Mesh Interpolation Graph Network for Dynamic and Spatially Irregular Global Weather Forecasting
Authors: Zinan Zheng
Yang Liu
Jia Li
Venue: NeurIPS 2025 poster ()
Keywords: Weather forecasting
Graph neural network
Earth sciences
Primary area: machine_learning_for_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
Zinan Zheng', Yang Liu?; Jia Li
'The Hong Kong University of Science and Technology (Guangzhou)
'The Chinese University of Hong Kong
zzheng078@cconect. hkust-gzz eeu.uu.z..uu.u. edu. en y yliuweather@gmaill com, jialefust. .k
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 fectively. Thus, in this work, we study global weather forecasting that is irregularly distributed and dynamically varying practice, requiring the model generalize to unobserved locations. To address such challenges, we propose 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 generalizing to previously unseen stations.
Introduction
Stations Distrbbutiee
(Bi0,000
0694
(5)
Number Your
Probsbattyy Danslly
082
Total Statiass
incressdd Stations
33.
0.000
0.00
Longitude
Latitude
50
0
2000
2004
2008
2002
2010
2000
2004
Figure 1: (A), Illustrations of spatially irregular station distribution. (B). The probability density of the station in terms of longitude and latitude. (C). The recorded number stations the up-to-date NOAA Global Surface Summary of the Day (GSOD) dataset for each year.
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 crucial role in safeguarding lives and property. To address these problems, multiple date-driven models have been
"Corresponding authors
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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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. 44
proposed for weather forecasting, A series of works [3 15 25
19 17 [8] 20, [9] 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.
In contrast, graph neural networks [46] 45 22
26
43 24
However, most of the work [66 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 's weather system is globally connected. As 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 (igure[[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[ 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 (C) shows the temporal variations in station data of NOAA GSOD datase This can be due to the establishment of metcorological 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 I 39] typically use a fixed number of meteorological stations for predictions. Training models on 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 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 predict weather variables in areas with sparse observations or finite historical records. We propose a Mesh Interpolation Graph Network (MIGN) framework that implements 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 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- ofttee---hh--rr art spatial-tempont 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: http:///gggttttaoooooooo/osaaaaa
com compasszzn....ccccc /MIGN
https: //www ncei. noa.. www..//www...eeeetttttaee///aaaa
nca:
000516

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2 Preliminary and Related Work
Weather Forecasting Traditional weather forecasting depends on Numerical Weather Prediction (NWP) [ 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 as FourCastNet [3), Pangu [2], and GraphCast (15]-hap 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 IT] 39] typically assume 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 graph G (V*, Et xx (X', (')), where y (of. , " is the set of nodes. gx = [(vf, of) of. of E VCJ is edge sets, which is con- structed via k-nearest neighbor and the edge attributes (e.g., node distances) are denoted by dıj. Each station collects a single weather feature, X 2zj.22.... =ffy] is a collection of node feature where xy E R, Ve E V*_ (X', Φ¹) denotes global geographic coordinate where longitude X [-x, π] and latitudes of E [- 2: J] Given the initial condition g', our objective is learn a neural network to predict the next day weather feature value, as shown in the following:
y = fe E X (")),
where O denotes the parameters of the neural network. Y denotes the predicted feature while Y1+1 denotes the label. Note that the label Y here is different to X 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 4 [ and heatwave prediction[I6: The above methods utilize GNNs capture the spatial correlation and use time series models model temporal dependency. GNNs are typically implemented using message passing mechanisms. Given a graph EV, X, (x, Φ)). the message from node u to node v at layer is given by:
(p()) (h!-1 .,t"") Vu E NV)).
m b
where (0()) can be instantiated as a multi-laye perception (MLP). The generated messages from all neighbors are aggregated the target node v and the aggregated message m! is used to update the state of the node with function UPDATE' as follows:
m ( () (1m2... : (()))) h4 UPDATE' (h"'),m")
where (G can be implemented functions like sum, mean, max pooling neural network [10 and UPDATE is a learnable function, such as MLP or a gated recurrent unit (GRU).
38 13
23
However, the above framework ignores that the number and spatial distribution of stations change over time. Such model often fails to predict features in unseen locations.
Mesh Interpolation Mesh interpolation is common approach in Earth science for using spatially irregular station observation data to reproduce regular mesh data [12 3 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 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 specified exponent. Meshes are not only used in
Traditional interpolation
3

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traditional numerical methods but have also been widely adopted in data-driven approaches, One of the most pioneering works in this area GraphCast [5]. It maps local regions of the imput to the nodes of the multi-mesh graph structure and performs message passing on mesh as well. However, focuses on the regular gridded data and the edges between mesh and nodes are statie, 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 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 [4]. To be specific, function f(,, 6) defined on the sphere can be represented by set of orthonormalized spherical harmonics Y (A, Φ) as follows:
f(),)) -
W Y (A,
where 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 E [-n, of the basis functions, governing the oscillations in the longitudinal direction. À and e are longitude and latitude respectively. We consider maximum degree of N, which results in total of (N + 1)2 basis functions and learnable weights w The spherical harmonics functions defined on the sphere as:
2n + 1 (n /ml)!
Y (A,
pm (cos Ame
(5)
4x (n + [m])!
where pm are associated Legendre polynomials:
PTT (+) (-1) 2. 23)1m//2 dlm
Pa (z))
(6)
dr mm
which involve derivatives of Legendre Polynomials (x) defined by the following recurrence: Po(z) 1, (x) x, (z) (2n 1)xPn-1(2)) (n 1) In practice, we consider the real spherical harmonics given as sin()mlø) 1 (ms)
Y (A, Φ) = p|mm pllm A) {
0
m = 0 >>> 0.
(8)
where p!!ml (cos 2)
2n++1 (n-m)! p! (cos A), following the work [32], we pre-compute the V 4x (a+mm)
spherical harmonics for each node in experiments. A related work is Geographic Location En- coder [32. Although Geographic Location Encoder utilizes spherical harmonics, it focuses on training neural network based on land-ocean classification tasks for coordinate embedding and spatial forecasting (i.e., ERA5 interpolation) of weather data 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 different variable-specin location embedding.
3 Method
Our MIGN architecture is illustrated in Figure following In the following, we elaborate on the MIGN framework including spherical harmonics location embedding and mesh interpolation.
framework.

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Inegulallppt
Interpolate Encoder
Message Presingon Mesh
Station Interpolate
ImeegdarOUlg Output
Figure 2: Framework of the model. MIGN architecture follows an encoder-processore processor-dccoderr framework.
3.1 Spherical Harmonics Location Embedding
Spherical harmonics are widely used in the analysis of global weather patterns 40 Since the Earth can approximated as sphere, many meteorological variables can be naturally modeled as functions defined on the spherical surface. Spherical harmonics provide 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 function f(), o) defined 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
the sphere. Instead of learning
7 Using real spherical harmonics, the function of the sphere could be represented by (X, (),, w refers to spherical harmonics coefficients, which I(A, En=n Em=-n W y (X,
00
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(A, Φ) directly. Specifically, consider the node with feature and coordinates X, 6. The location embedding is denoted as follows:
SH の)
W Ym (A, o))
h [x; SH(A,)))
122222
where ④ indicates the concatenation of these basis functions into large vector and [;] denotes the concatenation of the embedding vector. Based on the definition of spherical harmonics, the learnable w 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 g (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 located in the center of the pixels and uniformly distributed across the sphere. The base resolution consists of 12 quadrilateral pixels on the sphere. To generate 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 nigghr-resoluttonn HEALPix mesh. When the process is conducted times, which is also called refinement level k, the original quadrilateral pixel can be divided into (2*)² pixels, leading to 12* * (2 pixels and mesh nodes total.
Mesh Interpolate Encoder Irregular station distributions make it hard 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 G ((V, V E (X. e: X()). (Ag, Φ()), where label denotes station nodes while label denotes the mesh nodes. The feature of the mesh nodes X can be initialized with zero. Ele.)) {(V), of) |
5

