influence of broader areas and limiting their ability to generalize effectively. Thus, in this work, we study global weather forecasting that is irregularly distributed and dynamically varying in practice, requiring the model to generalize to unobserved locations. To address such challenges, we propose a general M esh I nterpolation G raph N etwork (MIGN) that models the irregular weather station forecasting, consisting of two key designs: (1) learning spatially irregular data with regular mesh interpolation network to align the data; (2) leveraging parametric spherical harmonics location embedding to further enhance spatial generalization ability. Extensive experiments on an up-to-date observation dataset show that MIGN significantly outperforms existing data-driven models. Besides, we show that MIGN has spatial generalization ability, and is capable of generalizing to previously unseen stations.

1 Introduction

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Weather forecasting is critical for human activities and extreme weather warning. For example, accurate short-term predictions of precipitation and snowfall are valuable for agriculture [37] and outdoor activities planning, while forecasting extreme weather phenomena, such as heatwaves [16] and typhoons, is vital to mitigating significant damage. Early warnings can play a crucial role in safeguarding lives and property. To address these problems, multiple date-driven models have been

proposed for weather forecasting. A series of works [30, 2, 15, 25] have been developed based on the gridded Earth Reanalysis 5 (ERA5) dataset. However, these models are specifically designed for regular, image-like data structures and cannot be directly applied to weather station data, which consists of precise, fine-grained meteorological observations collected at irregular spatial locations. In contrast, graph neural networks [46, 45, 22, 26, 43, 24, 44, 19, 17, 18, 20, 19] are naturally suited to model such irregular structures. To capture the spatial dependencies inherent in such irregular data, multiple studies have achieved promising results in weather forecasting with GNNs. These approaches typically represent stations as nodes, construct edges among them via radius distance or nearest neighbors, and perform message passing thereon.

However, most of the work [16, 4] focuses on regional forecasting, typically limited to areas such as Europe and North America, while overlooking the influence of external regions. This localized modeling approach overlooks the fact that weather patterns in one region are often influenced by conditions in distant parts of the world, as the Earth's weather system is globally connected. As a result, learning from only regional data often misses broader spatial patterns, leading to suboptimal forecast performance. Moreover, models overfitted to specific regions tend to lack generalization capability, making them less practical for deployment in diverse or unseen geographical areas. Thus, global weather forecasting is crucial and presents the following challenges:

Spatial irregularity . The distribution of weather stations across the Earth's surface is uneven. As illustrated in Figure 1(A), the majority weather stations are concentrated in North America and Western Europe. The spatial distribution of the stations exhibits significant variations in different longitudes and latitudes (shown in Figure 1(B)). Existing data-driven models often overlook the spatial irregularity of station placements, which results in varying scales of information. During training, models often face challenges in simultaneously learning patterns from regions with high and low data point densities.

Dynamic distribution . The number and spatial distribution of stations are changing over time. Figure 1(C) shows the temporal variations in station data of NOAA GSOD dataset 2 . This can be due to the establishment of meteorological stations in remote areas to compensate for limited observational coverage, as well as the decommissioning or abandonment of certain stations over time. Current studies [6, 4, 11, 39] typically use a fixed number of meteorological stations for predictions. Training models on a limited set of stations often results in overfitting of the dataset. Such models often struggle to predict features at unseen locations during training, as the lack of generalization capability limits their performance on previously unobserved points.

To address the above problems, we study a fundamental spatial generalization problem in spherical Earth surface. That is, the models are required to predict weather variables in areas with sparse observations or finite historical records. We propose a Mesh Interpolation Graph Network (MIGN) framework that implements a mesh interpolation strategy and parametric spherical harmonics location embedding. To alleviate the uneven distribution of the data, MIGN first maps the latent space of the irregular station to regular mesh by message passing. Such a process could be viewed as interpolation, where the points on the mesh are uniformly distributed. Message passing on mesh points can be implemented to ensure that the model does not only learn patterns from high-density data regions. Secondly, we do not treat the coordinates as position features. Instead, we consider the weather information of the stations as a function of the coordinates, encoding a learnable weather function that can be generalized to unseen points. Through extensive experiments on the up-to-date NOAA GSOD dataset, we find that:

MIGN outperforms state-of-the-art spatial-temporal models. Ablation studies demonstrate that the two proposed designs, mesh interpolation and spherical harmonic location embedding, significantly enhance the performance.

The generalization study shows that most methods hard to learn global patterns from existing data, limiting their ability to generalize to unobserved locations. In contrast, MIGN demonstrates strong generalization to unseen stations, highlighting its adaptability to dynamic scenarios.

Most methods struggle to perform well in regions with dense and sparse observations. In contrast, we show that MIGN consistently produces more robust results across different regional patterns at the same time. The code is available at the link: https://github.com/compasszzn/MIGN

2 Preliminary and Related Work

Weather Forecasting Traditional weather forecasting depends on Numerical Weather Prediction (NWP) [1] models, which aim to forecast future weather patterns by simulating the dynamics and physics of the atmosphere with the equation of thermodynamics, fluid dynamics, etc. However, NWP requires substantial computing resources and often exhibits deviations [28]. Thus, various data-driven models have been proposed to predict the weather. Currently, data-driven models can be categorized based on the underlying data structure. The first category deals with regular gridded data, with the ECMWF Reanalysis v5 (ERA5) dataset being a representative example. Based on such data, several pioneering works-such as FourCastNet [30], Pangu [2], and GraphCast [15]-have achieved impressive results. However, these models are not well-suited for a second category of data: observed irregular station data. To address this, existing methods often employ Graph Neural Networks (GNNs) to capture spatial dependencies. Nevertheless, these approaches [6, 4, 11, 39] typically assume a fixed set of observation stations over time, limiting their ability to generalize to dynamic scenarios. Motivated by this limitation, we consider a more challenging setting in which the observation stations are irregularly distributed and vary across different samples.

Problem Definition Specifically, we treat each observation station as a node. On day t , the global stations could be represented by a graph G t = ( V t , E t , X t , ( λ t , ϕ t )) , where V t = { v t 1 , v t 2 , · · · , v t |V t | } is the set of nodes. E t = { ( v t i , v t j ) | v t i , v t j ∈ V t } is edge sets, which is constructed via k-nearest neighbor and the edge attributes (e.g., node distances) are denoted by d ij . Each station collects a single weather feature, X t = [ x t 1 , x t 2 , · · · , x t |V t | ] is a collection of node feature where x t v ∈ R , ∀ v ∈ V t . ( λ t , ϕ t ) denotes global geographic coordinate where longitude λ t i ∈ [ -π, π ] and latitudes ϕ t i ∈ [ -π 2 , π 2 ] . Given the initial condition G t , our objective is to learn a neural network to predict the next day weather feature value, as shown in the following:

where Θ denotes the parameters of the neural network. ˆ Y t +1 denotes the predicted feature while Y t +1 denotes the label. Note that the label Y t +1 here is different to X t +1 because the stations in each step would be different.

Graph Neural Networks Recently, researchers used GNNs to capture spatial patterns of the regional stations, such as air quality estimation[6, 4, 11] and heatwave prediction[16]. The above methods utilize GNNs to capture the spatial correlation and use time series models to model temporal dependency. GNNs are typically implemented using message passing mechanisms. Given a graph G = ( V , E , X , ( λ , ϕ )) , the message from node u to node v at layer l is given by:

where φ ( l ) can be instantiated as a multi-layer perception (MLP). The generated messages from all neighbors are aggregated at the target node v and the aggregated message m ( l ) v is used to update the state of the node v with function UPDATE l as follows:

where AGG ( l ) can be implemented as functions like sum, mean, max pooling or neural network [10, 38, 13, 23] and UPDATE is a learnable function, such as MLP or a gated recurrent unit (GRU).

