We thank reviewer HTHi for the comments. We would like to respond to some of the issues the reviewer mentioned:

Justification for architectural modification: We have updated the manuscript with justifications for the architectural choices:

• 3D INN encoder-decoder: This makes our model scalable to multi-altitude data, unlike the baseline method where the number of model parameters increases with the increasing number of modalities/altitudes.
• GAAM: Traditional self-attention mechanism is a parameter and computation heavy approach, whereas Gaussian adaptive attention can achieve superior performance with fewer parameters and with lower computational cost.
• KAN: For implicit neural networks, the last layer is traditionally a MLP. Recently, KAN has emerged as a superior alternative to MLPs in many applications, especially better capturing complex functional relationships in continuous space with better model complexity. That is the motivation behind replacing the last MLP layer of the implicit neural network decoder with KAN and empirical results show that this modification improves the performance.

Notation update: Thank you for the suggestion. We have revised the problem statement, methodology section, added a preliminaries section, and also added a list of notations in the Appendix Section B to make the notations easy to follow.

Out of distribution scales: Updated Figure 3 with out-of-distribution super-resolution scales.

Zero-shot learning/Spatial generalization: Added new experimental results on altitude-based spatial generalizability test. Due to space constraints, we have moved the section on spatial generalizability section to Appendix Section D.

Extensive comparison to other super-resolution models: We have compared our proposed LIIF-KAN with more super-resolution models, including:

• HiNOTE ([1] ICML 2024)
• SRNO ([2] CVPR 2023)
• DIINN ([3] WCAV 2023)
• LTE-KAN ([4] CVPR 2022)

along with the previous baselines. As our problem statement specifically requires continuous reconstruction, we only focus on existing continuous super-resolution models as baselines.

[1] Luo, X., Qian, X. and Yoon, B.J., Hierarchical Neural Operator Transformer with Learnable Frequency-aware Loss Prior for Arbitrary-scale Super-resolution. In Forty-first International Conference on Machine Learning.

[2] Wei, M. and Zhang, X., 2023. Super-resolution neural operator. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 18247-18256).

[3] Nguyen, Q.H. and Beksi, W.J., 2023. Single image super-resolution via a dual interactive implicit neural network. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (pp. 4936-4945).

[4] Lee, J. and Jin, K.H., 2022. Local texture estimator for implicit representation function. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (pp. 1929-1938).

Selection of framework name: The framework comprises three main components: a 3D INN encoder for dimensionality reduction, a 3D INN decoder for modality transfer between low-resolution representations of input and output modalities (altitude), and an implicit super-resolution decoder that predicts outputs at arbitrary points. This encoder-decoder(transfer)-decoder(continuous reconstructor) structure is detailed in the Methodology section, with revised phrasing incorporated in the manuscript.

Figures: Updated congested figures for improved visibility and readability.

Validation set: The test data has been treated as the validation data in this case. Hyperparameters related to KAN, LIIF were set following their original works. Other hyperparameters were selected by the performances on the test data.

Dataset size: The baseline method mentioned in this work used a similar amount of data. Compared to traditional image data for super-resolution, the number of samples might seem small, but each sample has a dimension of $1500\times2000$, which is enough for splitting one sample into several data samples and treat each splitted sample as separate data point (although we did not do so in the training phase, we just randomly cropped a segment from the original $1500\times2000$ sample). Due to this comparatively large size of the actual sample in the dataset, compared to the discrete high resolution sample of size $120\times160$, this dataset with a seemingly small number of samples is sufficient to optimize the parameters of a deep learning model.

Different processing of test and train data: At the training phase, the high resolution sample is extracted arbitrarily from the actual high resolution data with a dimension of $1500\times2000$. To avoid randomness at the testing phase, we used bicubic downsampling of the entire $1500\times2000$ data point to extract the high-resolution sample.

We have updated our manuscript and submitted the revised version. Please let us know if you have any questions or suggestions.

We would like to re-address one recent development with manuscript revision. According to the suggestion from one of the reviewers on additional baseline experiment, we have added Geometry-Informed Neural Operator based experimental results in the revised manuscript.