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o E vf ) } 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 tf messages are generated by its neighbors station nodes ng E (. The hidden state of the station nodes and the message are given by:
hag = [mlg: SH( (. S1y),
(E) 15-+0 = ( (hay), Vug (WA)),
(10)
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:
(E)
AG
(Im (E) of - : E (((o))))).
(11)
For the processor part, we consider the mesh nodes graph 93P (V) E H (Ni, (f.).. The feature of the mesh nodes H., are the message aggregate from station nodes, denoted as h (B) each mesh node. The hidden state of the Oth layer processor and the message are denoted as
o% 1%, E V% for
(0) k
SS og 18 ;
m
- =p(0) (h/T). h Vog E
(12)
Messages are exchanged within the mesh nodes and aggregated follows:
= AGG
(1m²2-0) ())
-og o ((()))) h' = UPDATE' (
(13)
The output hidden state of the processor is denoted as 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 (c.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 de- coder follows reverse process of the encoder. Consider the graph in the decoder G
(()+1) " (H v (M, ())),, ÿ!+1 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 ÿ as follows:
h [h Nog Ligh,
m (D)
+ pD( (n. ) No EN((tt'),
(14)
Yott = (G)
((m (D)
of E N((o
Our framework is readily adaptable to multi-step input and output, as shown in Appendix A.4
Training Given the predicted feature Ý of the decoder. The model parameters can be optimized by minimizing the discrepancy between the prediction and ground truth: Curin = E>eDim +12
|
+++
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 B 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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DyGrAE
STAR
MIGN
Figure 3: The global MAE distribution of SLP in the generalization experiment testing set
Experiments
Dataset We evaluate model performance up-to-date daily NOAA Global Surface Summary of the Day (GSOD) dataset. We use 6 commonly daily observed variables, including maximum dew point (DEWP), mean
temperature (MAX TEMP), minimum temperature (MIN TEMP), sea level pressure (SLP), mean wind speed (WDSP) and maximum sustained wind speed (MXSPD). We 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 (3. dynamic graph models: DyGrAE [55)
ReDyNet 77 (4) graph pooling based models: HD-TTS [27] (5) Transformer based models: STAR 42 GTN [3), GPS (3T) 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 hyperpa- rameters: 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 2. All models are implemented based on Pytorch Lightning. trained on GeForce RTX3090 GPU. Baseline models are implemented with PyG library, while our model realized with the DGL. library. For 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 Tabe
spherical harmonics degree
4.1 Overall Performance
In this section, we evaluate the performance of our proposed model against several baseline methods. As summarized in Table 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 SL.,, respectively, compared to the strongest baseline. To further evaluate our model's performance across different time horizons, we conduct experiments using 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 multistep forecasting setting. Besides,
horizons, highlighting its robustness and effectiveness we further conduct series of studies, including varying input steps, autoregressive inference. The results are presented in Appendix A.8.T
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 SL.P 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
7

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Table I: Bold font indicates the best result, and Underline is the strongest baseline. We report both the mean and the standard deviation that are computed over runs.
Model
MAX TEMP (A) MSE
MAE
MIN TEMP (K)
NISE:
MAA
MSE
DEWP(A)
MAE
MSE
SLP
MAA
MISE
WDSP (6i)
MAE
MISE
MXSPD
(Er)
MIAE
Prnsistence
9.58
2.12
9.80
2.00
9.56
2.10
26,62
3.54
10.35
2.15
25.33
3.48
STOON
2017
0.76-0.00
2.23-0.00
0.00
0,44+0.0
3:1190000
9.28-0000
2.17-0.00
24 15+0000
342-0000
8.60-0.0
2.01-0.00
30.6-10.00
3:27++0.00
DyGraB
10:13-0.33
2.34-0.22
0.40+0.08
9.25-0.55
2.10-001
24/9-0005
1-40-0.00
8.77-0.3
2.04-0.01
70. 20-78+0.07
3:28+00.01
STAR 232
10:18-0000
2
2.26-0.00
9.65+0.01
1115+0.0
9.56=0.01
2.16=0.0
24 04:1400000000
3-2:00.00
9.31-0.0
2. 2,10=0.00
21 1.55++0.00
3.36+00.0
GTN ST20
9.58-0.00
2:22:01:0
9,40:00.00
2.14+0.00
0.51-0000
2.16:0000
24.49:00.00
3.4-00.00
8.82-0.0
2.01::0000
20 20.56:40.05
3.50:00.3
MPNNLS
LLS 2021
47.34-0.33
4.70
45
45.24410.01
4,46:10.00
40:94:00.00
4.33::0000
39, 28.74200.03
4.40:00.02
10.48:0000
2:304:0000
24.66:10.01
3,60:00000
GPS 20E2
10.91=11.49
2:42::0088
10.37:10.39
2.27..005
11.13:10:32
2.502:0006
24.24
3.57-20.16
8.79-0.11
2,04
20.59:10.006
3.30:1001
T&S TRP PUZE
12.1200.70
2:51
10,92110.58
2.33110.05
10.00_00.10
2.512:002
24.70:10.21
3.46:10.02
K.88_0.06
2.062.0022
20.93.10.10
16
3.30,10,01
T&S AME MIAA
10.16_09.10
2:28_0003
12,90122.65
2.5910.333
9.43-00.10
2.10.0003
24.38.10.16
3.42100022
K.88.0.10
2.04..0022
20.72,10.25
3.28
ITS IMP
10.40_0.110
2:32-003
11 11,59.10.96
2.4410.111
13369_.3.66
2.641:033
24.76.10.28
347-1000
9.05.0.16
2.0720022
21 11..6.00.50
3.40.10.05
ITS- AMI
0.32.00.2
2.25.002
9.8110.06
1.11100
9.11.0.00
2.19.000
24.43.0.11
145.0
8.74.006
2.051002
20:09.10:55
1.28.1..0
10:20-11:01
2:33-010
9.65
2.17:0001
9.77-0.02
2:11:001
34:27+0.10
34::0.00
511-0229
2:11::055
20:25:40:21
3:23-001
ReDyNee
10.33-11005
2.26:0001
10.85.0.15
2:3010004
10.11:0.12
20104
14.1510:11
340:10.02
8.750005
2.0620001
20:95:10.17
DusiCast
10.81-0000
10.11.10.09
2.36-10:03
9.22-0.01
2002
23.53.00044
3.39.00:0
K.3--013
2.030002
20:27:10.15
3:25.0001
MIGN
8.7.20.05
0.09
8.011:004
1.991:0011
7.92:0005
1971001
10.09
3.12::0.01
0.38.0011
0.01
19.7310000
3.19.00.01
Improsemnns
15%6
Table 2: Bold font indicates the best result, and Underline is the strongest baseline. We report the mean MSE that is computed over 5 runs.
Model
Stepl
Step2
MAX TEMPLA)
Stup3
Step4
Total
Stepl
Step2
MIN TEMP (IC)
Step3
Step4
Total
Stepl
Step2
DEWPUU) Step3
Step4
Total
Stepl
Slep2
SLP (nub) Step3
Step4
Total
Penistenne
9.96
18.5 18
23:43
26.36
19:60
9:80
17.94
22:21
24.41
18.3
9.56
19.82
24.87
27
20-40
26.62
58.70
74
$4.31
61
STGCN
2077
11.10
16.87
20.09
22.05
17.53
10.38
15.65
18.36
19.96
16.09
10.42
17.15
20.15
22.23
17:49
22.25
$2.93
$1.08
55.22
42.87
DyGrAL
2019
9.85
16.81
20.40
22.49
17.39
9.27
15.29
18.19
19.72
15.38
9.01
16.91
20.22
21.57
17:00
23.95
44.18
$2.30
36,58
61.25
STAR
9.92
16.75
20.43
24.45
17.88
9:75
15.81
18.46
20.36
16.55
11.06
17:94
21.32
32.96
18:45
22.86
4485
53.70
58,20
44.93
GTN 20017
10.43
16.94
20.76
22.86
17.75
9.94
16:25
22:27
22.65
17.78
9.74
18.01
21.07
23.35
18:04
23.09
47.23
53.78
57.21
45.33
MPN
STM
2023
45.40
50.04
52.18
$3.20
50.23
45.29
4781
49.10
49.59
47.95
40.55
44
46.62
46.86
44
$7.58
$4.59
61.73
65.89
$4.95
GPS
2012
12.29
18.45
21.36
19.15
10.65
16.37
19:26
20.30
11.46
18.21
21.66
23.15
18.2
22.45
43.79
51.88
56,03
43.54
21E3
12.28
20.79
17.68
23.54
43883
51.77
T&S-MMP
17.44
15.60
19.78
18.11
21.70
1864
TTS-IMP
3073
15.79
19447
24.17
44.52
TTS AME
19.70
2166
16.82
16.29
20.62
17.13
22.29
31.71
17:00
20.17
21.48
23.27
ReDyNer
17,89
21.72
20.14
21
22.06
20.36
18.71
22.61
25.54
22.87
47.97
62
63.10
$6.95
DaniCast
MUAN
10.05
16
1987
31
17008
15.78
14.44
19.78
1600
10.14
17.57
20.64
26.16
1763
23.41
43.17
51 21
54.00
42.97
MIGIN
8.41
18.27
15.47
9.20
14.88
19.30
1533
8.19
1547
19,02
21,23
15.98
39.93
53,37
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 bascline 9,81/9.55 .552 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 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 low-resource environments. Notably, in Asia, MIGN demonstrates significant improvements, reducing the MSE for MAX TEMP and MIN TEMP to below 8 and 6, pepeectvvel-threshollle 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.
8