However, the above framework ignores that the number and spatial distribution of stations change over time. Such a model often fails to predict features in unseen locations.

Mesh Interpolation Mesh interpolation is a common approach in Earth science for using spatially irregular station observation data to reproduce regular mesh data [12, 3]. Traditional interpolation methods include Inverse Distance Weighting (IDW), Kriging, and 3D-thin plate splines (TPS). Among them, IDW is widely used in earth science, which assumes that the influence of a given observation decreases with distance, typically following a power law. Mathematically, the estimated value at an unmeasured location is computed as a weighted average of nearby observations, where the weights are inversely proportional to the distance raised to a specified exponent. Meshes are not only used in

traditional numerical methods but have also been widely adopted in data-driven approaches. One of the most pioneering works in this area is GraphCast [15]. It maps local regions of the input to the nodes of the multi-mesh graph structure and performs message passing on mesh as well. However, it focuses on the regular gridded data and the edges between mesh and nodes are static, while our mesh interpolation lies in alleviating the spatial irregularity problem in station data by mapping the information to a regular space. In addition, the complex distribution of the stations motivates us to enhance the spatial generalization ability of the model. We further propose spherical harmonics location embedding to handle the dynamic data, while GraphCast is based on static data points, which means it lacks generalization capability for grid data with varying resolutions.

Spherical Harmonics The aforementioned GNNs do not incorporate the geometric information of the sphere to improve generalization ability. In contrast, we introduce mesh interpolation to alleviate the spatial irregular problem and spherical harmonics location embedding to enhance spatial generalization. Spherical harmonics have been wides used in earth science for magnetic field [36], weather patterns [40] and gravity field [14]. To be specific, a function f ( λ, ϕ ) defined on the sphere can be represented by a set of orthonormalized spherical harmonics Y n m ( λ, ϕ ) as follows:

where n denotes the degree, which controls the spatial scale of variation, with small n capturing coarse, global patterns and larger n resolving finer structures. m denotes the order with m ∈ [ -n, n ] of the basis functions, governing the oscillations in the longitudinal direction. λ and ϕ are longitude and latitude respectively. We consider a maximum degree of N , which results in a total of ( N +1) 2 basis functions and learnable weights w m n . The spherical harmonics are functions defined on the sphere as:

where P n m are associated Legendre polynomials:

which involve derivatives of Legendre Polynomials P n ( x ) defined by the following recurrence:

In practice, we consider the real spherical harmonics given as

where ˆ P | m | n (cos λ ) = √ 2 n +1 4 π ( n -m )! ( n + m )! P | m | n (cos λ ) , following the work [32], we pre-compute the spherical harmonics for each node in experiments. A related work is Geographic Location Encoder [32]. Although Geographic Location Encoder utilizes spherical harmonics, it focuses on training a neural network based on land-ocean classification tasks for coordinate embedding and spatial forecasting (i.e., ERA5 interpolation) of weather data to learn the coefficients of the spherical harmonics. However, MIGN aims to spatio-temporal forecast of irregular and dynamic distributed weather station data, therefore, it employs the spherical harmonic embedding as part of the input. Besides, considering that the variation patterns of different weather variables differ within the same region, we would learn a different variable-specific location embedding.

3 Method

Our MIGN architecture is illustrated in Figure 2, following an encoder-processor-decoder framework. In the following, we elaborate on the MIGN framework including spherical harmonics location embedding and mesh interpolation.

Irregular Output

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3.1 Spherical Harmonics Location Embedding

Spherical harmonics are widely used in the analysis of global weather patterns [40]. Since the Earth can be approximated as a sphere, many meteorological variables can be naturally modeled as functions defined on the spherical surface. Spherical harmonics provide a convenient basis for representing such functions, allowing us to capture spatial structures of the station. Besides, the non-parametric positional embedding provides limited location information, which restricts the model's ability to generalize to unseen areas. Inspired by this, we assume that the global location information could be represented by a function f ( λ, ϕ ) defined on the sphere. Instead of learning this function with a neural network directly, we decomposed this function into spherical harmonics to learn the spherical harmonics coefficients. The figure for the method is shown in the Appendix Figure 7. Using real spherical harmonics, the function of the sphere could be represented by f ( λ, ϕ ) = ∑ ∞ n =0 ∑ n m = -n w m n Y n m ( λ, ϕ ) , w m n refers to spherical harmonics coefficients, which are learnable weights.

Consider the expressive power of the location embedding. We concatenate the spherical harmonic basis function as features instead of learning the function f ( λ, ϕ ) directly. Specifically, consider the node with feature x and coordinates λ , ϕ . The location embedding is denoted as follows:

where ⊕ indicates the concatenation of these basis functions into one large vector and [; ] denotes the concatenation of the embedding vector. Based on the definition of spherical harmonics, the learnable w m n coefficient is shared across all nodes.

3.2 MIGN Framework

In this section, we first introduce the HEALPix which is employed to construct mesh. Then we would elaborate on the mesh interpolation framework with spherical harmonics location embedding.

HEALPix Mesh HEALPix [8] (Hierarchical, Equal Area, and iso-Latitude Pixelisation of the sphere) is a hierarchical structure for multi-resolution applications, which uniformly divides the sphere into equal-sized pixels. The data points are located in the center of the pixels and are uniformly distributed across the sphere. The base resolution consists of 12 quadrilateral pixels on the sphere. To generate a higher-resolution HEALPix grid, each pixel can be subdivided along the edge twice, resulting in 4 subgrids that represent the original quadrilateral pixel. We can do this recursively to get a higher-resolution HEALPix mesh. When the process is conducted k times, which is also called refinement level k , the original quadrilateral pixel can be divided into (2 k ) 2 pixels, leading to 12 ∗ (2 k ) 2 pixels and mesh nodes in total.

Mesh Interpolate Encoder Irregular station distributions make it hard to represent spatial patterns, and graph-based neighborhood aggregation becomes difficult due to the lack of consistent locality and connectivity. Motivated by this, MIGN first conducts a message passing from station nodes to regular mesh nodes within the encoder. Consider the graph in the encoder at day t G t E = (( V t s , V t h ) , E t ( s,h ) , ( X t s , X t h ) , ( λ t s , ϕ t s )) , where label s denotes station nodes while label h denotes the mesh nodes. The feature of the mesh nodes X h can be initialized with zero. E t ( s,h ) = { ( v t s , v t h ) |

v t s ∈ V t s , v t h ∈ V t h } is edge sets. Inspired by mesh interpolation in earth science, we only consider constructing the edges from station nodes to mesh nodes. Instead of interpolating the value of mesh nodes with fixed weight like IDW, we utilized message passing neural network to project the value into latent space. For each mesh node v t h , messages are generated by its neighbors station nodes v t s ∈ N ( v t h ) . The hidden state of the station nodes and the message are given by:

Message Passing The messages from the station nodes are aggregated to the target mesh nodes, and the hidden state of the mesh nodes would update with the message directly:

For the processor part, we consider the mesh nodes graph G t P = ( V t h , E t h , H t h , ( λ t h , ϕ t h )) . The feature of the mesh nodes H t h are the message aggregate from station nodes, denoted as h ( E ) v t h , v t h ∈ V t h for each mesh node. The hidden state of the 0 th layer processor and the message are denoted as

Messages are exchanged within the mesh nodes and aggregated as follows:

The output hidden state of the processor is denoted as H t +1 h , which refers to the latent space of the mesh in the next time step. On the regular mesh, spatial adjacency is clearly defined, and each node has a fixed position. This allows for standard modeling tools (e.g., CNNs, GNNs, Transformers) to be used effectively. Any existing GNN can be implemented in these phases, which makes MIGN a flexible method. Because the spatial layout of the mesh remains fixed over time, it provides a consistent data structure across time steps, enabling more stable and coherent temporal modeling.

