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

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.

Weakness 1 & Questions 1 & Limitation 1: Daily-resolution data is very smooth.

Thanks 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:

Mean Hourly Fluctuations

Mean Daily Fluctuations

Our 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.

We 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.

Weaknesses 2 & Questions 2: Why are the main experiments in the paper limited to a one-step history?

Thanks 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.

We acknowledge the importance of long input history analysis and have included these results in the revised manuscript's main text.

Weaknesses 3 & Questions 3 & Limitations 2: MIGN operates on static graph structures, which gives it a natural advantage under this setting.

Thanks for the question. As we have shown in Fig.8 and Fig.9, MIGN outperforms baselines even if increasing the length of history.

Weaknesses 4 & Questions 3: How are node-specific parameters handled in global-local architectures under a dynamic sensor network?

Thank 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.

Weaknesses 5 & Questions 5: Experimental details.

We 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.

Our hyperparameter search space included:

• Hidden sizes: [8,16,32,64,128]
• Batch sizes: [1,2,4,8,16,32] (limited to [1,2] for memory-intensive models like HDTTS)
• Learning rates: $1\times 10^{-4}$ to $1\times 10^{-2}$

The 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.

Weaknesses 6: Presentation.

We 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.

Questions 4: How does MIGN handle longer input sequences, and how do the baselines handle missing stations in those cases?

Thanks 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.

Questions 5: How were the baselines evaluated on this dataset, and do they share a unified codebase for training and evaluation?

All 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.

We sincerely thank the reviewer for the reply! In the following, we provide the additional results and experimental details to address your concerns.

Task significance 1: forecasting only one day ahead using solely the previous day’s data.

Thank you for raising this important point regarding task significance. To address your concern, we conducted additional experiments evaluating spatio-temporal forecasting performance.

As shown in our input length study (Appendix), increasing the input steps from 3 to 4 yields only marginal performance gains. Therefore, we maintained the 3-step input configuration for both our model and baselines, while predicting 4 output steps (representing a medium-range 4-day forecast horizon). During inference, all 4 steps are generated in a single forward pass.

Table 1(MSE computed over the entire 4-step sequence as a whole):

Table 2(MSE for step 1)：

Table 3(MSE for step 2)：

Table 4(MSE for step 3)：

Table 5(MSE for step 4)：

The results demonstrate that MIGN maintains superior performance compared to baseline methods in this multi-step forecasting setting.

Task significance 2: the limited scope of the forecasting horizon.

We appreciate the reviewer's insightful comment regarding forecasting horizons. We would like to clarify that our framework can indeed support extended forecasting horizons, as demonstrated in our comparative analysis (see Experimental Evaluation 2). Below we address the relationship between hourly and daily forecasting:

• Empirical applications: We acknowledge the critical importance of high-temporal-resolution (e.g., hourly) forecasts for applications requiring immediate response, such as severe weather warnings. Simultaneously, daily aggregated forecasts (e.g., max/min temperature, SLP) remain essential for strategic planning in agriculture, energy management, and climate adaptation.
• Task difficulty: Daily extreme-value predictions (e.g., MAX/MIN temperature) represent physically integrated extremes over full diurnal cycles, not temporal averages. This constitutes a fundamentally different and often more challenging task than predicting instantaneous hourly values within smoother temporal sequences.

We fully agree that multi-step hourly forecasting represents an important research direction. We have explicitly committed to extending MIGN for high-resolution temporal forecasting in future work and will strengthen this discussion in the revised manuscript.

Experimental evaluation 2: results explanation.

We sincerely appreciate the reviewer's careful attention to detail regarding to Autoregreeisve Table. We acknowledge that the original caption may have caused confusion about the evaluation metric. The caption “Table 7 (MSE): Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively” refers to the MSE computed over the entire 4-step sequence as a whole, rather than the MSE for the first predicted step alone. The step-1 prediction results are already presented in the main table. For clarity, we also include the step-1 results again in the subsequent table for direct comparison.

Table 7 (MSE) (Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively (showing the MSE computed over the entire 4-step sequence as a whole))：

Table 8(MSE for step 1)：

Table 9(MSE for step 2)：

Table 10(MSE for step 3)：

Table 11(MSE for step 4)：

Experimental evaluation 3: how temporal dynamics are integrated into MIGN’s otherwise static architecture?

We sincerely appreciate this insightful technical question regarding MIGN's temporal integration mechanism. Our architecture processes temporal dynamics through the following sequence:

Independent Time-step Processing:
For each input time step $t_{-N+1}, \ldots, t_{-1}, t_0$, we apply the Mesh Interpolation Encoder and perform Message Passing independently, generating static mesh graph features as defined in Equations (9)-(12).

Temporal Tensor Construction:
With $K$ mesh nodes and $H$-dimensional features per time step, we obtain $N$ tensors of shape $[K, H]$. These are concatenated along the temporal dimension to form an input tensor of shape $[K, H, N]$.