---

Table 3: Ablation studies.
Model Varienc
MSS:
MAX TEMPIR)
MAE
MSE
MIN TEMP
MAR
MSS
DEWP (K)
MAK
MISE:
SLP (mdo
MAR
NISIC
WDSP dea
MAE
MISK
MIXSPD (év)
MAE
we itesh (MPNN)
2.16=0.01
2,00:0.011
211120011
(MPNN-SFH)
9.43:10.02
2.0:0000
9,045:000
9.000065
2.08:0001
23:03:0055
3.50:0..1
8.3400001
3.36=0.01
9.06-40.00 DK
21550001
0.71::005
2:0:0001
6.7190004
2.0:30001
23:01-0077
3.33:0001
8.765000
2,00--001
20.330000
3.27--0.01
encoddr
K.00:40077
21240001
8.52-1006
2/M-1001
8 1:5090007
2.
22.57-0011
3:29:0001
8.594004
2.01
1944000
3.25-00001
we docoder SRI
5.0010.05
3.2200.01
3.20-0.04
2.01-0.01
7 1.99:0.33
1.0-00.01
23.07-0.00
3.20-0.01
8.39:0004
L961:0.01
19 19.76:0.00
3.22-:0.11
Derbuit
3:01 0.04
1,09 0.01
1.92-005
1,07 10.01
20.00 10.07
8,36 18.01
1.96 11.11
10.751
3.19-0001
Improvenennts
14%
616
1816
876
10%
806
1774
548
5%
7.5
.1%
Table 4: Bold font indicates the best result and Underline is the strongest baseline. We report the mean results that are computed over 5 runs. Global generalization experiments.
Model
MAX TEMPLNY
MSS
MAR
MIN TEMP (N) MSF
MAF
MSR
DEWP(R))
MAR
MSR
(md))
MAF
MIS
WRSS (Gr)
MXSPD (Er)
Prrsiteese
9.98
2,02
9,65
2.11
3.54
10:31
2.15
987720.00
9.5210000
2.1310:00
945-0000
2,1420000
25.8140.00
3.5910.00
8.62-0.0
20:9011.:00
3.32,0:00
DyGrAlipog PO
10083-00.22
9.5510002
9.5.20.22
24.0010:46
341::0:0
0.76-000
2.0
21:01:10:01
3.28:0:00
STAR
2.1510000
9.5..0.00
24.25
342.10:00
2,06.0000
21:67:10:00
16,10:00
2.23.00.00
9.66.0:00
3:4.10:00
21.01:10:09
2:41:0000
13.00-3.22
0.79-045
3.45-0.19
2.36-0155
34.97+1.19
6.82-.17
2.06-005
TAS-TNT 2013
12.1100.75
2.46-0.66
12.11+40.85
11.45-0:22
2.34-0022
34.99-0.54
893-0.66
2.06-0000
31 11:29+0:07
9.33+0.00
TAS-AME
10.38-0.17
2.30-003
10:07+0.300
2.55+0003
9.78-0006
217=0002
34.70=0.17
347-0000
885-004
20050001
20:52+0004
3.29-0.01
TTS-MP
10.53=0:233
2.33=003
10,42:00.61
2.27:0.09
16.22=3.83
2.36:1112
24.58+0.56
347:0003
896=004
2:0=0001
21 21:66:0006
3.52=0002
TTS-AMP
11.30=1.50
2.43::1221
9,80:-005
2.1740.12
10:15=00:0
2:23::0000
24.51
345:0000
884=0.12
2.06
20.21
3.32:001
931:0.19
2 2:55:004
9,711-0044
2.18:0003
9.50:0007
2.14::002
24.39
3.4:00011
896=001
209
21 11.55:0,100
3.36
ReDyNee
10:41-0004
2:31:0011
10,97110006
2.38.10022
10.97-0.18
1000
24.31
3.52
K93-0.04
2.1300011
21.09110.09
3.32.0044
DualCast
10.11-0.02
2.4
10,38.10.07
2
9.09
2.17 10004
40.572
3:42.0.00
0.0
20000001
20.32.10.11
3.29.1001
MIGG
8.55
2.10:0011
8,05.0.14
2,0010902
7.95 0.08
1.99.0011 0.01
30.9010.133 0,04
834
1.90
19,82,0,077
3.20.0001
Table 5: Spherical Harmonics degree analysis.
Degree
Order
MSE
MAX TEMPLK)
MAE
MISE
MIN TEMP (K)
MAE
MSE
DEWP (A)
MAE
MSE
SIP (nu1)
MAE
MSE
WDSP (en)
MAE
MSE
MXSPD (Er)
MAE
8.99:0.05
2.3420.01
8.71=0.144
2.06=0.00
=f.00
2.06-0011
23 23.00+0066
3,33
8. 8.76:0002
2.02:0.00
21136=00.24
3 3.36-00.01
8.22-0.01
2.1310:01
6.44.00112
202:10.01
8.29
0.004
201000.11
21 11/751005
3.2410001
5,641002
1.9910001
19.22.0.00
3:20.10.01
9
8.47-0.05
2
00.01
0.01:0.04
1.99:0.01
7.22-0.6
197-0001
200911007
3.1280.01
8.381001
1.98:6.01
19.73-0:05
3.19:10.01
0.38 16
0.10
2,09-0.60
8.16-00.08
2.01 +0.01
7.56
-0.04
0.6-
20.06 +0.05
3.11-8.02
8.35+0003
1,90+0.01
19.67-0.88
3.19-0.01
3072 and 12288 number of nodes) for mesh interpolation. The results are shown in Figure 5(A)). A). As the refinement level increases from 1 to 3, the MSE loss of the MIGN model exhibits 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 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 neighbors yields the lowest loss for almost all variables. In con- trast, 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 O 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 more precise representation of the location. When the degree increases from 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 significant regional variation 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 (c.g.. northern Asia), indicating their limited ability to capture the underlying

---

MSE
Figure 6: The global MAE distribution of MAX TEMP in testing set MAX TEMP
MSE
MIN TEMP
MSE
S1.P
18
10
16
9
6.0
12
8-
10
5.0
6
Australia
South America
South America
America
Persistence
STGCN
DyGrAE
HD-TTS
MIGN
Figure 4: Comparison of different models data-scarce regions.
1.)
Variable
Varlable MAX TERPP
0.8
08
MSE
0.6
SL. Whop MAAPD
0.4
0.2
02
Refinement Level
2
5
Neightors
Figure 5: Comparison of model performance with different mesh hyperparameter settings
STGCN
HD-TTS
MIGN
MAE
MAX TEMP
patterns 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 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. Exten- sive 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, plan 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 marine areas, 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.
10