Station Interpolate Decoder After modeling on the mesh, the results need to be mapped back to the observation stations to enable comparison with real-world measurements. The decoder follows a reverse process of the encoder. Consider the graph in the decoder G t +1 D = ( ( V t +1 h , V t +1 s ) , E ( h,s ) , ( H t +1 h , ˆ Y t +1 s ) , ( λ t h , ϕ t h )) , ˆ Y t +1 s denotes the predicted feature in next step. The decoder would aggregate the message from the hidden state of a L layer processor directly to update the ˆ Y t +1 s as follows:

Our framework is readily adaptable to multi-step input and output, as shown in Appendix A.4.

Training Given the predicted feature ˆ Y t +1 s of the decoder. The model parameters can be optimized by minimizing the discrepancy between the prediction and ground truth: L train = ∑ s ∈D train || ˆ Y t +1 s -Y t +1 s || 2 .

3.3 Generalization Empirical Verification

To illustrate our motivation, we conduct global generalization experiments. Specifically, we randomly sample half of the stations from 2017-2023 for training and validation, while reserving the unseen half from 2024 as the test set. Detailed experimental settings are provided in Section 4.3. The results for mean sea level pressure (SLP) are visualized in Figure 3. As shown, predictions from both DyGrAE and STAR exhibit higher MAE values across large regions of Europe and North America, indicating that these baseline models struggle to generalize to previously unobserved areas. In contrast, MIGN achieves lower errors in these regions, demonstrating superior generalization performance. A complete numerical comparison is provided in Table 4.

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4 Experiments

Dataset We evaluate model performance on a up-to-date daily NOAA Global Surface Summary of the Day (GSOD) dataset. We use 6 commonly daily observed variables, including maximum temperature (MAX TEMP), minimum temperature (MIN TEMP), mean dew point (DEWP), mean sea level pressure (SLP), mean wind speed (WDSP) and maximum sustained wind speed (MXSPD). We use the 2017-2022 data for training, the 2023 data for validation, and the 2024 for testing. The detailed information of the dataset is shown in the Appendix A.6.

Baselines and metric We compare our MIGN with the following 13 spatial-temporal baselines: (1) global-based models: STGCN [41], MPNNLSTM [29], DualCast [34]. (2) global-local based models: T&S-IMP, T&S-AMP, TTS-IMP, TTS-AMP [5]. (3) dynamic graph models: DyGrAE [35], ReDyNet [7]. (4) graph pooling based models: HD-TTS [27]. (5) Transformer based models: STAR [42], GTN [33], GPS [31]. We adopt Mean Squared Error (MSE) and (Mean Absolute Error) MAE to evaluate the model performance. We run each method five times and report the average metric of all models.

Implementation We utilize Adam optimizer to train our model and use the following hyperparameters: Batch size 4, hidden state 64, and learning rate 0.001. The model is set to 2 layers. The mesh refinement level is set to 3, and we use 10-nearest neighbor to construct the graph, and the spherical harmonics degree is set to 2. All models are implemented based on Pytorch Lightning, trained on GeForce RTX3090 GPU. Baseline models are implemented with PyG library, while our model is realized with the DGL library. For a fair comparison, we tune different hyperparameters for all baselines, finding the best setting for each. The detailed information can be found in the Appendix A.7 and Tabel7.

4.1 Overall Performance

In this section, we evaluate the performance of our proposed model against several baseline methods. As summarized in Table 1, our approach consistently outperforms all baselines across every variable. In particular, MIGN achieves relative MSE improvements of 13%, 15%, and 15% on MAX TEMP, MIN TEMP, and SLP, respectively, compared to the strongest baseline. To further evaluate our model's performance across different time horizons, we conduct experiments using a three-day multistep input and a four-day multistep output training setup. The results, summarized in Table 2, show that our proposed MIGN consistently outperforms all baselines across both short- and long-term horizons, highlighting its robustness and effectiveness in the multistep forecasting setting. Besides, we further conduct a series of studies, including varying input steps, autoregressive inference. The results are presented in Appendix A.8.1.

4.2 Ablation Study

To demonstrate the effectiveness of each model design, we compare the default configuration of MIGN with four variants that differ in their use of spherical harmonics location embedding and mesh interpolation. As shown in Table 3, we observe that: (1) adopting mesh interpolation consistently improves performance; for example, DEWP and SLP MSE decrease from 9.00/23.93 to 7.92/20.09. (2) spherical harmonics embedding further enhances performance when applied to both the encoder and decoder, as the encoder embedding captures station node locations while the decoder embedding represents mesh node locations. This validates the effectiveness of spherical harmonics embeddings

in learning geometric geographic information from data. For completeness, we also compare our SH embedding with the commonly used coordinate-based embedding, with results reported in Appendix A.8.2.

4.3 Global Generalization Analysis

To evaluate model performance in a global and dynamic setting, we further conduct an experiment to validate the generalization ability of MIGN. We randomly sample half of the stations from the year 2017-2022/2023 for training and validation, while using the remaining stations from 2024 as the test set. Although the global distribution of stations is similar between the training and test sets, the test stations are entirely unseen during training. As shown in Table 4, We can find that MIGN outperforms all baselines across all variables, achieving the lowest MSE and MAE consistently. For example, MIGN achieves an MSE of 8.55/8.05 in MAX TEMP and MIN TEMP, outperforming the closest baseline 9.81/9.52 respectively. These results highlight MIGN's superior ability to generalize to unobserved stations in dynamic, real-world scenarios.

4.4 Sparse region analysis

To investigate the model performance in area with sparse weather station coverage, we analyze the model performance in data-scarce regions, including Africa, Asia, Australia, and South America, as shown in Figure 4. Across all regions and variables, MIGN consistently achieves the lowest MSE, highlighting its strong generalization capability in low-resource environments. Notably, in Asia, MIGN demonstrates significant improvements, reducing the MSE for MAX TEMP and MIN TEMP to below 8 and 6, respectively-thresholds that other models fail to surpass. These findings suggest that MIGN effectively captures variable patterns even under sparse observational conditions.

4.5 Mesh Analysis

Refinement level analysis To validate the effect of different refinement level mesh on the MIGN performance. We compare the metric of 5 different refinement levels (corresponding 48, 192, 768,

3072 and 12288 number of nodes) for mesh interpolation. The results are shown in Figure 5(A). As the refinement level increases from 1 to 3, the MSE loss of the MIGN model exhibits a decline. For WDSP and MXSPD, the model achieves optimal performance at refinement level 4. In contrast, for the other four variables, the best performance is observed at refinement levels 3. From an empirical perspective, the optimal refinement level is typically chosen based on a mesh node count that is on the same order of magnitude or one order of magnitude lower than the number of station points.

Mesh neighbors analysis Figure 5(B) illustrates the MIGN perfomance across different mesh neighbor. We observe that using 10 neighbors yields the lowest loss for almost all variables. In contrast, performance significantly degrades when using only 2 neighbors due to limited information, and again when using 40 neighbors, likely due to the inclusion of distant or irrelevant nodes introducing noise.

4.6 Spherical Harmonics Degree Analysis

To evaluate the effectiveness of spherical harmonics, we conduct experiments with varying degrees of location embedding. The results are displayed in Table 5. We discover that, with the degree of spherical harmonics increasing from 0 to 2, MIGN achieves relatively better performance. For example, the MSE of the SLP and MXSPD decreases from 23.01/20.56 to 20.09/19.73. Because the rise of degree could make embedding approximate the higher-frequency harmonic, indicating a more precise representation of the location. When the degree increases from 2 to 3, the improvement in spherical harmonics embedding becomes marginal.

