Paper: Continuous Field Reconstruction from Sparse Observations with Implicit Neural Networks (ICLR-2024)

Reviewer comment (Experiments):
The dataset is not that challenging. This paper considers monthly-averaged temperature data in **Simulation-based data** CESM2. The dynamics are relatively smoother than daily-averaged data. Please see some of my follow-up concerns regarding the dataset in **Questions**.

Author response:
Response: Thank you for pointing it out. In the data preprocessing step, we intentionally remove the temporal mean and seasonal cycle as they are straightforward to predict. This clarification has been included in the revised paper.

Reviewer comment (Experiments):
The sparsity setting considers some random sampling schemes. If the authors can discuss a bit on the optimal sensor placement, it would enhance the paper.

Author response:
Response: Thank you for the excelelnt suggetion. We included some discussion in the revised paper. In utilizing MMGN for optimal sensor placement, two approaches can be considered. Firstly, employing state-of-the-art methods such as POD-QDEIM [1,2] to identify optimal sensor locations and then utilizing MMGN trained with measurements from these identified locations for field reconstruction. Secondly, given the challenging combinatorial nature of optimal sensor placement, MMGN can serve as a surrogate model to expedite the search process.

[1] Drmac, Z. and Gugercin, S., 2016. A new selection operator for the discrete empirical interpolation method---improved a priori error bound and extensions. SIAM Journal on Scientific Computing, 38(2), pp.A631-A648.

[2] Manohar, K., Brunton, B.W., Kutz, J.N. and Brunton, S.L., 2018. Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns. IEEE Control Systems Magazine, 38(3), pp.63-86.

Reviewer comment (Experiments):
Since the authors use the Gabor filter instead of other common schemes to learn high-frequency signals, it would be good to have an ablation study on the Gabor filter part.

Author response:
Response: Thank you for your valuable suggestion. We agree that an ablation study on the Gabor filter part would provide further insights into its contribution to learning high-frequency signals. In response to your feedback, we will conduct additional experiments to assess the impact of the Gabor filter. The comprehensive results are now detailed in the revised paper, specifically in Section 4. High-level summaries of these findings are presented here (The evaluation metric used for performance measurement is Mean Squared Error): (1) The elimination of filters leads to a significant drop in model performance, highlighting the indispensability of filter designs; (2) Substituting the Gabor filter with a Fourier filter not only diminishes accuracy ($9.0\%$'s drop in simulation-based data and $18.1\%$'s drop in satellite-based data) but also increases the model size. This shows the adaptability of Gabor filters to different bandwidths for capturing patterns at various scales.


| Designs | $\#$ Param | Simulation |  Satellite |
| ----------- | ----------- | ----------- |  ----------- | 
|  None | $577 K$ | $1.758e^{-2}$ |$6.883e^{-3}$ |
|  Fourier | $601 K$ | $4.439e^{-3}$ |$7.422e^{-4}$ |
|  Gabor | $581 K$ | $4.073e^{-3}$ |$6.290e^{-4}$ |

Reviewer comment (Writing):
The 4th line of Page 6, “we uniformly distributing” should be “we uniformly distribute”.

Author response:
Response: Thank you for pointing it out. Fixed.

Reviewer comment (Experiments):
How do the authors determine the sparse setting (Page 7) for those two datasets? The sparsity ratios for those two datasets are quite different. Also, can the authors provide an estimate or convergence analysis of the smallest number of sensors needed for reconstruction?

Author response:
Response: We have conducted additional analysis to determine the minimum number of sensors required for effective reconstruction. Specifically, for each dataset, we employed proper orthogonal decomposition (POD) and sorted the computed eigenvalues in descending order. The cumulative energy was calculated, and for the simulation-based data, approximately 750 POD modes are needed to reconstruct $90\%$ of the total kinetic energy. Note this calculation is based on the snapshot POD method [1, 2]. Using this information, we defined an interval $[0.1\%, 1\%]$ and conducted a grid search to estimate the minimal sampling rate. Our findings, both quantitative and qualitative, indicate that $1\%$ is the closest approximation to the minimum sampling rate for the simulation data. Below is the MSE comparison between MMGN and POD reconstruction, with detailed results and descriptions included in the revised paper.