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6 Acknowledgement
This work is supported by the Guangzhou Industrial Information and Intelligent Key Laboratory Project (No. 2024A03J0628) and Guangdong S&T "1+1+1" Joint Funding Program C019.
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Question:
Does the research conducted in the paper conform, in every respect. with the
NeurIPS Code of Ethics
Answer: [Yes]
Justification: This research conducted in the paper conforms with the NeurlPS Code of Ethics.
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: See Appendix A.1 Broader Impacts part.
Safeguards
Question: Does the paper deseribe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, scraped datasets)?
Answer: [NA]
Justification: This paper poses no safety risks.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: The materials in this paper are used with permission and properly cited.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification: The code is provided in /tttts://gtthiii com comm/comasszzn///////ocooooo MIGN
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: This paper does not involve crowdsourcing or research with human subjects. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
16. Declaration LLM usage
Does the paper describe the usage of LLMSs if it is important, original, or non-standard component of the core methods this research? Note that the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: The core method development in this research does involve LLMs any important, original, or non-standard components.
15

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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 sociocconomic 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 coarse grrsseegr 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-sii accurately modeled. This limitation restricts the ability station-based approaches, especially when applied to areas far from the existing observational network.
remote oceans, deserts, polar areas-remain unobserved and cannot be
A.3
Additional Related Work
GINO 21 also leverages latent regular grid. However, GINO aims simulate computational fluid dynamics. Such different application scopes result in distinctive model designs. GINO utilizes regular 3D grid for variable input geometry. In contrast, MIGN employs 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 spherical harmonie 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 multi-step imput-outuut setting. We define the input steps as a sequence of past observations from n to f. and the output steps as the sequence of future predictions from + to + m. Formally, the input consists of station nodes y(-n 1 V and their corresponding mesh nodes v/= ..... VX V/--
For each input step, we independently apply the Mesh Interpolation Encoder and message passing as defined in Eq. (10) [33, producing hidden states h' -......... h/ for mesh nodes and aggregated
processor states H
H
We then concatenate the temporal mesh representations H H and project them through a linear layer to obtain the output latent states [H] :...; H for the mesh. Finally, for each output step, the Station Interpolation Decoder (Eq. 14 maps mesh states back to station
predictions, yielding p y Vt+m
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 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 Daylattps://wwn, ncwi. nww.. moooaa.go//ettaattaa/// Begporaalress//ettaaataa//// iettitaaitem////// www.e/ooaa.ommmmm cesaaaandd::COOo c00516/htm1) is derived from The Inte- grated Surface Hourly (ISH) dataset. The latest daily summary data are normally available 2 days after the date-time of the observations used in the daily summaries. The updated fre-
16

---

1= 0
[=0
1= 1
SH
Lon. Lat.
[ ==
Feature
1
Concatenate
Pre-computed Spherical Harmonics
Shared Learnable Weight
Spherical Harmonics Embedding
Figure 7: Spherical harmonics location embedding.
quency and reference time are daily and Greenwich Mean Time. The data can be download in the link httpps//www ncei nwa.. gow..gooooao httv/dta//ae hasaa/gllllllsseeeee htlggalllluuaarrrr...e summary-ooooooo of tte-tthhaaya wwad/aaciveeeeeee We accessed the data on 2025/4/7. We computed both the total count data points per variable over the period 2017-2024 in the Table
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-rand maangg 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.
Table 6: Dataset variable statistic.
VARIABLES
MAX TEMP
MIN TEMP
DEWP
SLP
WDSP
MXSPD
TOTAL NODE
13260
13259
12709
9725
13014
12895
Table 7: The optimal training hyperparameter of baseline models for each variable.
MAX TEMP
Model
SSTGNN DyGrAE STAR
GTN
MPNNESTM
GPS
T&S-IMP
T&S-AMP TIS-IMP
TIS-AMP HD-TTS ReDyyNet DualCast
Learning rule
0,0066
0.0084
0.0033
0,0054
0.0040
0,0026
0.0059
0.0093
0,0044
0.0057
0,0007
0.0024
0.005
Batch
16
Hidden
64
Model
ISTGNN DyGrAE STAR
MPNNESTM
GPS
T&S-IMP
T&S-AMP TTS-IMP TIS-AMP HD-TTS
ReDyNet DualCast
Learning rate
0,0002
0.083
0.0023
0,0034
0L0099
D.0076
0,0086
0.0667
0.0056
0.0087
0.0074
0.0033
(L007
Batch size
16
16
16
8
Hidden size
32
128
64
128
32
32
64
64
32
64
64
32
64
DEWP
Model
SIGCN DyGrAE STAR GTN MPNNESTM
GPS
T&S-IMP
T&S TIS-MMP ITS-MPP HD-TTS ReDyNNet DualCust
0.0065
0.0905
0,011
0.0071
0.0004
0.0043
SLP
Model
STGGN DyGrAR STAR GTN MPNNESTM
GPS
T&S-IMP
T&S-AMP TTS-IMP TIS-AMP HD-TTS
ReDlyNet DualCast
Learning rate
0,0089
0.0073
0.0065
0,0071
OL00S8
0.0092
000063
0.00R0
0.0039
0,0060
0,0044
0,0078
0.002
Ratch size
16
2
16
16
Hidden size
64
128
66
128
32
64
32
61
WDSP
STGUN DyGrAE
T8S-AMP
ReDyNet
Learning
0.0004
0.0998
0.00K2
0.0041
0.0094
0,0061
0.0081
0.0045
0.0038
0.0012
0.003
Batch size
16
8
16
Hidden size
32
128
64
64
128
32
64
64
64
32
52
64
MXSPD
Model
STGGN
DyGrAE
STAR
GTN MPNNESTM
GPS
T&S-IMP
T&S-AMP TTS-IMP TTS-AMP HD-TTS
ReDyNet DualCast
Learning
0.0061
0.0015
0.0090
0,0023
0.0032
0.0018
0.0090
0.0012
0,0071
0.0023
0.0012
0.0035
0.0022
Batch size
16
Hidden
17

---

A.7 Baselines Implementation
STGCN
implemented base on Pytorch Geometric Temporal library
The graph convolution kernel
size is set to
DyGrAE (9] is implemented base on Pytorch Geometrie Temporal httpp:://ithut:t com/ We use mean convolution aggregate
aatngearroccctrreeee///r/ooreeee method.
STAR 42
is implemented base on Pytorch Geometric. The attention head is set to 4.
GTN 33
is implemented base Pytorch Geometric. The attention head is 4.
GPS is implemented base on Pytorch Geometric. The attention head is set to 1.
MPNNLSTM 29
implemented base on Pytorch Geometric Temporal library https: / http///enddarrrreemmerr......ooo
temporal The dropout rate is set to 0.5
and the Window is set to 1.
T&S-IMP, A&S-AMP, TTS-IMP, TTS-AMP 5 are implemented base official code https:// //
github cwwmcooo//////o. Graph- Maachine--Lerrinng
We use
elu' activation and mean normalization.
implemented base official code https////gttbbb..o//maaaaaattttt com/marshka/hhtt..... We use anisoconv message passing method and kmis pooling method, with the dilation and kernel
HD-TTS [7
size are
ReDyNet (2025) are implemented base on official code codd:/ttte///gghhbbb.///aaa/zz/
ReDyNet
DualCast 2025
are implemented base on official code hutpps://gtthbb.com//suu
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 Tabe
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, conduct additional experiment using different input step settings. The results of MAX TEMP, MIN TEMP and DEWP, presented in Figure g 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 more significant reduction in loss, particularly on DEWP, where the MSE drops from 8 to 7. Notably, increasing the input step from to 4 yields only marginal gains for most models, indicating 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 lower MSE loss, indicating that incorporating more historical information improves prediction accuracy. MIGN demonstrates the best overall performance. Its advantage becomes especially pronounced the input step increases, achieving the lowest MSE 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 learning- based approaches.
Autoregressive inference analysis We evaluate the autoregressive forecasting performance of our model (MIGN) against series of competitive baselines. Results are reported in and 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 2.72, outperforming the strongest baseline (STGCN, 2.96). For MIN TEMP, MIGN achieves 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 total MAE of 4.36.
18