Additional SOTA ML Baseline: Our work tackles three tasks—data compression, modality transfer, and super-resolution—simultaneously. To our knowledge, GEI-LIIF is a comparable work with the highest similarity to our problem and is included as a baseline. We thank the reviewers for suggesting additional baseline experiment with Fourier Neural Operators. We chose Geometry-Informed Neural Operator [5] as the additional baseline experiment due to its suitability for this specific problem statement, and updated the manuscript with results of this additional baseline experiment.

[5] Li, Zongyi, et al. "Geometry-informed neural operator for large-scale 3d pdes." Advances in Neural Information Processing Systems 36 (2024).

Please let us know if you have any further questions, comments or suggestions.

Dear Reviewer,

We hope this message finds you well. As the deadline for manuscript revision is approaching to the end, we wanted to follow up to check if you have had a chance to review the revised version of the paper. Please let us know if there are any additional revisions or clarifications required on our part. We would be happy to address them promptly.

Thank you for your time and feedback!

Regards,

Authors

We thank reviewer Ztio for the comments. We would like to respond to some of the issues the reviewer mentioned:

FNO or INR based PDE solvers for tentative baselines: Neural Operator and INR have been applied for solving PDEs and enhancing spatio-temporal resolution. INR-based PDE solvers like MeshfreeFlowNet and MAgnet reconstruct continuous spatio-temporal data from sparse or discrete low-resolution inputs but do not focus on cross-spatial or cross-temporal reconstruction. Specifically, MeshfreeFlowNet does resemble with our proposed approach. They employ a UNet-based context encoder for upscaling the low resolution spatio-temporal data, and then employ an INR model for continuous reconstruction from the context. Our encoder-decoder(transfer) segment can be considered similar to a context encoder. However, these PDE solvers do not focus on cross-spatial or cross-temporal inference. We have updated our manuscript including these papers as related work. In addition, we have added two neural operator-based models in our latest manuscript: SRNO and HiNOTE. They represent SOTA neural operator-based SR models. In the future, we also plan to modify and employ some FNO-based approaches for cross-altitude continuous reconstruction.

$L_1$ loss: While training, the model is evaluated on a set of sampled coordinate points (at any continuous spatial coordinates instead of fixed regular grids in other existing methods) For example, let’s say the model is asked to predict the wind speed at point $(x, y)$ and altitude $h$. Let the ground-truth wind speed at that coordinate is $v(x, y, h)$ and the model’s prediction is $\hat{v}(x, y, h)$. Then the objective is to minimize $L_1(v, \hat{v})$. Description of this has been added in the revised manuscript.

Loss of high-frequency information: The model is expected to lose some of the high-frequency information even when the super-resolution scale is 1. The reason behind this is the fact that we are reconstructing the data from a reduced dimensional data. So even if the super-resolution scale is 1.0, technically the super-resolution scale is 8.0 because we first downscale the data and then reconstruct the data from this reduced dimensional space. This very feature of simultaneous dimension reduction and arbitrary scale super resolution makes the problem even more challenging.

Neat figures: Thank you for the suggestion. We have updated all the figures in the revised manuscript with hopefully improved figures for more visibility and readability.

According to the valuable comments and suggestions from all the reviewers, we have updated our manuscript and submitted the revised version. Please let us know if you have any questions or suggestions.

We would like to thank the reviewer for suggesting experiment with Fourier Neural Operators. In the encoder-decoder(transfer) segment of our proposed model, we replaced the Residual Implicit Neural Block with Fourier Neural Operator and trained the model with similar optimization procedures. Here, attached the results on the two evaluation metrices for different super resolution scales:

10m to 160m

PSNR:

SSIM:

10m to 200m

PSNR:

SSIM:

60m to 160m

PSNR:

SSIM:

60m to 200m

PSNR:

SSIM:

Please let us know if you have any questions or suggestions.

We thank reviewer Pjzg for the comments. We would like to respond to some of the issues the reviewer mentioned:

Lack of state-of-the-art ML baselines for comparison: Dear reviewer, thank you for your comment on state-of-the-art (SOTA) baselines. Our work tackles three tasks—data compression, modality transfer, and super-resolution—simultaneously. To our knowledge, GEI-LIIF is the only comparable work and is included as a baseline. We have also extended our ablation study (Section 6.6 in the revised manuscript) to evaluate each task individually against SOTA methods specific to those tasks. If you have additional baseline suggestions, we would greatly appreciate them.