| Sampling rate | POD | MMGN |
| ----------- | ----------- | ----------- | 
|  $0.1 \%$ | $0.40475$ | $0.01672$ |
|  $0.3 \%$ | $0.14760$ | $0.01312$ |
|  $0.5 \%$ | $0.07922$ | $0.01199$ |
|  $0.7 \%$ | $0.04830$ | $0.00972$ |
|  $0.9 \%$ | $0.03106$ | $0.00796$ |


[1] Taira, K., Brunton, S.L., Dawson, S.T., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V. and Ukeiley, L.S., 2017. Modal analysis of fluid flows: An overview. Aiaa Journal, 55(12), pp.4013-4041.

[2] Sirovich, L., 1987. Turbulence and the dynamics of coherent structures. I. Coherent structures. Quarterly of applied mathematics, 45(3), pp.561-571.

Reviewer comment (Experiments):
Why not consider the ERA5 dataset for climate modeling since the recent climate modeling papers [1-3] all use this benchmark dataset?

Author response:
Response: The model is designed to be data-agnostic, making it applicable to the ERA5 dataset. We plan to release the implementations as open source, enabling readers to reproduce the presented results and apply the model to datasets of their research interest.

Reviewer comment (Experiments):
What are the spatial resolutions for the datasets used in this paper?

Author response:
Response: Thank you for bringing this to our attention. For the simulation-based data, it's 1024 (time) $\times$ 192 (latitude) $\times$ 288 (longitude); and for the satellite-based data, it's 360 (time) $\times$ 901 (latitude) $\times$ 1001 (longitude). We have also accordingly revised the data description section in the paper.

Reviewer comment (Presentation):
Presentation is quite weak and I cannot follow it and understand what they are doing. First, mathematical notation is ambiguous and does not seem to follow standard mathematical principles, which creates confusion (Examples are in Questions section). Furthermore, background and description seems to be more like a list of different things and without clear connection. To make presentation more clear, authors could perhaps try to express their method with less mathematically rigorous notations, but explaining their method with words (e.g. by providing example).

Author response:
Dear reviewer, we appreciate your comments and acknowledge your concerns regarding our manuscript. We have carefully considered your feedback and made substantial revisions to enhance the clarity and coherence of our work. We have simplified the mathematical notations, providing more intuitive explanations and examples to facilitate better understanding. A simple walkthrough is provided below.

**Problem** Our objective is to accurately reconstruct a spatiotemporal continuous physical field, denoted as $\boldsymbol{u}$, representing quantities such as temperature, velocity, or displacement. This field, $\boldsymbol{u}$, is inherently a function of both spatial coordinates ($\boldsymbol{x}$) and time ($t$).

**Prior approaches**  Traditional implicit neural networks (INRs) incorporate the time index ($t$) along with spatial coordinates. For instance, in 2D data, the network features three input nodes for the $x$-coordinate, $y$-coordinate, and time $t$, where the time index ($t$) is employed to direct the model to a particular time instance.

**Our approach** Motivated by the desire for a more context-aware indexing mechanism, we posed the question: Can the indexing process be improved? We propose utilizing measurements observed $u_t^{(1)}, u_t^{(2)}, \dots$ at time t for implicit model guidance, rather than relying on $t$ for explicit pointing. More precisely, our approach involves constructing an encoder $E_\varphi(\cdot)$ to transform the observed measurements $u_t^{(1)}, u_t^{(2)}, \dots$ into a latent code ($\boldsymbol{z}_t$). Subsequently, we employ a decoder $D_\phi(\cdot)$ that utilizes both the encoded information ($\boldsymbol{z}_t$) and spatial coordinates ($\boldsymbol{x}$) to reconstruct the underlying physical field. Below are detailed explanations of the model components we designed and the rationale behind each choice.