---

MAX TEMP
MIN TEMP
DEWP
11.0
105
9.5
MSE LOS5
100
9.0
8.5
90
85
8.0
7,5
Impat Step
Imqat Stop
Input Step
Prndstmme
STOCN
DyGrAE
TTSAMP
HISHTS
MIGN
Figure 8: Comparison of different input steps on three key variables: MAX TEMP, MIN TEMP, and DEWP. MIGN achieves the best performance.
SLLP
WDSP
MXSPD
26
25
10.0
21
NSE 1,055
9.5
23
20
22
8.5
20
16
2
3
4
Stpot:Stpp
Stop
SteptStepp
Persstence
STOCN
DyGrAE
TIS-MP
HD-ITS
MIGN
Figure 9: Comparison of different input steps three key variables: SLP, DEWP, and MXSPD. MIGN achieves the best performance.
Notably, the performance gain becomes more pronounced at longer forecasting horizons (Step 2 and Step 3), indicating that MIGN is particularly effective capturing temporal dependencies in extended autoregressive prediction. These results demonstrate that integrating mesh-station interactions enables our model generalize better across different meteorological variables and forecasting horizons, thereby enhancing both short-term and long-term prediction accuracy.
Table 8: Bold font indicates the best result and Underline is the strongest baseline.
MAX TEMP(K)
MIN
TEMP (K)
Model
Step1 MSE MAE
Step2
Step3
Total
Step1 MSE
Step2
Step3
Total
MAE
MSE
MAE
MAE
MSK
MAE
STGON 2017
9.74
2.22
17.75
3.11
22.55
3.54
16.68
2.96
9.44
2.11
17.21
2.98
21.32
3.36
16.04
2.83
DyGrAE
2019
10.31
2.24
18.68
3.16
24.27
3.63
17.56
3.11
9.49
2.11
17.81
3.01
22.56
3.41
16.75
2.86
GIN 2020
9.88
2.22
18.40
23.72
17.34
9.49
2.14
17.43
21.95
3
16.37
T&SS IMPT 2023
12.12
2.51
20.84
3.42
27.47
3.5
20.25
3.31
10.92
2.33
18.57
3.15
23,86
3.61
17.91
3.05
T&P-AMP
2023
10.16
2.28
18.74
3.18
24.48
3.69
17.76
3.04
12.90
2.59
31.31
4.15
44.89
4.96
30.57
3.99
ITS-IMP 2025
10.40
2.32
19.34
3.27
25.22
3.79
18.31
3.12
11.58
244
26.69
3.78
38.75
4.61
25.91
3.65
TTS-AMP
2013
9.88
2.25
19.06
3.23
24.96
3.73
18.05
3.07
9.80
2.18
18.02
3.10
23.12
3.58
17.03
2.96
HD-TTS
2024
10.20
2.33
18.64
3.27
24.18
3.78
17.67
9.65
2.17
17.13
21.35
3.43
16.07
2.88
MIGN
8.47
2.09
15.08
2.84
19.28
3.24
14.33
2.72
8.01
1.99
14.63
2.75
18.18
3.09
13.83
2.63
A.8.2 Further Ablation study of location encoding
To investigate the impact of different location embedding strategies, we compare our proposed Spher- ical Harmonics (SH) embedding method with three commonly used coordinate- approaches: Direct Coordinate, WRAP, and Cartesian 3D. The formulations of these methods are as follows. Let longitude X E [-T, π and latitude E [-T/2, 2/2
Direct Coordinate:
PE(A, (2,
(15)
WRAP:
PE 0) [cos X, sin X, cos 0, sin 8
(16)
19

---

Table Bold font indicates the best result and Underline is the strongest baseline.
DEWP(K)
SLP (mb)
Model
Step1 MSE
Step2
Step3
Total
Step1
Step2
Step3
Total
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
STGON
2017
23.13
3.49
4.85
55.79
4.55
DyGrAE
2019
9.25
2.10
18.68
3.05
23.18
3.45
17.12
2.88
24.09
3.40
45.99
4.81
55.11
5.30
41.71
4.51
GIN
2020
9.51
2.16
19.66
3.19
25.25
3.68
18.22
3.02
24.49
3.44
47.57
4,93
57.31
5.47
43.11
4.61
T&S-IMIP
2023
10.80
2.31
19.87
3.22
25.26
3.69
18.67
3.08
24.70
3.46
46.39
4.88
55.75
5.40
42.31
4.59
T&S AMP 20223
9.43
2.16
19.25
3.18
24.35
3.65
17,80
3.02
24.38
3.4
45.59
4,82
54,66
5.31
41.47
4.52
TTS-IMP
13.69
2.64
3.79
46.86
28.57
24.76
491
57.41
4.62
TTS-AMI
2025
9.91
2.19
20.28
3.21
26.09
3.70
18.83
3.04
24.43
3.45
46.43
4.89
55.84
5.40
42.22
4.58
9.77
24.54
3.69
24.27
3,4
5.38
MIGN
7.92
1.97
16.35
2.90
20.45
3.31
15.21
2.76
20.09
3.14
43.29
4.64
55.29
5.26
39.93
4.36
Cartesian 3D:
PE 0) = [cos cos X, cos sin X, sin
(17)
We apply the above three location embeddings as well as our SH embedding method to both the encoder and decoder of our model. Table 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.
Table 10: Comparison different location embeddings in terms of mean squared error (MSE). Lower values indicate better performance.
Model Variant
MAX TEMP
MIN TEMP
DEWP
SLP
WDSP
MXSPD
W/O
9.04+0008
8.71-0.05
8.71+0.04
23.01+0.07
8.76+0.02
20.63+00.04
Direct WRAP
8.8811.06
8.57=0.09
8.35
21.89:1.06
8.6410.01
19.8511.04
8.70++0.05
8.32-0.08
8.23+0.03
21.14++0.05
8.52+00.3
19.83+00008
Cartesian 3D
8.67+00.9
8.34-0.05
8.19=0003
21.21++0.05
8.48+00.33
19.86+0.06
SH Embedding
8.47:10.05
8.01
7.92.0.05
20.09110.07
8.38.10.01
19.73
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, shown in Figure [T Across all variables and regions, MIGN consistently achieves the lowest MSE, demonstrating robust performance in regions with limited observational data. Particularly 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 A). performance improves as the refinement level increases from 1 to 3, reaching the lowest MAE level 3. Beyond this point, error increases again, suggesting that too fine a mesh may introduce noise. In Figure [2),,,
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 40) may introduce irrelevant information, hurting performance.
A.8.5
Visualization of Spherical Harmonics embedding
Since our spherical harmonics designed to learn the coefficients w we compute the spherical function f(), ()) = Σ Σm w. y (X, Φ). where wilt are the learned coefficients in 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.
20

---

2.0
1.5
1.0
0.0
Figure 10: The visualization of learned spherical harmonics embedding.
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 fair comparison with our station-based model (MIGN), MIGN was trained on station observations from 2017-2020, validated on 2021, and tested 2022. Results are summarized in Table
Table 1l: Comparison with gridded model
VARIABLES
MAX
MIN
WDSP
PANGU(0.25 RESOLUTION)
10.84
9.95
9.76
MIGN
8.71
9.02
8.60
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 itation-based learning are threefold: (i) directly leverages ground-truth observations without inheriting potential biases smoothing artifacts introduced during reanalysis assimilation, (ii) it preserves fine-scale spatial variability essential for capturing extremes and urban microclimates, and (iii) 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-Locatin Generalization To further investigate the spatial generalization of station- based 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 summarized in Table 12
Table 12: Comparison with gridded model
VARIABLES
MAX
MIN
WDSP
PANGU(0.225 2 RESOLUTION)
11.45
10.45
9.98
PANGU(1 RESOLUTION)
14.12
13.25
12.87
MIGN
13.84
12.87
13.14
MIGN outperforms Pangu 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 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 total
21

---

MSE
DEWP
MSE
WDSP
MSE
MX
18
16
14
Asia
South
Africa
Australo
South
South America
America
Americ
Persistence
STGCN
DyGrAE
HD-TTS
MIGN
Figure 11: Comparison of different models in data-scarce regions (Africa, Asia, Australia, South America) on three key variables: DEWP, WDSP, and MXSPD.
1.)
Verall MAX TEMP
(F) 1.0
Visiides
MAX TEMP
TEEMP
MAE
WDFF
MAE
066
MA.9
MAXPO
0.4
041
02
2
Refinement
3
4
2
Noks
10
20
40
(a) Comparison of model performance with different refinement level
Comparison of model performance with different
neighbors
Figure 12: Comparison of model performance with different mesh hyperparameter settings
data volume of roughly 10 GB, highlighting its efficiency and accessibility under data-scarce resource-constrainee 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 brute-force approach, the computational complexity is O(MN). For baselines, the computational complexity of nearest neighbor connections among the feature nodes is ((M?). Mesh analysis experiments Figure 12a indicate that the optimal number Healpix nodes is smaller than the number of feature nodes M, meaning that N M, thus O(MN) O(M2). Furthermore, by utilizing a KD-Tree algorithm, we can further reduce the complexity from O(MN) 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, 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 demonstrates that degree is sufficient for location embedding. Thus, its computational cost linear to the node number, which is not the primary time-consuming component. We measure the processing speed for MAX TEMP data with a degree-3 SH embedding (13260 nodes) on AMD EPYC 75F3 32-Core Processor. The computation completes in just 2s.
22