4.7 Empirical analysis

We visualize the global loss of MAX TEMP in Figure 6. The results reveal a significant regional variation in the difficulty of the prediction. For maximum temperature, inland areas of North America and northern Asia exhibit higher prediction errors compared to western Europe and Africa. STGCN and HD-TTS consistently show increased losses in both data-rich regions (e.g., North America) and data-scarce regions (e.g., northern Asia), indicating their limited ability to capture the underlying

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patterns in these regions. In contrast, our model demonstrates superior performance, which indicates that the mesh design can capture the pattern of data-dense and data-sparse regions at the same time.

5 Conclusion and Future Works

In this work, we propose a MIGN framework for dynamic and spatially irregular global weather forecasting. It mitigates the spatially irregular problem by using mesh interpolation. We propose parametric spherical harmonics location embedding to learn the global weather information. Extensive experiments show that MIGN outperforms existing spatial-temporal models. Ablation studies demonstrate the effectiveness of the model designs and we further explored the hyperparameters in the mesh construction and the degree of spherical harmonics. Empirical analysis and generalization studies further illustrate the superior generalization ability. Due to the sparse distribution of weather stations over marine areas, our dataset primarily focuses on land-based observations. In future work, we plan to incorporate marine observation data to further enhance the robustness and generalization of our model in ocean-related scenarios.

Limitations Due to the sparse distribution of weather stations over marine areas, our dataset primarily focuses on land-based observations. However, incorporating additional data sources covering global oceans could further improve the performance of MIGN. Since the Earth operates as an interconnected system, integrating marine data would provide a more complete representation of global weather patterns.

References

P. Bauer, A. Thorpe, and G. Brunet. The quiet revolution of numerical weather prediction. Nature , 525(7567):47-55, 2015.

K. Bi, L. Xie, H. Zhang, X. Chen, X. Gu, and Q. Tian. Accurate medium-range global weather forecasting with 3d neural networks. Nature , 619(7970):533-538, 2023.

C. Camera, A. Bruggeman, P. Hadjinicolaou, S. Pashiardis, and M. A. Lange. Evaluation of interpolation techniques for the creation of gridded daily precipitation (1 × 1 km2); cyprus, 1980-2010. Journal of Geophysical Research: Atmospheres , 119(2):693-712, 2014.

L. Chen, J. Xu, B. Wu, and J. Huang. Group-aware graph neural network for nationwide city air quality forecasting. ACM Transactions on Knowledge Discovery from Data , 18(3):1-20, 2023.

A. Cini, I. Marisca, D. Zambon, and C. Alippi. Taming local effects in graph-based spatiotemporal forecasting. In Proceedings of the 37th International Conference on Neural Information Processing Systems , pages 55375-55393, 2023.

P. S. S. Ejurothu, S. Mandal, and M. Thakur. Forecasting pm2. 5 concentration in india using a cluster based hybrid graph neural network approach. Asia-Pacific Journal of Atmospheric Sciences , 59(5):545-561, 2023.

Q. Gao, Z. Wang, L. Huang, G. Trajcevski, G. Liu, and X. Chen. Responsive dynamic graph disentanglement for metro flow forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 39, pages 11690-11698, 2025.

K. M. Gorski, B. D. Wandelt, F. K. Hansen, E. Hivon, and A. J. Banday. The healpix primer. arXiv preprint astro-ph/9905275 , 1999.

S. Guo, Y. Lin, N. Feng, C. Song, and H. Wan. Attention based spatial-temporal graph convolutional networks for traffic flow forecasting. In Proceedings of the AAAI conference on artificial intelligence , volume 33, pages 922-929, 2019.

W. Hamilton, Z. Ying, and J. Leskovec. Inductive representation learning on large graphs. Advances in neural information processing systems , 30, 2017.

K. H. Hettige, J. Ji, S. Xiang, C. Long, G. Cong, and J. Wang. Airphynet: Harnessing physicsguided neural networks for air quality prediction. arXiv preprint arXiv:2402.03784 , 2024.

N. Hofstra, M. Haylock, M. New, P. Jones, and C. Frei. Comparison of six methods for the interpolation of daily, european climate data. Journal of Geophysical Research: Atmospheres , 113(D21), 2008.

T. N. Kipf and M. Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907 , 2016.

S. Klosko and C. Wagner. Spherical harmonic representation of the gravity field from dynamic satellite data. Planetary and Space Science , 30(1):5-28, 1982.

R. Lam, A. Sanchez-Gonzalez, M. Willson, P. Wirnsberger, M. Fortunato, F. Alet, S. Ravuri, T. Ewalds, Z. Eaton-Rosen, W. Hu, et al. Graphcast: Learning skillful medium-range global weather forecasting. arXiv preprint arXiv:2212.12794 , 2022.

P. Li, Y . Yu, D. Huang, Z.-H. Wang, and A. Sharma. Regional heatwave prediction using graph neural network and weather station data. Geophysical Research Letters , 50(7):e2023GL103405, 2023.

Y. Li, P. Wang, Z. Li, J. X. Yu, and J. Li. Zerog: Investigating cross-dataset zero-shot transferability in graphs. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 1725-1735, 2024.

Y. Li, P. Wang, X. Zhu, A. Chen, H. Jiang, D. Cai, V. W. Chan, and J. Li. Glbench: A comprehensive benchmark for graph with large language models. Advances in Neural Information Processing Systems , 37:42349-42368, 2024.

Y. Li, Y. Wang, J. Tang, H. Chang, Y. Ren, and J. Li. Advancing graph foundation models: A data-centric perspective. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.2 , KDD '25, page 1635-1646, New York, NY, USA, 2025. Association for Computing Machinery. ISBN 9798400714542. doi: 10.1145/3711896.3736833. URL https://doi.org/10.1145/3711896.3736833 .

Y. Li, X. Zhang, L. Luo, H. Chang, Y. Ren, I. King, and J. Li. G-refer: Graph retrievalaugmented large language model for explainable recommendation. In Proceedings of the ACM on Web Conference 2025 , pages 240-251, 2025.

Z. Li, N. Kovachki, C. Choy, B. Li, J. Kossaifi, S. Otta, M. A. Nabian, M. Stadler, C. Hundt, K. Azizzadenesheli, et al. Geometry-informed neural operator for large-scale 3d pdes. Advances in Neural Information Processing Systems , 36:35836-35854, 2023.

Y. Liu, Z. Zheng, Y. Rong, and J. Li. Equivariant graph learning for high-density crowd trajectories modeling. Transactions on Machine Learning Research .

Y. Liu, L. Chen, X. He, J. Peng, Z. Zheng, and J. Tang. Modelling high-order social relations for item recommendation. IEEE Transactions on Knowledge and Data Engineering , 34(9): 4385-4397, 2020.

Y. Liu, J. Cheng, H. Zhao, T. Xu, P. Zhao, F. Tsung, J. Li, and Y. Rong. Segno: Generalizing equivariant graph neural networks with physical inductive biases. arXiv preprint arXiv:2308.13212 , 2023.

Y. Liu, Z. Zheng, J. Cheng, F. Tsung, D. Zhao, Y. Rong, and J. Li. Cirt: Global subseasonalto-seasonal forecasting with geometry-inspired transformer. arXiv preprint arXiv:2502.19750 , 2025.

Y. Liu, Z. Zheng, Y. Rong, D. Zhao, H. Cheng, and J. Li. Equivariant and invariant message passing for global subseasonal-to-seasonal forecasting. In KDD , page 1879-1890, 2025.

I. Marisca, C. Alippi, and F. M. Bianchi. Graph-based forecasting with missing data through spatiotemporal downsampling. arXiv preprint arXiv:2402.10634 , 2024.

S. Mouatadid, P. Orenstein, G. Flaspohler, J. Cohen, M. Oprescu, E. Fraenkel, and L. Mackey. Adaptive bias correction for improved subseasonal forecasting. Nature Communications , 14(1): 3482, 2023.

G. Panagopoulos, G. Nikolentzos, and M. Vazirgiannis. Transfer graph neural networks for pandemic forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 35, pages 4838-4845, 2021.