- **Encoder**: The *auto-decoder* is chosen as the backbone for the encoder due to its ability to accommodate free-formed observations, considering that the number and positions of available observations vary over time.
- **Gabor filter**: INRs often encounter challenges in learning high-frequency functions, a phenomenon known as "spectral bias." To mitigate this issue, we employ the Gabor filter to transform spatial coordinates ($\boldsymbol{x}$) before the decoding process.
- **Decoder**: The decoder incorporates a multiplicative filter network (MFN) as its backbone network. This choice is motivated by the recursive mechanisms embedded in MFN, enabling improved fusion of spatial coordinates ($\boldsymbol{x}$) and latent codes ($\boldsymbol{z}$).

**Results** We conducted comprehensive experiments to demonstrate the superior performance of the proposed model.

In response to your feedback, we have extensively revised our paper, ensuring that the updated manuscript is more accessible and comprehensible. In the meantime, we provide point-to-point response to your comments below:

Reviewer comment (Experiments):
While numerical results look good when compared to other INR methods, there is no comparison to other non-INR based methods such as GP (if applicable) or methods that are currently used in meteorology and climate studies.

Author response:
**Comparison with Non-INR Methods**:
We conducted a comparison of our method with Gaussian processes. Specifically, we employed Gaussian process regression using scikit-learn, testing various kernels such as RBF, DotProduct, WhiteKernel, and their combinations. These experiments were conducted on simulation-based data, with 5% of the data points utilized for training. The Mean Squared Error (MSE) for our MMGN is $4.244e^{-3}$, and the performances of the GPs are provided below:


| Data size | 10 Instances |  20 Instances |  30 Instances |  40 Instances |  50 Instances |
| ----------- | ----------- | ----------- | ----------- | ----------- | ----------- |
|  GPs Performance | $7.76032e^{-2}$ | $6.86319e^{-2}$ | $7.26413e^{-2}$ | $6.26571e^{-2}$ | $5.31689e^{-2}$ |

Overall, it's important to note that GPs come with a cubic cost in terms of dataset size. We conducted our experiments on an A100 GPU with 40 GB of memory, limiting the maximum number of time instances to less than 50, even with a low sampling rate. Additionally, we noticed that GPs are known to be sensitive to conditioning. Given the time constraints between the reviewer's comment update and the discussion period closing, we were only able to achieve the presented performance. However, we acknowledge that further improvements could be achieved with a more efficient implementation and hyperparameter tuning, although the computational cost remains high.

Reviewer comment (Theory):
Problem setup: They assume that the system is governed by a PDE (1). I don't see any reason why this assumption is made, or is that assumption used somewhere?

Author response:
Comment $\#$ 1: Problem setup: They assume that the system is governed by a PDE (1). I don't see any reason why this assumption is made, or is that assumption used somewhere?

Response: We used PDE to facilitate the transition to functional variable separation and subsequently to our proposed context-aware indexing mechanism. However, we acknowledge the potential confusion it may have caused, and as a result, we have removed this equation from the paper.

Reviewer comment (Writing):
Section 2.1, "Objective" (and throughout the paper): They use notation $\mathcal U_k=(u(t_k,x)|\mathcal X_k)_{x\in X_k}$. What does this "$|\mathcal X_k$" stand for in the set? "$|$" is usually used for conditioning but I cannot see such a thing here. I would rather remove this extra part to avoid confusion (I use '(' and ')' instead of curly brackets as this form does not seem to render those (and actually, strictly speaking, '(' and ')' would be preferred if order of measurements has meaning $\mathcal U_k$)).

Author response:
Comment $\#$ 2: Section 2.1, "Objective" (and through the paper): They use notation . What this " " stands in the set? usually is used for conditioning but I cannot see such a thing here. I would rather remove this extra part here to avoid confusion (I use '(' and ')' instead of curly brackets as this form does not seem to render those (and actually, strictly speaking, '(' and ')' would be preferred if order of measurements have meaning ))

Response: Conditioning is employed in our notation because the latent codes are derived from all available position-dependent measurements at time $t$.