---

Table 13:
Training and inference time per step for different models.
Model
STGCN
TGCN
DyGrAE
MPNNLSTM
GPS
HD-TTS
MIGN
Training time per step (s)
0.013
0.014
0.016
0.012
0.048
3.25
0.013
Inference time per step (s)
0.004
0.010
0.012
0.011
0.019
3.03
0.006
A.10 Empirical analysis
To further illustrate our motivation, we visualize the global loss of MAX TEMP, MIN TEMP, 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 degrade in high-latitde regions, including Western Europe and North America. This pattern suggests increased difficulty in capturing surface-level pressure dynamics in these areas, possibly due more complex atmospheric interactions and variability at higher latitudes. Baseline models exhibit particularly high prediction errors in these regions, reinforcing their limitations 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.
DEWP. SLP in Figure 13 14 T6
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Figure 13: The global MAE distribution of MAX TEMP in testing set
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Figure 14: The global MAE distribution of MIN TEMP in testing set
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Figure 15: The global MAE distribution of DEWP in testing set
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Figure 16: The global MAE distribution of SLP in testing set
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Reviews:
[{"review_id": "gJrHUaxnMb", "reviewer": "Reviewer_AvaR", "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.", "questions": "See weaknesses.", "limitations": "The authors have adequately addressed the limitations of their work.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well-organized and clearly written, with a logical flow from problem formulation to experimental validation.\n- 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.\n\nWeaknesses:\n- 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.\n- 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.", "was_revised": true, "final_justification": "The rebuttal has partially addressed my concerns. Hence, I keep my rating unchanged, which is borderline accept.", "rebuttal": "Thank you for recognizing the novelty, comprehensive experiments, and writing of our paper. We have addressed all questions below.\n\n>Weaknesses 1: Multi-step autoregressive assessment.\n\nThank you for raising the issue. We agree that multi-step autoregressive assessment would provide stronger evidence of the model's predictive capabilities. Thus, we additionally evaluate the iterative performance of single-step models in generating 4-day predictions. The results are presented in the following table. \n\nTable 1(MSE):Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|Persistence|19.60|18.63|20.49|61.16|14.77|33.91|\n|STGCN| 19.01|18.12|19.41|46.73|11.46|26.54|\n|DyGrAE| 18.77|19.15|19.38|46.24|11.83|26.87|\n|TASAMP| 20.59|37.27|20.39|45.92|12.71|28.30|\n|MIGN|15.79 |14.81|16.62|45.69|11.31|25.43\n\nTable 2(MSE) for step 2：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|17.83 |17.26|18.75 |46.29 | 11.46| 26.21|\n|DyGrAE| 17.77| 17.98| 18.84| 46.07| 11.76| 26.52|\n|TASAMP|18.84 |31.44 |19.39 | 45.67|  12.80|28.42 |\n|MIGN| 14.51| 13.97| 15.70|42.57 |11.27 | 25.13|\n\nTable 3(MSE) for step 3：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|22.56 |21.43|23.31 |55.55 |12.47 | 28.69|\n|DyGrAE|22.29 | 22.74| 23.34|54.90 | 12.91| 28.99|\n|TASAMP|24.49 | 44.94|24.50 | 54.43|13.97 |30.45 |\n|MIGN|18.60 | 17.33| 19.90| 53.75| 12.39|27.55 |\n\nTable 4(MSE) for step 4：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|25.96|24.04|26.00 | 60.87| 13.34|30.64 |\n|DyGrAE| 25.23|25.95 |25.77 | 58.77|13.94 |31.05 |\n|TASAMP| 28.90|57.11 |27.84 | 59.27| 14.83|32.11 |\n|MIGN| 21.74|19.66 |22.63|58.12 | 13.30|29.37 |\n\nFrom the results, we can find that:\n\n* MIGN achieves the lowest MSE on the total errors, demonstrating its effectiveness.\n* MIGN demonstrates superior performance compared to the baselines when evaluated under conditions of error accumulation.\n\nThese results further verify the effectiveness of MIGN. Thank you for motivating us to conduct the experiments.\n\n>Weaknesses 2: Clarification for Global Generalization Analysis.\n\nThanks for raising these observations. Global generalization experiments evaluate the generalization ability to unseen stations. Several models perform worse when trained on the full dataset compared to the global generalization setting. This suggests that training on the full dataset may lead to overfitting to specific station patterns, resulting in poor generalization even to future observations at those same stations. For example, HD-TTS constructs downsampled graphs based on graph pooling. In dense regions (e.g., North America/EU), the full training set creates highly connected subgraphs. Models may over-rely on redundant local messages while underutilizing global patterns. Randomly subsampling stations thins these dense subgraphs, inadvertently encouraging longer-range dependency learning."}, {"review_id": "9aHiwGaRlq", "reviewer": "Reviewer_fq8d", "summary": "This 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.", "questions": "1. 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)?", "limitations": "No, please see questions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "The rebuttal has addressed most of my concerns, and I find the proposed method well justified within this problem setup. However, I remain uncertain about its practical value, given its weak performance in non-stationary areas.", "rebuttal": "We sincerely thank the reviewer for recognizing the studied problem, method effectiveness, and extensive experiments. We have addressed all questions below.\n\n>Weaknesses 1: Extend to Multi-step forecasting.\n\n\nThank you for raising the issue. We agree that multi-step autoregressive assessment would provide stronger evidence of the model's predictive capabilities. Thus, we additionally evaluate the iterative performance of single-step models in generating 4-day predictions. The results are presented in the following table. \n\n\nTable 1(MSE):Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|Persistence|19.60|18.63|20.49|61.16|14.77|33.91|\n|STGCN| 19.01|18.12|19.41|46.73|11.46|26.54|\n|DyGrAE| 18.77|19.15|19.38|46.24|11.83|26.87|\n|TASAMP| 20.59|37.27|20.39|45.92|12.71|28.30|\n|MIGN|15.79 |14.81|16.62|45.69|11.31|25.43\n\nTable 2(MSE) for step 2：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|17.83 |17.26|18.75 |46.29 | 11.46| 26.21|\n|DyGrAE| 17.77| 17.98| 18.84| 46.07| 11.76| 26.52|\n|TASAMP|18.84 |31.44 |19.39 | 45.67|  12.80|28.42 |\n|MIGN| 14.51| 13.97| 15.70|42.57 |11.27 | 25.13|\n\nTable 3(MSE) for step 3：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|22.56 |21.43|23.31 |55.55 |12.47 | 28.69|\n|DyGrAE|22.29 | 22.74| 23.34|54.90 | 12.91| 28.99|\n|TASAMP|24.49 | 44.94|24.50 | 54.43|13.97 |30.45 |\n|MIGN|18.60 | 17.33| 19.90| 53.75| 12.39|27.55 |\n\nTable 4(MSE) for step 4：\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|STGCN|25.96|24.04|26.00 | 60.87| 13.34|30.64 |\n|DyGrAE| 25.23|25.95 |25.77 | 58.77|13.94 |31.05 |\n|TASAMP| 28.90|57.11 |27.84 | 59.27| 14.83|32.11 |\n|MIGN| 21.74|19.66 |22.63|58.12 | 13.30|29.37 |\n\nFrom the results, we can find that:\n\n* MIGN achieves the lowest MSE on the total errors, demonstrating its effectiveness.\n* MIGN demonstrates superior performance compared to the baselines when evaluated under conditions of error accumulation.\n\nThese results further verify the effectiveness of MIGN. Thank you for motivating us to conduct the experiments.\n\n\n\n\n> Weekness 2: Visualization of the learned SH embeddings.\n\nThanks for the comment. Due to limitations of the rebuttal format, we are unable to include visualizations here. However, in the revised version of the paper, we will include comprehensive visualizations of each SH embedding dimension by projecting station locations onto the sphere and coloring them according to the corresponding embedding values. \n\n> Weekness 3:  Training/inference cost.\n\nThank you for the suggestion. The computational cost analysis is provided in Appendix A.7.  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 $O(M^2)$. We found that the optimal number of Healpix nodes $N$ is smaller than the number of feature nodes $M$, meaning that $N<M$, thus $O(MN)<O(M^2)$. Therefore, this interpolation step is more efficient than directly constructing nearest-neighbor graphs among all station nodes, as typically done in standard GNN baselines. Moreover, our model adopts a simple message passing mechanism; thus, from a theoretical perspective, its overall complexity is comparable to that of commonly used spatio-temporal GNNs.