J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, et al. Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. arXiv preprint arXiv:2202.11214 , 2022.

L. Rampášek, M. Galkin, V. P. Dwivedi, A. T. Luu, G. Wolf, and D. Beaini. Recipe for a general, powerful, scalable graph transformer. Advances in Neural Information Processing Systems , 35: 14501-14515, 2022.

M. Rußwurm, K. Klemmer, E. Rolf, R. Zbinden, and D. Tuia. Geographic location encoding with spherical harmonics and sinusoidal representation networks. In Proceedings of the International Conference on Learning Representations (ICLR) , 2024. URL https://iclr.cc/virtual/ 2024/poster/18690 .

Y. Shi, Z. Huang, S. Feng, H. Zhong, W. Wang, and Y. Sun. Masked label prediction: Unified message passing model for semi-supervised classification. arXiv preprint arXiv:2009.03509 , 2020.

X. Su, F. Liu, Y. Chang, E. Tanin, M. Sarvi, and J. Qi. Dualcast: A model to disentangle aperiodic events from traffic series. IJCAI , 2025.

A. Taheri, K. Gimpel, and T. Berger-Wolf. Learning to represent the evolution of dynamic graphs with recurrent models. In Companion proceedings of the 2019 world wide web conference , pages 301-307, 2019.

E. Thébault, G. Hulot, B. Langlais, and P. Vigneron. A spherical harmonic model of earth's lithospheric magnetic field up to degree 1050. Geophysical Research Letters , 48 (21):e2021GL095147, 2021.

K. E. Ukhurebor, C. O. Adetunji, O. T. Olugbemi, W. Nwankwo, A. S. Olayinka, C. Umezuruike, and D. I. Hefft. Precision agriculture: Weather forecasting for future farming. In Ai, edge and iot-based smart agriculture , pages 101-121. Elsevier, 2022.

P. Veliˇ ckovi´ c, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903 , 2017.

H. Wu, H. Zhou, M. Long, and J. Wang. Interpretable weather forecasting for worldwide stations with a unified deep model. Nature Machine Intelligence , 5(6):602-611, 2023.

Z. Xie, R. Black, and Y. Deng. Daily-scale planetary wave patterns and the modulation of cold season weather in the northern extratropics. Journal of Geophysical Research: Atmospheres , 122(16):8383-8398, 2017.

B. Yu, H. Yin, and Z. Zhu. Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. arXiv preprint arXiv:1709.04875 , 2017.

C. Yu, X. Ma, J. Ren, H. Zhao, and S. Yi. Spatio-temporal graph transformer networks for pedestrian trajectory prediction. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part XII 16 , pages 507-523. Springer, 2020.

H. Zhao, A. Chen, X. Sun, H. Cheng, and J. Li. All in one and one for all: A simple yet effective method towards cross-domain graph pretraining. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 4443-4454, 2024.

H. Zhao, C. Zi, Y. Liu, C. Zhang, Y. Zhou, and J. Li. Weakly supervised anomaly detection via knowledge-data alignment. In WWW , page 4083-4094, 2024.

Z. Zheng, Y. Liu, J. Li, J. Yao, and Y. Rong. Relaxing continuous constraints of equivariant graph neural networks for broad physical dynamics learning. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 4548-4558, 2024.

C. Zi, H. Zhao, X. Sun, Y. Lin, H. Cheng, and J. Li. Prog: A graph prompt learning benchmark. Advances in Neural Information Processing Systems , 37:95406-95437, 2024.

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A Appendix

A.1 Broader Impacts

Climate change has enhanced weather variability and extreme event frequency, such as heatwaves, droughts, and heavy rainfall, resulting in enormous socioeconomic loss. Accurate weather forecasting, especially in urban areas, is crucial for mitigating their impacts and benefiting various aspects of human life, including transportation management, agricultural planning, and resource allocation. Although multiple weather foundation models have been proposed, they focus on coarse-grained global forecasting of reanalysis data. Accurate predicting weather station observations, which are closer to urban areas and with fewer biases, are critical to weather forecasting applications.

A.2 Limitations

Direct observation prediction models are inherently limited by the spatial distribution of stations. Unlike gridded models, which provide uniform coverage across the globe, station-based models can only make predictions at locations where observation data exists. Consequently, regions with sparse or no stations-such as remote oceans, deserts, or polar areas-remain unobserved and cannot be accurately modeled. This limitation restricts the ability of station-based approaches, especially when applied to areas far from the existing observational network.

A.3 Additional Related Work

GINO [21] also leverages latent regular grid. However, GINO aims to simulate computational fluid dynamics. Such different application scopes result in distinctive model designs. GINO utilizes a regular 3D grid for variable input geometry. In contrast, MIGN employs a HEALPix mesh as the regular grid, which is aligned with the inherent spherical geometry of Earth. To model the spatially irregular and dynamic station distribution, we further incorporate a spherical harmonic embedding to enhance the spatial generalization ability of the model, which is not considered in GINO.

A.4 Temporal Format of MIGN

MIGN can be naturally extended to a multi-step input-output setting. We define the input steps as a sequence of past observations from t -n to t , and the output steps as the sequence of future predictions from t +1 to t + m . Formally, the input consists of station nodes V t -n s , . . . , V t -1 s , V t s and their corresponding mesh nodes V t -n h , . . . , V t -1 h , V t h .

For each input step, we independently apply the Mesh Interpolation Encoder and message passing as defined in Eq. (10)-(13), producing hidden states h l v t -n , . . . , h l v t h for mesh nodes and aggregated

h

We then concatenate the temporal mesh representations [ H t -n h ; . . . ; H t -1 h ; H t h ] and project them through a linear layer to obtain the output latent states [ H t h ; . . . ; H t + m h ] for the mesh. Finally, for each output step, the Station Interpolation Decoder (Eq. (14)) maps mesh states back to station predictions, yielding ˆ Y t +1 s , ˆ Y t +2 s , . . . , ˆ Y t + m s .

A.5 Spherical Harmonics Location Embedding

The illustration of spherical harmonics location embedding is shown in Figure 7. We regard the weather information of the station nodes as a learnable spherical harmonics function. The spherical harmonics can be precomputed according to the coordinates and we learn the weight in the model directly.

A.6 More Details on Datasets

Data source Global Surface Summary of the Day( https://www.ncei.noaa.gov/metadata/ geoportal/rest/metadata/item/gov.noaa.ncdc:C00516/html ) is derived from The Integrated Surface Hourly (ISH) dataset. The latest daily summary data are normally available 12 days after the date-time of the observations used in the daily summaries. The updated fre-

Lon. Lat.

SH{

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quency and reference time are daily and Greenwich Mean Time. The data can be download in the link https://www.ncei.noaa.gov/data/global-summary-of-the-day/archive/ . We accessed the data on 2025/4/7. We computed both the total count of data points per variable over the period 2017-2024 in the Table 6.

Dataset Selection Rationale We focus on daily maximum/minimum temperature and maximum sustained wind speed, which are inherently defined at daily timescales and not well captured by hourly aggregation. Daily forecasting also better aligns with medium-range horizons, whereas hourly data primarily serve short-term operations. Spatially, daily observations offer broader coverage, averaging over 10,000 stations per day compared to about 6,000 per hour (NOAA 2022), which enhances spatial pattern learning and supports global generalization. This expanded coverage provides a stronger foundation for capturing diverse geographical features and complex spatial dependencies essential to our modeling approach.

A.7 Baselines Implementation

STGCN [41] is implemented base on Pytorch Geometric Temporal library https://github. com/benedekrozemberczki/pytorch_geometric_temporal . The graph convolution kernel size K is set to 1.

DyGrAE [9] is implemented base on Pytorch Geometric Temporal library https://github.com/ benedekrozemberczki/pytorch_geometric_temporal . We use mean convolution aggregate method.