Reviewer comment (Theory):
The section 2.1 does not talk about noise or randomness, and formulations in this section would indicate that $u(t,x)$ is deterministic. However, later sections seem to talk about noise and assume that $u$ includes noise or is random (e.g., Eq (5) which does not have meaning if $u$ is deterministic). This is contradictory. I would suggest that authors use separate notations for a noiseless quantity $u$ and the measurements, e.g., say something like $y_k$ is a vector of measurements or $u(t_k,x)$ plus noise.

Author response:
Comment $\#$ 3: The section 2.1 does not talk about noise or randomness and formulations in this section would indicate that is deterministic. However, later sections seems to talk about noise and assume that is includes noise or is random (e.g. Eq (5) which does not have meaning if is deterministic). This is contradicting. I would suggest authors that authors would use separate notations for a noiseless quantity and the measurements, e.g., say something like is a vector of measurements or plus noise.

Response: Thank you for your feedback. We have incorporated the suggested changes into the revised paper, and additional details regarding the model's performance under various noise levels have been included.

Reviewer comment (Evaluation):
Section 2.3, "Neural pinpointing": They mention that learning-based methods can be computationally demanding, but there are two sides to this—train time computational complexity and run time/inference computational complexity. GANs have high training complexity but inference may not be so much, and perhaps even less than the proposed INR depending on the architectures.

Author response:
Comment $\#$ 4: Section 2.3, "Neural pinpointing". They mention that learning-based methods can be computationally demanding. But there are two sides in this: train time computationally complexity and run time/inference computational complexity. GANs have high training complexity but inference may not be soo much, and perhaps even less as the proposed INR depending on the architectures.

Response: Yes, we agree with the reviewer. What we meant by "computationally demanding" was the challenge of adapting Generative Adversarial Networks (GANs) to irregular mesh and arbitrary-scale super-resolution settings.

Reviewer comment (Writing):
Section 2.4, "$\alpha=g(x)\circ \eta(t)$": The standard meaning for $\circ$ is composition of functions, i.e., $(f\circ g)(x)=f(g(x))$. Here this probably is not the case as I cannot see how you would compose $g$ and $\eta$ in this case. Of course, authors can define their own notations, but then they should state how they use the notation. If $\circ$ is meant to be multiplication, I would just remove $\circ$.

Author response:
Comment $\#$ 5: Section 2.4, " . The standard meaning for is composition of functions, i.e. . Here this probable is not the case as I cannot see how you would compose and in this case. Of course, authors can define their own notations, but then they should say that how they use the notation. If is meant to be the multiplication, I would just remove.

Response: Thank you for the comment. The introduction of the method has been rewritten for clarity, and a new figure has been added to facilitate a comparison between the proposed approach and current INRs.

Reviewer comment (Presentation):
Section 3 seems to be a list of different things and it is not clear how the things are connected. For example, how do you use auto-decoder? Do you have an (auto-)decoder as encoder? How are Gabor filters used? Are these somehow connected to $g$ and $\eta$ earlier? Is the multiplicative filter network the decoder?

Author response:
Comment $\#$ 6: Section 3 seems to be a list of different things and it not clear to me that how the things are connected. For example, how you use auto-decoder? You have (auto-)decoder as encoder? How Gabor filters are used? Are these somehow connected to and earlier? Multiplicative filter network is decoder?

Response: Please refer to the previously provided walkthrough for detailed information.

Reviewer comment (Theory):
About latent code z. The paper claims to use the trainable latent code to represent temporal information instead of using time index. What is the motivation for this? Because the recorded time index is not reliable in this application? Or the latent code can also represent other semantic information? Previous works have shown INR can represent spatial-temporal information in a good way by adding temporal index as input coordinates.

Author response:
Yes, the latent code has the advantage of capturing not only temporal information but also other semantic features of the underlying physical process. Previous studies have indeed demonstrated the capability of INR to effectively capture spatialtemporal information by incorporating a temporal index as additional input coordinates. But it should be noted that different from forecasting problems, the time index ($t$) is primarily used as a reference to indicate a specific time instance in the context of field reconstruction. Motivated by the desire for a more **context-aware indexing mechanism**, we introduce a new perspective to the modeling process by utilizing a trainable latent code. Since the latent code is learned using the available measurements at time $t$ via auto-decoding, it is a more context-aware representation, which improves the performance of the INR-based decoder. We have expanded the methodology section to elaborate on these points, providing a detailed discussion on the motivation behind our model design choices and a visual comparison between the traditional INR and our proposed approach. We sincerely invite the reviewer to check the updated paper and provide any further feedback or suggestions you may have.