\n\nTo further demonstrate the training and inference efficiency of our model, we compare the training and inference times per step of all models 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.\n\n|Model|STGCN|TGCN|DyGrAE |MPNNLSTM |GPS |HD-TTS|MIGN|\n|-------------|--------|--------|--------|--------|--------|--------|--------|\n|Training time per step(s)|0.013|0.014|0.016|0.012|0.048|3.25|0.013|\n|Inference time per step(s)|0.004|0.010|0.012|0.011|0.019|3.03|0.006|\n\n\n\n> Questions 1: Model performance w.r.t.  longer historical input sequences.\n\nThanks for your comment. The further input step analysis are shown in Appendix A.6. MIGN consistently outperforms all other models across input steps ranging from 1 to 4. We find that increasing the input step from 3 to 4 yields only marginal gains for most models.\n\n> Questions 1: Is the current architecture capable of forecasting multiple steps into the future?\n\nYes, please see Weeknesses 1.\n\n> Questions 1: How the model handles situations where the spatial distribution shifts during multi-step inference?\n\nWe appreciate the reviewer's valuable comment regarding our inference process. In our multi-step inference framework, we employ the following procedure: (1) dynamically varying station observations are first projected onto static mesh nodes through mesh-based interpolation; (2) spatial message passing is then performed across the mesh nodes; and (3) finally, the decoder transforms the mesh node embeddings back to the target station distribution for prediction.\n\n\n> Questions 2: Model performance w.r.t. regions with no stations.\n\nWe sincerely appreciate the reviewer's insightful question regarding model performance in unobserved regions. To systematically evaluate this capability, we conducted rigorous regional generalization experiments (detailed in Appendix A.6), where models were trained exclusively on data from observed regions and tested on completely unobserved areas. Baseline approaches including TGCN, DyGrAE, and MPNN-LSTM exhibit limited forecasting capability in unobserved regions, with minimum temperature prediction MAE values reaching 7.61°C, 5.32°C, and 13.60°C respectively - performance levels that may not meet practical application requirements. In contrast, MIGN consistently outperforms the baseline models, demonstrating its stronger ability to generalize to areas without direct observations.\n\n> Questions 2: 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? \n\nIn weather forecasting, approaches are generally categorized into two paradigms based on their input data: (1) Gridded reanalysis data: This is the foundation for models like Pangu-Weather and GraphCast, which operate on high-resolution global grids analysis data with 6-hour forecast intervals. (2) Discrete station observations: This task, which our work addresses, focuses on modeling sparse observational data using spatiotemporal techniques.\n\nWhile a technical comparison between GraphCast and MIGN could be attempted, such a comparison is not practically meaningful due to fundamental differences in their input data and application settings. GraphCast relies on gridded reanalysis data derived through data assimilation, whereas MIGN is designed to work with raw station-based observational data. Additionally, the two models differ in temporal resolution—GraphCast performs 6-hour forecasts, while MIGN operates at a daily timescale. Our work specifically addresses the station-based forecasting scenario, which involves unique challenges such as sparse spatial coverage, irregular station distribution, and observational noise. These characteristics distinguish it from grid-based forecasting and highlight the need for a tailored modeling approach."}, {"review_id": "U4mg614m88", "reviewer": "Reviewer_Fzb7", "summary": "This 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.", "questions": "1. 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.", "limitations": "Yes. The authors discussed the limitations of their work in the conclusion section and elaborated on the Broader Impacts in the Appendix A.1.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The rebuttal has partially addressed my concerns. However, the baseline methods included are still not representative of the most widely adopted state-of-the-art approaches in spatiotemporal forecasting. Additionally, several of the baselines are designed for classification rather than forecasting, which undermines the relevance and strength of the experimental comparisons. The proposed method also lacks originality, as it primarily applies existing techniques without significant novel contributions. Given these considerations, I will maintain my original score.", "rebuttal": "We appreciate the listing of our paper's strengths and have addressed all questions below.\n\n>Weaknesses 1 & Questions 1: Latest spatial-temporal baseline\n\nThank you for the suggestion. We have incorporated additional recent spatiotemporal forecasting baselines, including DualCast [1] and ReDyNet [2]. MIGN consistently outperforms these methods as well, further demonstrating its effectiveness.\n\nMSE\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|DualCast|9.92|9.68|9.53|23.98|8.54|19.92\n|ReDyNet|10.65|9.42|9.36|24.34|8.77|21.03\n|MIGN|8.55 |8.05|7.95|20.90|8.34|19.82|\n\nMAE\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|DualCast|2.24|2.17|2.15|3.41|2.02|3.26|\n|ReDyNet|2.37|2.12|2.11|3.41|2.06|3.31|\n|MIGN|2.10 |2.00|1.99|3.14|1.98|3.20|\n\n[1] DualCast: A Model to Disentangle Aperiodic Events from Traffic Series. IJCAI 2025.\n[2] Responsive Dynamic Graph Disentanglement for Metro Flow Forecasting. AAAI 2025.\n\n>Weaknesses 2 & Questions 2: Ablation study of location encoding.\n\nThanks for the suggestion. We further compare our SH embedding method with three commonly used coordinate-based location embedding approaches, including Direct Coordinate, WRAP, and Cartesian 3D. The details are as follows:\nConsider longitude $\\lambda \\in [-\\pi,\\pi]$ and latitude $\\theta \\in [-\\pi/2,\\pi/2]$\nDirect: $PE(\\lambda, \\theta) = (\\lambda, \\theta)$\nWRAP: $PE(\\lambda, \\theta) = [cos\\lambda,sin\\lambda, cos\\theta, sin\\theta]$\nCARTESIAN3D:  $PE(\\lambda, \\theta) =[cos\\theta cos\\lambda,cos\\theta sin\\lambda,sin\\theta ]$\n\nWe apply the three location embedding methods to both the encoder and decoder of our model. Our proposed SH embedding strategy outperforms the other three baseline methods, demonstrating its effectiveness.\n\n\n\nMSE\n|Model Variant|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|W/O|9.04|8.71|8.71|23.01|8.76|20.63|\n|Direct|8.88|8.57|8.35|21.89|8.64|19.85|\n|WRAP|8.70|8.32|8.23|21.14|8.52|19.83|\n|CARTESIAN3D|8.67|8.34| 8.19|21.21|8.48|19.86|\n|SH Embedding|8.47|8.01|7.92|20.09|8.38|19.73|\n\n\n\n>Weaknesses 3 & Questions 3: Some of the equations in the paper are either redundant or not fully explained.\n\nThanks for the suggestion. We will revise the equations in the revised version of the paper.\n\n> Questions 4: Indicating the specific percentage of improvement.\n\nThank you for the suggestion. We have included the specific percentage improvements shown in the following table in the revised version of the paper.\n\n\nMain table\n|MSE|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|Overall Improvement|13%| 15%| 14%| 16%| 2%|1% |\n\n\n\nAblation study\n|MSE|MAX TEMP|MIN TEMP|DEWP|SLP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|Mesh Improvement|10%|11% |12% |16%  |4% |4% |\n|SH Embedding Improvement|6%|8.03%|9%|12%|4%|4%"}, {"review_id": "vb31BZVu0M", "reviewer": "Reviewer_yq1V", "summary": "The 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.", "questions": "1. 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?", "limitations": "While 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "Following the rebuttal and further discussion, I have revised my recommendation to borderline accept. The authors have provided thoughtful clarifications and additional results that address several key concerns, particularly regarding the task formulation and experimental design. My final evaluation is based on the following considerations:\n\n* **Resolved issues**\n\n  * The authors clarified the rationale behind using daily-resolution data and justified the forecasting setup. The additional multi-step forecasting results are more aligned with common practices and significantly strengthen the paper's relevance.\n  * The authors addressed concerns regarding implementation details (e.g., temporal extension, batch sizes, and handling of spatial embeddings in baselines). These clarifications improve understanding of the method and confidence in the fairness of the comparisons.