STAR [42] is implemented base on Pytorch Geometric. The attention head is set to 4.

GTN [33] is implemented base on Pytorch Geometric. The attention head is set to 4.

GPS [31] is implemented base on Pytorch Geometric. The attention head is set to 1.

MPNNLSTM [29] is implemented base on Pytorch Geometric Temporal library https://github. com/benedekrozemberczki/pytorch_geometric_temporal . The dropout rate is set to 0.5 and the Window is set to 1.

T&S-IMP, T&S-AMP, TTS-IMP, TTS-AMP [5] are implemented base on official code https:// github.com/Graph-Machine-Learning-Group/taming-local-effects-stgnns . We use 'elu' activation and mean normalization.

HD-TTS [27] are implemented base on official code https://github.com/marshka/hdtts . We use anisoconv message passing method and kmis pooling method, with the dilation and kernel size are set to 1

ReDyNet (2025) are implemented base on official code https://github.com/wangzz-yyzz/ ReDyNet .

DualCast (2025) are implemented base on official code https://github.com/suzy0223/ DualCast .

We use Wandb Sweeps to automate hyperparameter search for each baseline and each varibales, utilizing the Bayesian sweep method. The hyperparameters are shown in Tabel7.

A.8 Additional Results

A.8.1 Time Horizon Analysis

Input step analysis To evaluate the models' ability to leverage varying lengths of historical input for accurate forecasting, we conduct an additional experiment using different input step settings. The results of MAX TEMP, MIN TEMP and DEWP, presented in Figure 8, demonstrate that MIGN consistently outperforms all other models across all three variables and input steps. While the baseline models show slight improvements as the number of input steps increases, MIGN achieves a more significant reduction in loss, particularly on DEWP, where the MSE drops from 8 to 7. Notably, increasing the input step from 3 to 4 yields only marginal gains for most models, indicating a diminishing return from longer input histories. The results of SLP, WDSP and MXSPD are presented in Figure 9. Across all variables, a general trend is observed where increasing the input step consistently leads to lower MSE loss, indicating that incorporating more historical information improves prediction accuracy. MIGN demonstrates the best overall performance. Its advantage becomes especially pronounced as the input step increases, achieving the lowest MSE in all three variables when the input step reaches 4. In contrast, Persistence, which serves as a naive baseline, maintains a high and constant error across all settings, emphasizing the benefits of using learningbased approaches.

Autoregressive inference analysis We evaluate the autoregressive forecasting performance of our model (MIGN) against a series of competitive baselines. Results are reported in Tables 8 and 9. Across all variables, MIGN consistently achieves the best performance in terms of both MSE and MAE. In particular, for MAX TEMP, MIGN reduces the total MAE to 2.72, outperforming the strongest baseline (STGCN, 2.96). For MIN TEMP, MIGN achieves a total MAE of 2.63, significantly lower than previous methods. Similar improvements are observed for DEWP, where MIGN yields a total MAE of 2.76, and for SLP, where MIGN achieves a total MAE of 4.36.

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Notably, the performance gain becomes more pronounced at longer forecasting horizons (Step 2 and Step 3), indicating that MIGN is particularly effective at capturing temporal dependencies in extended autoregressive prediction. These results demonstrate that integrating mesh-station interactions enables our model to generalize better across different meteorological variables and forecasting horizons, thereby enhancing both short-term and long-term prediction accuracy.

A.8.2 Further Ablation study of location encoding

To investigate the impact of different location embedding strategies, we compare our proposed Spherical Harmonics (SH) embedding method with three commonly used coordinate-based approaches: Direct Coordinate , WRAP , and Cartesian 3D . The formulations of these methods are as follows. Let longitude λ ∈ [ -π, π ] and latitude θ ∈ [ -π/ 2 , π/ 2] .

· Direct Coordinate:

· WRAP:

· Cartesian 3D:

We apply the above three location embeddings as well as our SH embedding method to both the encoder and decoder of our model. Table 10 reports the mean squared error (MSE) on six meteorological variables. Our proposed SH embedding consistently outperforms the baseline methods, highlighting the effectiveness of modeling spherical positional information using harmonics.

A.8.3 Sparse region analysis

To assess the generalization ability of the models in data-scarce regions, we conduct experiments across Africa, Asia, Australia, and South America, focusing on three key meteorological variables: DEWP, WDSP, and MXSPD, as shown in Figure 11. Across all variables and regions, MIGN consistently achieves the lowest MSE, demonstrating robust performance in regions with limited observational data. Particularly in South America, MIGN significantly outperforms other baselines for DEWP, achieving an MSE below 3.5, while other models yield considerably higher errors. Similarly, for WDSP and MXSPD, MIGN maintains stable and low error rates across all continents, showcasing its ability to generalize well across diverse climatic conditions. These results further confirm MIGN's effectiveness in learning reliable patterns even when data availability is limited.

A.8.4 Mesh analysis

The MAE result of refinement level analysis and mesh neighbors analysis is shown in Figure 12. In Figure 12(A), performance improves as the refinement level increases from 1 to 3, reaching the lowest MAE at level 3. Beyond this point, error increases again, suggesting that too fine a mesh may introduce noise. In Figure 12(B), the optimal number of neighbors is around 5-10, where MAE is minimized. Too few neighbors (e.g., 2) lack spatial context, while too many (e.g., 20 or 40) may introduce irrelevant information, hurting performance.

A.8.5 Visualization of Spherical Harmonics embedding

Since our spherical harmonics are designed to learn the coefficients w m n , we compute the spherical function f ( λ, ϕ ) = ∑ 3 n =0 ∑ n m = -n w m n Y n m ( λ, ϕ ) , where w m n are the learned coefficients in the MAX task with degree 3. As illustrated in the Figure 10, different regions on the globe exhibit distinct colors: North America appears purple, South Africa black, Europe yellow, and Asia red. This indicates that the spherical harmonics embedding can capture location-specific information.

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A.8.6 Comparison with gridded model

Grid-to-Station Evaluation To further examine the necessity of station-based approaches, we conducted a direct comparison against gridded reanalysis models. Specifically, we evaluated Pangu's 2022 gridded forecasts from WeatherBench2 by bilinearly interpolating them to station locations, ensuring a fair comparison with our station-based model (MIGN). MIGN was trained on station observations from 2017-2020, validated on 2021, and tested on 2022. Results are summarized in Table 11:

Despite reanalysis data offering broader spatial coverage and higher temporal frequency, MIGN consistently outperforms Pangu across all three variables when evaluated at station locations. The advantages of station-based learning are threefold: (i) it directly leverages ground-truth observations without inheriting potential biases or smoothing artifacts introduced during reanalysis assimilation, (ii) it preserves fine-scale spatial variability essential for capturing extremes and urban microclimates, and (iii) it avoids the computational burden of handling full 4D gridded datasets, enabling efficient training and deployment even under limited hardware resources.

These strengths make station-based approaches particularly valuable in scenarios where high-quality observations are available, offering sharper local accuracy and practical efficiency that complement the broad coverage of reanalysis methods.

Unmonitored-Location Generalization To further investigate the spatial generalization of stationbased methods, we conducted an experiment where 10% of observation sites were withheld as unmonitored ground truth points. The remaining 90% of stations were used for training our MIGN model. For a fair comparison, both Pangu and MIGN predictions were bilinearly interpolated to these unmonitored sites. Results are summarized in Table 12:

MIGN outperforms Pangu at 1.00° resolution on MAX TEMP and MIN TEMP, and achieves competitive WDSP accuracy. However, Pangu at 0.25° grid spacing remains superior, which can be attributed to its massive data and computational advantages. Pangu is trained on over one million global reanalysis points per snapshot, leveraging rich multi-variable inputs (geopotential height, temperature, humidity, wind, etc.) across multiple vertical levels, requiring more than 200 TB of training data. By contrast, MIGN operates with only 10,000 data points per day and a total

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data volume of roughly 10 GB, highlighting its efficiency and accessibility under data-scarce or resource-constrained conditions.