Temporal Projection:
A specialized multilayer perceptron (MLP) with parameters conditioned on $(N, M)$ maps the input tensor $[K, H, N]$ to decoder-ready features of shape $[K, H, M]$, effectively distributing temporal information across output steps.

Decoder Output Generation:
After obtaining the output mesh tensor $[K, H, M]$, we independently apply the Station Interpolation Decoder (Equation 13) to each $[K, H]$ feature slice corresponding to output time steps $t_1, t_2, \ldots, t_M$ to generate station-level predictions.

This design maintains spatial processing efficiency while enabling flexible temporal modeling through the parameter-conditioned MLP bridge. We thank the reviewer for highlighting this architectural detail and will enhance the method part in the revised manuscript to clarify this temporal integration mechanism.

Experimental evaluation 4: how positional or spatial embeddings are implemented?

We appreciate the reviewer's insightful question regarding our positional embedding approach.

In baseline methods (T&S-AMP and HD-TTS), models learn station-specific parameters, yielding an embedding tensor of shape $[N, H]$ where $N$ is station count and $H$ is embedding dimension.

Our approach differs fundamentally:

1. Geographic Encoding:
   We compute positional embeddings using geographic coordinates:

   • Longitude $\lambda \in [-\pi, \pi]$
   • Latitude $\theta \in [-\pi/2, \pi/2]$ Yielding $PE(\lambda, \theta) = (\lambda, \theta)$ → tensor shape $[N, 2]$

2. Shared Projection:
   Rather than station-specific embeddings, we apply a global linear transformation:
   $\text{Linear}(2, H): [N, 2] \rightarrow [N, H]$

This design:

• Learns a universal embedding function
• Enhances cross-station generalization
• Maintains spatial awareness without station-specific parameters

We thank the reviewer for this valuable inquiry and will clarify these architectural distinctions in the revised manuscript.

We appreciate the listing of our paper's strengths and have addressed all questions below.

Weaknesses 1 & Questions 1: Latest spatial-temporal baseline

Thank 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.

MSE

MAE

[1] DualCast: A Model to Disentangle Aperiodic Events from Traffic Series. IJCAI 2025. [2] Responsive Dynamic Graph Disentanglement for Metro Flow Forecasting. AAAI 2025.

Weaknesses 2 & Questions 2: Ablation study of location encoding.

Thanks 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: Consider longitude $\lambda \in [-\pi,\pi]$ and latitude $\theta \in [-\pi/2,\pi/2]$ Direct: $PE(\lambda, \theta) = (\lambda, \theta)$ WRAP: $PE(\lambda, \theta) = [cos\lambda,sin\lambda, cos\theta, sin\theta]$ CARTESIAN3D: $PE(\lambda, \theta) =[cos\theta cos\lambda,cos\theta sin\lambda,sin\theta ]$

We 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.

MSE

Weaknesses 3 & Questions 3: Some of the equations in the paper are either redundant or not fully explained.

Thanks for the suggestion. We will revise the equations in the revised version of the paper.

Questions 4: Indicating the specific percentage of improvement.

Thank you for the suggestion. We have included the specific percentage improvements shown in the following table in the revised version of the paper.

Main table

Ablation study

Thank you for recognizing the novelty, comprehensive experiments, and writing of our paper. We have addressed all questions below.

Weaknesses 1: Multi-step autoregressive assessment.

Thank 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.

Table 1(MSE):Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively：

Table 2(MSE) for step 2：

Table 3(MSE) for step 3：

Table 4(MSE) for step 4：

From the results, we can find that:

• MIGN achieves the lowest MSE on the total errors, demonstrating its effectiveness.
• MIGN demonstrates superior performance compared to the baselines when evaluated under conditions of error accumulation.

These results further verify the effectiveness of MIGN. Thank you for motivating us to conduct the experiments.

Weaknesses 2: Clarification for Global Generalization Analysis.

Thanks 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.

We sincerely thank the reviewer for recognizing the studied problem, method effectiveness, and extensive experiments. We have addressed all questions below.

Weaknesses 1: Extend to Multi-step forecasting.

Thank 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.

Table 1(MSE):Trained with 1 input step and 1 output step; during inference, 4 output steps are generated autoregressively：

Table 2(MSE) for step 2：

Table 3(MSE) for step 3：

Table 4(MSE) for step 4：

From the results, we can find that:

• MIGN achieves the lowest MSE on the total errors, demonstrating its effectiveness.
• MIGN demonstrates superior performance compared to the baselines when evaluated under conditions of error accumulation.

These results further verify the effectiveness of MIGN. Thank you for motivating us to conduct the experiments.

Weekness 2: Visualization of the learned SH embeddings.

Thanks 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.

Weekness 3: Training/inference cost.

Thank 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.

To 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.

Questions 1: Model performance w.r.t. longer historical input sequences.

Thanks 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.

Questions 1: Is the current architecture capable of forecasting multiple steps into the future?

Yes, please see Weeknesses 1.

Questions 1: How the model handles situations where the spatial distribution shifts during multi-step inference?

We 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.

Questions 2: Model performance w.r.t. regions with no stations.

We 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.

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?

In 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.

While 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.