\n\n* **Remaining concerns**\n\n  * The evaluation remains limited to a dataset with relatively coarse temporal resolution, which may reduce the generalization and impact of results. However, this is mitigated by the challenging nature of the dynamic sensor setting considered.\n  * While improved in the rebuttal, the manuscript would benefit from a thorough revision to improve clarity, organization, and notation in the final version.\n\nOverall, the paper addresses a relevant and timely problem, introduces a novel and interesting approach, and demonstrates promising performance. While certain limitations remain, the authors' responses and proposed revisions sufficiently alleviate my initial concerns, warranting a borderline accept recommendation.", "rebuttal": "We sincerely thank the reviewer for listing the strengths. After carefully reading the review, we are afraid that the reviewer significantly **Underestimates** the problem challenges. We have tried our best to eliminate the misunderstanding as follows.  If you are satisfied with the response, we kindly request your approval by considering an improved score.\n\n> Weakness 1 & Questions 1 & Limitation 1: Daily-resolution data is very smooth.\n\nThanks for your suggestion. We would like to clarify that the daily data is **NOT** smoother than the hourly data in our one-day head prediction setting, especially for the extreme variables (i.e., daily maximum, minimum temperature, and maximum sustained wind speed). To verify it, we compared the Mean Absolute First Difference (MAFD) of the hourly records in NOAA with their corresponding NOAA Global Daily station data. Due to time constraints, we were only able to download a randomly selected subset of 1,000 stations from the NOAA Global Hourly dataset. While this is a limited sample, we are in the process of downloading more hourly data to support a broader comparison. The current results are as follows:\n\nMean Hourly Fluctuations\n|Dataset|TEMP|DEWP|WDSP|\n|-------------|--------|--------|--------|\n|Hourly|1.44 °C|0.936 °C |0.96m/s |\n\nMean Daily Fluctuations\n|Dataset|MEAN TEMP|MAX TEMP|MIN TEMP|DEWP|WDSP|MXSPD|\n|-------------|--------|--------|--------|--------|--------|--------|\n|Daily|1.96°C|2.40 °C|2.35°C| 2.22 °C|1.14m/s | 1.76m/s| \n\nOur analysis reveals that the daily time series exhibits significant non-smooth characteristics, which present substantial challenges for one-day-ahead prediction tasks. The relatively strong performance of the Persistence baseline in this context should not be interpreted as indicating task triviality. Rather, it demonstrates the inherent difficulty in capturing meaningful temporal patterns from daily-scale weather data. It highlights the need for specialized architectural designs (i.e., MIGN) tailored to daily prediction scenarios.\n\nWe appreciate the reviewer’s insightful suggestion regarding higher-frequency data. While MIGN could theoretically be extended to hourly forecasting, our current work focuses on daily resolution due to its greater relevance to our target application. Specifically, station configurations in hourly settings often exhibit less dynamic variability, whereas the daily scale better captures the spatial-temporal patterns critical to our study. Future work could explore adapting MIGN to finer temporal resolutions, but we argue that daily prediction remains a practically significant and scientifically valuable benchmark for our proposed framework.\n\n> Weaknesses 2 & Questions 2: Why are the main experiments in the paper limited to a one-step history?\n\nThanks for the comment. While our primary focus in this work is on demonstrating MIGN's spatial generalization capabilities, we acknowledge the importance of investigating multi-step historical inputs. Due to space constraints in the original submission, we initially presented only single-step history results. As we have shown in Figures 8 and 9 in the Appendix, MIGN maintains superior performance even with extended input histories. The illustration would not affect our contribution. \n\nWe acknowledge the importance of long input history analysis and have included these results in the revised manuscript's main text.\n\n> Weaknesses 3 & Questions 3 & Limitations 2: MIGN operates on static graph structures, which gives it a natural advantage under this setting.\n\nThanks for the question. As we have shown in Fig.8 and Fig.9, MIGN outperforms baselines even if increasing the length of history.\n\n> Weaknesses 4 & Questions 3: How are node-specific parameters handled in global-local architectures under a dynamic sensor network?\n\nThank you for your question. Using node-specific parameters deteriorates performance in our dynamic setting, as the test set contains nodes that are not seen during training. To ensure fair comparison and enable generalization to unseen nodes, we replace node-specific parameters with direct position embeddings, defined as $PE(\\lambda, \\theta) = (\\lambda, \\theta)$。 This allows the models to retain spatial awareness without overfitting to specific nodes. \n\n> Weaknesses 5 & Questions 5: Experimental details.\n\n\nWe appreciate the reviewer's inquiry regarding implementation details. For comprehensive information about baseline implementations, please refer to Appendix A.5. Regarding hyperparameter optimization, we employed a rigorous Bayesian search approach using Weights & Biases (WandB) rather than manual tuning, as documented in Table 6 (Appendix A.5). This systematic process ensured optimal configurations for all models.\n\nOur hyperparameter search space included:\n* Hidden sizes: [8,16,32,64,128]\n* Batch sizes: [1,2,4,8,16,32] (limited to [1,2] for memory-intensive models like HDTTS)\n* Learning rates: $1\\times 10^{-4}$ to $1\\times 10^{-2}$\n\nThe final reported results reflect the best-performing configurations identified through this exhaustive search, with some models achieving optimal performance at smaller batch sizes (1-2). This methodology guarantees fair and reproducible comparisons across all evaluated approaches.\n\n> Weaknesses 6: Presentation.\n\nWe sincerely appreciate the reviewer's careful reading and apologize for the typographical errors in our current draft. We will conduct thorough proofreading to ensure all notations are consistent and the manuscript's readability is significantly improved in the revised version.\n\n> Questions 4: How does MIGN handle longer input sequences, and how do the baselines handle missing stations in those cases?\n\nThanks for the question. For different input time steps, station features are passed to the corresponding Healpix nodes through message passing. An MLP is then used to map the features from the encoder's Healpix nodes to those in the decoder. For the baselines, we follow the approach used in HD-TTS: if a sensor (node) is not observed at time t, but appears at t+1, it is treated as missing data at t and initialized with the mean value. This allows the construction of a complete graph at each timestep.\n\n> Questions 5: How were the baselines evaluated on this dataset, and do they share a unified codebase for training and evaluation?\n\nAll baseline models were implemented, trained, and evaluated using a unified PyTorch Lightning framework to ensure consistency with MIGN's experimental setup. We use the Adam optimizer with an early stopping strategy (patience = 3). For graph construction in the baselines, we use the 20 nearest neighbors. We will open-source the complete baseline code implementation."}]

Author rebuttal:

Meta-review: The paper proposes a method for global weather forecasting based on graph neural networks.

The strengths of the paper include a well-justified model based on message passing among both regular and irregular nodes, supporting direct observational data from irregularly located stations, an additional spherical harmonics coupling that enables improved generalization to locations with sparse observations, extensive empirical evaluation, state-of-the-art performance, and generally speaking good presentation.

In the reviews, the reviewers unanimously considered only 1-day ahead forecasting a weakness.

In the rebuttal, the authors provided extended results covering up to 4-day ahead forecasting task, where the method was still found to achieve state-of-the-art performance.

All reviewers found the rebuttal satisfactory and no remaining major issues remained. On the other hand, no reviewer was particularly enthusiastic about the work and all suggest only "borderline accept" suggesting that there is still room for improvement.

Lastly, it must be emphasized that formatting rules must be followed exactly without any tricks with negative vspace, overly small font-sizes, etc. Violating the rules risks desk-rejecting the paper. If the page limit becomes an issue, content that isn't essential to the main contribution can be included in appendices.
Decision: Accept (poster)