A.9 Computational Cost Analysis

Analysis of mesh interpolation Mesh interpolation is achieved by constructing a nearest neighbors graph between feature nodes and Healpix nodes. Suppose there are M feature nodes and N Heslpix nodes. Using a brute-force approach, the computational complexity is O ( MN ) . For baselines, the computational complexity of nearest neighbor connections among the feature nodes is O ( M 2 ) . Mesh analysis experiments Figure 12a indicate that the optimal number of Healpix nodes N is smaller than the number of feature nodes M , meaning that N < M , thus O ( MN ) < O ( M 2 ) . Furthermore, by utilizing a KD-Tree algorithm, we can further reduce the complexity from O ( MN ) to O ( NlogN + MlogN ) .

Training and inference efficiency To further demonstrate the training and inference efficiency of our model, we compare the training and inference times per step of several baselines with MIGN on an NVIDIA RTX 3090 GPU, as shown in the following table. We observe that the training and inference time of MIGN is comparable to that of STGCN and MPNNLSTM, demonstrating its efficiency and practical effectiveness.

Analysis of spherical harmonics Spherical harmonic basis functions are precomputed and stored for efficiency. The Spherical Harmonics (SH) Degree Analysis Experiment 5 demonstrates that a degree of 2 is sufficient for location embedding. Thus, its computational cost is linear to the node number, which is not the primary time-consuming component. We measure the processing speed for MAXTEMPdata with a degree-3 SH embedding (13260 nodes) on an AMD EPYC 75F3 32-Core Processor. The computation completes in just 2s.

A.10 Empirical analysis

To further illustrate our motivation, we visualize the global loss of MAX TEMP, MIN TEMP, DEWP, SLP in Figure 13, 14, 15, 16. The results reveal that prediction difficulty varies significantly across regions. For MAX TEMP, MIN TEMP and DEWP, inland areas of North America and northern Asia exhibit higher prediction errors compared to Western Europe and Africa, highlighting distinct regional characteristics and suggesting that different regions follow different weather patterns. Baseline models consistently show higher losses in North America and northern Asia, indicating their limited ability to capture the underlying patterns in these regions. In contrast, our model demonstrates superior performance across both the United States and the Asian continent. This suggests that mesh interpolation and spherical harmonics facilitate the learning of global patterns and effectively capture regional features. For SLP, model performance tends to degrade in high-latitude regions, including Western Europe and North America. This pattern suggests increased difficulty in capturing surface-level pressure dynamics in these areas, possibly due to more complex atmospheric interactions and variability at higher latitudes. Baseline models exhibit particularly high prediction errors in these regions, reinforcing their limitations in modeling such complexity. In comparison, our model maintains relatively stable performance, indicating its enhanced capacity to learn intricate spatial patterns through mesh interpolation and spherical harmonics.