Reviewer comment (Experiments):
As described in the paper, during training, both latent code and INR are trained. While during training, INR model parameters are fixed and only latent code is optimized. Since the given information is quite sparse during testing, how reliable it can be to optimize latent code? How robust this method can be when generalized across different objects? By only changing the latent code in the input of INR decoder and freezing all other model weights, can this change the decoded image in an efficient way?

Author response:
Thank you for the insightful question regarding the generalizability of latent codes. To investigate this aspect, we conducted additional experiments by partitioning the simulation-based data set into two subsets, each comprising 512 instances. The temporal splitting was performed in an ascending order, with the initial 512 instances assigned to subset $\mathcal{D}_1$ and the subsequent 512 instances to subset $\mathcal{D}_2$. Consequently, due to extrapolation in the time direction, the distributions of $\mathcal{D}_1$ and $\mathcal{D}_2$ could vary significantly. We trained MMGN using $\mathcal{D}_1$. Then, keeping the decoder parameters fixed, we performed inference on $\mathcal{D}_2$, where the only trainable parameters were the newly initialized latent codes. The results indicate a marginal increase in error, demonstrating that the proposed MMGN avoids the catastrophic forgetting issue encountered by other time-index-based INR methods when re-trained using different data set.

| Data set | MSE |
| ----------- | ----------- |
|   $\mathcal{D}_1$ | $3.997e^{-3}$ |
|   $\mathcal{D}_2$ | $4.012e^{-3}$ |

Admittedly, one limitation of our current work is its focus on reconstructing a selected trajectory. In the future, we plan to investigate MMGN's potential for generalization to multiple trajectories or even across various climate properties, with the goal of recovering arbitrary underlying flow maps. We have incorporated this discussion into the conclusion section of the revised paper.

Reviewer comment (Experiments):
For the comparison with baseline methods as in Table 1, it seems all the baseline methods use a different encoding than the proposed model, which uses Gabor filters over Fourier bases to transform the coordinates as Fourier transforms emphasize a global frequency representation. Usually, the INR models are quite sensitive with different encoding methods. Can the paper also report the comparison with baseline methods using the same Gabor filters as encoding to disclose how impactful the encoding method is? Similar with the backbone model architecture, are all the comparisons based on the same backbone model? Only in this way, it may be fair to say how efficient the latent code can be in this case.

Author response:
We appreciate the reviewer's suggestion regarding the encoding method and backbone model architecture in our comparison with baseline methods. To further study the impact of the Gabor filter, we conducted additional ablation experiments, encompassing the removal of filter designs and their replacement with other types. The comprehensive results are now detailed in the revised paper, specifically in Section 4. High-level summaries of these findings are presented here (The evaluation metric used for performance measurement is Mean Squared Error): (1) The elimination of filters leads to a significant drop in model performance, highlighting the indispensability of filter designs; (2) Substituting the Gabor filter with a Fourier filter not only diminishes accuracy ($9.0\%$'s drop in simulation-based data and $18.1\%$'s drop in satellite-based data) but also increases the model size. This shows the adaptability of Gabor filters to different bandwidths for capturing patterns at various scales.

| Designs | $\#$ Param | Simulation |  Satellite |
| ----------- | ----------- | ----------- |  ----------- | 
|  None | $577 K$ | $1.758e^{-2}$ |$6.883e^{-3}$ |
|  Fourier | $601 K$ | $4.439e^{-3}$ |$7.422e^{-4}$ |
|  Gabor | $581 K$ | $4.073e^{-3}$ |$6.290e^{-4}$ |

Reviewer comment (Experiments):
The compared methods are all the object-specific or scene-specific models, while the proposed model seems to be a generalized model across multiple images or objects. The method may also need to be compared with: (1) other generalized models based on INR such as SRN [1], etc. (2) other spatial-temporal models using time index such as D-NeRF [2].