Availability of codes: We will release the codes upon acceptance of our paper. However, we are open to provide specific code implementations, e.g. model architecture, before the official release.

How fine is continuous? Thank you for highlighting this concern. By "continuous super-resolution," we refer to our decoder's ability to perform inference at any super-resolution scale or arbitrary coordinate point, unlike traditional models with fixed upsampling ratios. This flexibility justifies the term "continuous super-resolution."

Multi-altitude vs. multi-modality: We define multimodality when data from different modalities are acquired from different sensors. Under this definition, multi-altitude wind data can be attributed to multi-modal data. “Implicit neural representations for simultaneous reduction and continuous reconstruction of multi-altitude climate data”, which we use as the baseline method also follows this definition of multi-modality and attributes multi-altitude data as multimodal data. This definition of multi-modality is added in the revised manuscript. Furthermore, inspired by your insightful comment, we are conducting additional experiments where the input and output consist of distinct climate variables (e.g., temperature and water vapor), representing dual modalities in the earth system. While these experiments are still ongoing, we will update the manuscript with the latest results as soon as they are available.

Comparison to generative model based downscaling (e.g. diffusion, GAN): Traditional downscaling tasks focus on reconstruction to the same dimension as to that of the input. On the contrary, we focus on discrete downscaling and consecutive continuous reconstruction. The problem can be seen as two separate tasks and two separate models can be employed (obviously solved separately). However, we only focus on those approaches that treat both tasks as one single problem. Additionally, in this work, we did not focus on generative model based downscaling. We hope to deal with the same problem of this work with generative models in future.

One for all scales: One model for any arbitrary scale super resolution, offering flexibility over fixed scale super resolution models.

Description of metrics: Added the details in Appendix of the revised manuscript.

Stable performance across different super resolution scales: This is indeed the purpose of designing continuous reconstruction with implicit neural networks, which is expected to capture the functional relationships better in the continuous spatial coordinates compared to traditional regular discrete grid based super-resolution. Implicit neural networks reconstruct the data in a pixel-by-pixel or coordinate-by-coordinate manner in arbitrary spatial coordinates, unlike other super-resolution models with the fixed regular grids. Also, the PSNR metric evaluates the average performance of the reconstruction over all the pixels. That’s why the PSNR evaluations do not change that much over the different super-resolution scales. However, at OOD (out-of-distribution) super-resolution scales, the result might change and it is indeed an area for exploration.

Out of distribution scales: Updated Figure 3 with out-of-distribution super-resolution scales.

Zero-shot learning/Spatial generalization: Added extra experimental results on altitude-based spatial generalization test. Due to space constraints, we have moved the section on spatial generalizability section to Appendix Section D.

Clarity on figures: On all the figures of results, the $x$-axis shows different super resolution scales. In some figures, fractional super-resolution scales may not be shown in the $x$-axis labels to keep the figure cleaner and clearer.

According to the valuable comments and suggestions from all the reviewers, we have updated our manuscript and submitted the revised version. Please let us know if you have any questions or suggestions.

We would like to re-address one of the issues raised by the reviewer. According to the suggestion from one of the reviewers on additional baseline experiment, we have added Geometry-Informed Neural Operator based experimental results in the revised manuscript.

Lack of state-of-the-art ML baselines for comparison: Our work tackles three tasks—data compression, modality transfer, and super-resolution—simultaneously. To our knowledge, GEI-LIIF is a comparable work with the highest similarity to our problem and is included as a baseline. We thank the reviewers for suggesting additional baseline experiment with Fourier Neural Operators. We chose Geometry-Informed Neural Operator [5] as the additional baseline experiment due to its suitability for this specific problem statement, and updated the manuscript with results of this additional baseline experiment.

[5] Li, Zongyi, et al. "Geometry-informed neural operator for large-scale 3d pdes." Advances in Neural Information Processing Systems 36 (2024).

Please let us know if you have any further questions, comments or suggestions.

Dear Reviewer,

We hope this message finds you well. As the deadline for manuscript revision is approaching to the end, we wanted to follow up to check if you have had a chance to review the revised version of the paper. Please let us know if there are any additional revisions or clarifications required on our part. We would be happy to address them promptly.

Thank you for your time and feedback!

Regards,

Authors