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MAE

STAR

T&S-IMP

DyGrAE

T&S-AMP

TTS-IMP

TTS-AMP

MIGN

5

3

2

1

0

4

6

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes MIGN, a deep learning model for weather forecasting that handles irregularly distributed sensor data, different from previous work, which typically assumes input data is arranged on a regular grid. Sensors are also allowed to be missing for some time steps. The model encodes sensor observations onto a global mesh, performs message passing across the mesh, and decodes the results back to the sensor locations. A key design element is the use of spherical harmonics as positional encodings to improve spatial generalization. Experimental results show that MIGN outperforms prior graph-based approaches on the considered task.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n1. The paper tackles a highly relevant and timely task (weather forecasting) and contributes to the broader effort to reduce the computational cost associated with traditional numerical weather prediction (NWP) models.\n2. The setting adopted in this work falls within the Direct Observation Prediction (DOP) paradigm, which focuses on forecasting weather directly from meteorological observations. This is a particularly challenging setup due to the spatial sparsity of the data and the dynamic nature of the sensor network (i.e., sensors may appear or disappear). This makes the problem formulation both realistic and impactful.\n3. While there is room for improvement in the clarity of the text and notation, the core ideas of the proposed approach are communicated effectively. In particular, the figures are well-designed and convey the intended messages clearly and concisely.\n4. The paper includes a wide experimental analysis, comparing MIGN against both standard baselines and state-of-the-art spatiotemporal graph neural networks. Ablation studies are thoughtfully conducted and demonstrate the contribution of each architectural component. Additionally, confidence intervals are reported, which strengthens the reliability of the results.\n\n## Weaknesses\n\n1. The proposed approach is evaluated on a single dataset, derived from the NOAA Global Surface Summary of the Day service. As the name suggests, this dataset contains daily-resolution data, resulting in very smooth time series. Furthermore, the chosen task — predicting one day ahead — is of limited interest in this context (compared to, e.g., hourly data). Given the coarse temporal resolution, such forecasting is often trivial, as evidenced by the strong performance of the persistence model (which simply copies the previous day's value), which frequently ranks among the best-performing baselines.\n2. The problem is set as next-day prediction based solely on data from the previous day. This setup imposes an overly restrictive condition on sequence-processing models, such as those used in several baselines. Moreover, figures 8 and 9 in the appendix show that increasing the look-back window improves performance across all models, including MIGN. Therefore, it is unclear why the main experiments in the paper are limited to a one-step history.\n3. Most of the baselines rely on recurrent neural networks (RNNs) to model temporal dynamics. These models are inherently designed to process sequences, and are thus disadvantaged when constrained to a single static input matrix, as is the case here. In contrast, MIGN operates on static graph structures (see Equation 9), which gives it a natural advantage under this setting.\n4. Many of the baselines, particularly those labeled “global-local” and the HD-TTS model, are equipped with node-specific parameters. This design choice introduces new challenges in scenarios with dynamic sensor networks. Since these parameters are learned during training, sensors not seen in the training set retain randomly initialized weights. As a result, performance may degrade significantly when many sensors are encountered for the first time during testing or appear sporadically in the training set. This issue is particularly relevant for the Global Generalization Analysis, in which half the sensors are held out during training. The paper should explicitly address how this challenge is handled.\n5. The code repository provided with the paper includes only the implementation of the proposed architecture. Precise experimental details for the baselines are not specified, either in the repository or in the paper. This omission prevents a full understanding of how experiments were conducted and undermines reproducibility. Additionally, several baselines were trained using very small batch sizes (1–2 samples), which can significantly hinder their performance.\n6. While the core ideas are conveyed adequately, the paper suffers from issues in organization, notation, and language. Some symbols and operations are not properly introduced (e.g., “[ ; ; ]”), and Section 3.3 references results and acronyms from the following section without first describing the experimental setup. Encoding and decoding procedures are initially described without spherical harmonics in Section 3.1, only to be redefined with harmonics in Section 3.2, creating confusion. The use of \"would\" in lines 158–162 is also grammatically incorrect. These are just a few examples, but overall, the manuscript would greatly benefit from a thorough revision.\n\nQUESTIONS:\n1. Why did the authors choose to evaluate only on a dataset with daily resolution, and why is the forecasting task limited to one-day-ahead predictions? Given the smoothness of daily-resolution data, this setup seems to trivialize the forecasting task. Could the authors clarify whether MIGN could be tested on higher-frequency data (e.g., hourly), where forecasting would be more challenging and meaningful?\n2. Why is the input limited to a single time step? Results in the appendix suggest that increasing the look-back window improves performance across all models, including MIGN. Could the authors explain the rationale for restricting the input window to one step in the main experiments?\n3. How are node-specific parameters handled in global-local architectures under a dynamic sensor network? In scenarios where sensors appear only at test time, node-specific parameters (which are trained end-to-end) remain randomly initialized. How is this issue addressed, particularly in the Global Generalization Analysis, where many sensors are unseen during training?\n4. How does MIGN handle longer input sequences, and how do the baselines handle missing stations in those cases? For example, in a longer input window, how is MIGN modified to integrate multiple time steps? For the baselines, how is prediction handled when a sensor is only observed at time t+1, but not at time t? Which graph is used for the baselines?\n5. How were the baselines evaluated on this dataset, and do they share a unified codebase for training and evaluation? The code repository only includes MIGN. To ensure a fair comparison, it's important to understand whether all baselines were implemented and evaluated under consistent experimental conditions. Could the authors provide more details on the training and evaluation protocols used for the baselines?\n\nLIMITATIONS:\nWhile the authors do mention some limitations, two important points deserve more attention:\n\n1. The forecasting task focuses on daily summary weather prediction, which, due to its low temporal resolution, may not fully capture the complexity or practical demands of real-world weather forecasting applications.\n2. The setup of using only the previous day's data to predict the next day imposes a restrictive constraint on forecasting, and is especially limiting for sequence models.\n\nThese limitations should be more explicitly acknowledged and discussed in the manuscript.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel Mesh Interpolation Graph Network (MIGN) to address the issue of limited generalization ability in existing weather forecasting models, which is caused by finite and local areas for training and the neglect of the irregular and dynamic distribution of meteorological stations. MIGN effectively mitigates the spatial irregularity of meteorological station data through mesh interpolation and enhances global weather learning via parametric spherical harmonics location embedding, thereby endowing the model with robust generalization capabilities. Extensive experiments demonstrate MIGN's superior performance, spatial generalization ability, and capacity to generalize to previously unseen stations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The spatial correlations among irregularly and dynamically distributed meteorological stations are modeled through message passing between meteorological station nodes and regular grid nodes.\n2. By employing parametric spherical harmonics location embedding to encode the coordinates of meteorological stations, the model is empowered to generalize to unobserved areas.\n3. Through extensive comparative and ablation experiments, this paper comprehensively validates the superior performance and robust generalization capability of MIGN, especially in sparse regions and newly added meteorological stations. \n\nWeaknesses:\n1. As the authors mentioned in line 259, using the value from the previous day as the prediction for the next day has already outperformed most deep-learning models. Moreover, the benchmark methods chosen for the comparative experiments do not seem to include more novel approaches from the past two years, which somewhat undermines the persuasiveness of the experimental results.\n2. Both in sections 3.2 and 4.2, the authors fail to provide a detailed rationale for choosing parametric spherical harmonics to learn geographical geometric information, instead of directly inputting coordinate information into the neural network to learn the location embedding. A more thorough explanation, supported by additional ablation studies, is needed to validate this choice.\n3. Some of the equations in the paper are either redundant or not fully explained. For example, the significance of  $n$ and $m$ in Equation 4 is not explained; Equations 9 and 11 contain redundant superscript $t$ with the same meaning, which could be further simplified. Moreover, Equation 9 does not explain how to determine the neighboring meteorological station nodes for the regular grid, and this needs to be clarified.\n\nQUESTIONS:\n1. As the authors mentioned in line 259, using the value from the previous day as the prediction for the next day has already outperformed most deep-learning models. Moreover, the benchmark methods chosen for the comparative experiments do not seem to include more novel approaches from the past two years, which somewhat undermines the persuasiveness of the experimental results.\n2. Both in sections 3.2 and 4.2, the authors fail to provide a detailed rationale for choosing parametric spherical harmonics to learn geographical geometric information, instead of directly inputting coordinate information into the neural network to learn the location embedding. A more thorough explanation, supported by additional ablation studies, is needed to validate this choice.\n3. Some of the equations in the paper are either redundant or not fully explained. For example, the significance of $n$ and $m$ in Equation 4 is not explained; Equations 9 and 11 contain redundant superscript $t$ with the same meaning, which could be further simplified. Moreover, Equation 9 does not explain how to determine the neighboring meteorological station nodes for the regular grid, and this needs to be clarified.\n4. When presenting the model performance, the authors should indicate the specific percentage of improvement rather than simply listing the corresponding metric results.\n\nLIMITATIONS:\nYes. The authors discussed the limitations of their work in the conclusion section and elaborated on the Broader Impacts in the Appendix A.1.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Mesh Interpolation Graph Network (MIGN), a new framework for global weather forecasting using irregular and dynamically varying weather station data. The key contributions are (1) a mesh-based encoder-decoder architecture that interpolates irregular station data onto a regular mesh and performs message passing there, and (2) the use of parametric spherical harmonics to embed station locations, enabling better spatial generalization. The model is evaluated on the NOAA GSOD dataset and shows strong performance against 13 baselines. It also demonstrates superior generalization to unseen stations and robustness in data-scarce regions.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The paper addresses a meaningful problem in weather forecasting — how to model irregular and dynamic sensor data at global scale.\n2. The encoder–processor–decoder design is intuitive and effective for this task. \n3. The empirical section is extensive. The model consistently outperforms all baselines across a wide range of metrics. The generalization experiments to unseen stations and performance breakdowns by region are especially interesting.\n\n\n## Weaknesses\n1. The method forecasts one step ahead, and the architecture appears to focus almost entirely on spatial modeling. It’s not clear how well this would extend to multi-step forecasting or longer temporal horizons, which are important for operational forecasting use cases.\n2. While spherical harmonics are theoretically sound, the paper does not provide much intuition or visualization of what the learned SH embeddings capture. \n3. It’s not fully clear how the model scales with increasing station density or mesh resolution. A brief introduction on training/inference cost would help contextualize the method’s practicality.\n\nQUESTIONS:\n1. How does the model perform when provided with longer historical input sequences (e.g., 3–5 days)? Is the current architecture capable of forecasting multiple steps into the future? Additionally, since the paper emphasizes the challenge of dynamically changing station distributions, could the authors elaborate on how the model handles situations where the spatial distribution shifts during multi-step inference?\n2. How does the model perform in regions with no stations, such as oceans (as you mentioned in limitations)? Would MIGN still provide meaningful forecasts in these areas, and how would its performance compare to grid-based models like GraphCast? More broadly, can station-based models like MIGN be better compared to grid-based models? For example, given historical data or overlapping regions, is it feasible to compare their forecasts either locally (e.g., at specific locations) or globally (e.g., averaged over the domain)?\n\nLIMITATIONS:\nNo, please see questions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper presents a novel framework for global weather forecasting that addresses two fundamental challenges: spatially irregular weather station distributions and temporal variability in station availability. The approach combines mesh interpolation for mapping irregular observations onto a regular HEALPix grid with spherical harmonics location embedding for enhanced spatial generalization. Evaluated on the NOAA GSOD dataset against 13 baseline methods, MIGN demonstrates superior performance across multiple meteorological variables.

STRENGTHS AND WEAKNESSES:
Strengths:
- The paper is well-organized and clearly written, with a logical flow from problem formulation to experimental validation.
- The experimental evaluation and ablation analysis are comprehensive. The model's superior performance in data-scarce regions (e.g., Africa, South America) is particularly noteworthy for operational forecasting applications.

Weaknesses:
- The evaluation is limited to single-step forecasting, while multi-step autoregressive assessment would provide stronger evidence of the model's predictive capabilities, especially regarding error accumulation over extended horizons.
- The Global Generalization Analysis presents counterintuitive results that require clarification. Despite training on only half of the randomly sampled stations, several baseline models (HD-TTS, T&S-IMP, STAR) demonstrate improved performance compared to full dataset experiments (Table 3 vs. Table 1). For instance, HD-TTS shows reduced MSE for MAX TEMP (10.20 → 9.81), along with improvements in DEWP and WDSP metrics.

QUESTIONS:
See weaknesses.

LIMITATIONS:
The authors have adequately addressed the limitations of their work.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03