Author response:
Thank you for your valuable suggestion. We appreciate your insightful comments and tried our best to implement the suggested comparisons. It is still a work in progress. In the current revision, we have included some discussion in the related work section, acknowledging the importance of these comparisons.

Reviewer comment (Evaluation):
The visualization of latent code is quite interesting but it is not clear what it means for the Figure 5. Can the author explain more about what information is learned or embedded by the latent code from these plots?

Author response:
Thank you for the question. We acknowledge the need for improved clarity in our presentation, and we have refined the descriptions in the revised paper accordingly.

Reviewer comment (Writing):
I find the paper very hard to read. Notation and expressions in Section 3 and the last paragraph of Section 2 are confusing and it takes a long time to understand what is going on (Figure 2 helps here but the math could be clearer). Another reason is very florid prose which I'd hesitate to use in a machine learning paper. Examples: "The endeavor to tackle the conundrum of global field reconstruction from sparse observations is currently shaping two distinct paradigms of thought", "raging ocean waves", "intricate physical fields" (what are they?), "we present an _innovative_ neural network approach", "... a singular viable option", ...

Author response:
Response: We thank for the valuable feedback. We have made several improvements to the paper. Specifically, we corrected the reference in the introduction and provided a clearer explanation of the contributions. Moreover, we enhanced the methodology section by introducing the model more explicitly and illustrating the design process as well as adding a new figure.

Reviewer comment (Novelty):
While the narrative insists on strong novelty, all ingredients (and various combinations thereof) are well known. The separation of variable idea is based on Dona et al 21, multiplicative filter networks are known, local activations (such as Gabor or wavelet) have been used in many papers, e.g. https://openaccess.thecvf.com/content/CVPR2023/html/Saragadam_WIRE_Wavelet_Implicit_Neural_Representations_CVPR_2023_paper.html. In some sense the present paper is a multiplicative filter network with a separate treatment of time through an encoder–decoder. This is perfectly fine but in my view it limits the novelty, and there is no shortage of all sorts of implicit networks.

Author response:
Response: We truly appreciate this comment, recognizing that our initial presentation lacked clarity regarding the proposed model. We acknowledge the multifaceted nature of the model, as highlighted by the reviewer. In response, our revised paper offers a more explicit discussion, elucidating the rationale behind the selection of each component. Notably, we emphasize the pivotal innovation introduced in our work: the **context-aware indexing mechanism**. 

Previous Implicit Neural Representation (INR) models incorporated a temporal index ($t$) as additional input coordinates. Different from scenarios involving forecasting, where the time index plays a more active role, in the context of field reconstruction, we argue it primarily serves as a reference for indicating a specific time instance. Consequently, our aim is to devise an improved pointing method. The latent codes, derived from available context information through the utilization of measurements from the underlying physical process at time t, encapsulate richer semantic information. This is exemplified in extreme cases, such as fixing the latent size to 1, where our proposed MMGN consistently outperforms other INR baselines, as demonstrated in the following experimental results.

| Model | MSE (Simulation-based data) |  MSE (Satellite-based data) |
| ----------- | ----------- | ----------- |
|  ResMLP | $0.01950$ | $0.00154$ |
|  FFN+P |  $0.02516$ | $0.00303$ |
|  FFN+G |  $0.03239$ | $0.00510$ |
|  SIREN |  $0.27304$ | $0.31291$ |
|  MMGN (latent size = 1) |   $\boldsymbol{0.01898}$ | $\boldsymbol{0.00152}$ |

Reviewer comment (Theory):
In real fields space and time _are_ intertwined so one needs to go beyond the rank-one decomposition f(x) g(t); Dona et al mentions a low rank version \sum_k f_k(x) g_k(t). It is not clear to me from the current description how this required expressiveness is achieved. It would be nice to explain that more clearly.

Author response:
Response: Yes, achieving a clear separation of spatial function $f(x)$ and temporal function $f(x)$ in modeling poses significant challenges. In our approach, we reconsider the roles of spatial coordinates $x$ and temporal index $t$. Broadly, the time index $t$ serves as a means to guide the model to a specific time instance. In field reconstruction, we can (1) leverage $t$ for explicit pointing or (2) use measurements observed at time $t$ for implicit model guidance. We adopted the second approch. Consequently, there is no explicit need for variable separation.

Reviewer comment (Theory):
Generally, there is a lack of insight on why this strategy is supposed to perform well and why other approaches are supposed to perform poorly. This is a more general problem: there are now hundreds of implicit networks with different hyperparameters, training strategies, etc, and very little theoretical or other understanding of their mechanics. What if you compare your approach with an optimally tuned adaptive kernel estimator? How about spline interpolation? How about a simple low pass filter? It will surely not mess up as badly as ResMLP in Figure 3.

Author response:
Response: We conducted supplementary experiments to examine the enhancements facilitated by the context-aware indexing mechanism, as demonstrated in the extreme case results presented earlier. Additionally, we explored the effects of the Gabor filter through additional experiments, the outcomes of which are detailed below. Our observation of MMGN's superior performance is attributed to the synergistic impact of the context-aware indexing mechanism and the Gabor filter-based decoder utilized in our model.

| Designs | $\#$ Param | Simulation |  Satellite |
| ----------- | ----------- | ----------- |  ----------- | 
|  None | $577 K$ | $1.758e^{-2}$ |$6.883e^{-3}$ |
|  Fourier | $601 K$ | $4.439e^{-3}$ |$7.422e^{-4}$ |
|  Gabor | $581 K$ | $4.073e^{-3}$ |$6.290e^{-4}$ |

Reviewer comment (Experiments):
Along the same lines, in datasets description you mention that wavelets were used for initial interpolation of satellite-based data on a grid; why not compare the described method with a sparse wavelet interpolation?

Author response:
Response: The second dataset utilized in our study is sourced from GHRSST (Group for High-Resolution Sea Surface Temperature). Various data levels are available, with higher levels signifying increased spatial coverage. In the initial stages of our project, we aimed for greater data control to assess different INR models. Consequently, we opted for level 4, wherein the input data has already undergone preprocessing via sparse wavelet interpolation.

Reviewer comment (Reproducibility):
In experiments where you recover fields on the surface of a sphere, are you using Cartesian or spherical polar coordinates?

Author response:
Response: In experiments involving the recovery of fields on the surface of a sphere, we utilize spherical polar coordinates for modeling.

Reviewer comment (Experiments):
Why not include (different levels of) random noise in observations? This is an important test of robustness.

Author response:
Response: Thank you for the suggestion. We have conducted additional experiments considering the introduction of noise to the input data. The noise level in the dataset is quantified by the channelwise standard deviation specific to that dataset. Furthermore, we introduced customized noise ratios, including scenarios with noise levels set at 1\%, 5\%, and 10\%. The quantitative results are presented below. Additionally, we made a new figure to enhance the visualization of performance distinctions among various models in the presence of noise. The detailed discussion and presentation of these findings have been incorporated into the revised paper.

| Model | Noise free | $1\%$ | $5\%$ | $10\%$ |
| ----------- | ----------- | ----------- | ----------- | ----------- | 
|  ResMLP | $0.01852$ | $0.01895$ | $0.01969$ | $0.02214$ |
|  FFN+P  | $0.02357$ | $0.02463$ | $0.03020$ | $0.04324$ |
|  FFN+G  | $0.02982$ | $0.03239$ | $0.03774$ | $0.05335$ |
|  MMGN   | $0.00448$ | $0.00452$ | $0.00629$ | $0.01387$ |

Reviewer comment (Novelty):
I am not sure that (Izacard et al., 2019) is the right reference for sparse seismic networks and small earthquakes (or even for sparse spatial coverage in scientific data; it addresses a super-resolution problem).

Author response:
Response: We thank for the valuable feedback. We have made several improvements to the paper. Specifically, we corrected the reference in the introduction and provided a clearer explanation of the contributions. Moreover, we enhanced the methodology section by introducing the model more explicitly and illustrating the design process as well as adding a new figure.