W1: Handling Unstructured Grids

We sincerely appreciate the reviewers’ insightful comments and suggestions. In our paper, we have evaluated three irregular datasets, achieving state-of-the-art (SOTA) performance on two of them, which can be attributed to our new padding approach. It is worth noting that traditional transformation methods, such as FNO, are not directly applicable to irregular datasets (Table 1). Regarding the Elasticity dataset used in Geo-FNO[1], we would like to clarify that, as discussed in Section 2, the Radon transform is not suitable for this one. It is because that the Radon transform inherently reduces dimensionality (e.g., from 2D to 1D), whereas the Elasticity dataset is simply one-dimensional.

We greatly value the reviewer’s suggestion regarding point cloud data, which indeed presents a more challenging scenario. To address this, we propose two potential solutions. The first involves incorporating a GNO module, as introduced in [2], which we will further elaborate on in our response to the three-dimensional dataset question below. The second approach leverages a latent space representation. We have conducted experiments on the Elasticity dataset in the latent space, and the results, presented below, demonstrate superior performance compared to both FNO and Transolver. These findings further highlight the advantageous properties of the Radon transform.

batch size 20, single RTX-4090, angles 32

W2/Limitations: About real-world and 3D datasets

We sincerely appreciate the reviewer’s insightful feedback. First, in Appendix C.1.2, we rigorously prove the bilipschitz property of the Radon operator in arbitrary dimensions, which provides theoretical support for solving high-dimensional PDEs. Additionally, as thoroughly demonstrated in Section 2, the Radon transform inherently possesses strong dimension-reduction properties, making it particularly suitable for such problems.

We fully acknowledge that the exploration of 3D PDEs and real-world datasets remains an important and promising direction. We sincerely appreciate you bringing these two excellent works to our attention, which will be added as new citations in later submission. While FactFormer[3] studies Kolmogorov flow and buoyant smoke on uniform grids, Transolver++ [4] tackles million-scale car and aircraft CFD meshes plus six standard benchmarks. These papers on 3D PDEs and real-world datasets have indeed broadened our perspective, and we plan to explore them further - we agree this represents a particularly interesting and promising research direction. Regarding the reviewer’s concerns about 3D PDEs and real-world datasets (which are often point-cloud-based), we also propose a feasible solution inspired by [2], where the integration of a GNO module could effectively address these challenges. Thank you again for highlighting these valuable references.

As suggested for the real-world evaluation, we have conducted relevant experiments on the ERA5 dataset referenced in our work [5]. We provide error results as follows. We are currently considering including additional visualization of these results in later version to better demonstrate these findings and expand our discussion on these aspects to provide further insights.

Q1: About sinogram domain convolution

We appreciate the opportunity to clarify this point. Using Darcy flow as an illustrative example, the Radon transform converts the original data shape [B,C,W,H] into [B,C,max(W,H),A], where A represents the number of selected angles. While the convolution operation in the sinogram domain bears similarity to conventional convolution, it operates on angle-transformed coordinates.

The left subplot of Figure 5 demonstrates this process through red dashed arrows that visually depict the angle information extraction. Concerning the sinusoidal pattern shown in the right subplot, we would like to direct readers to our experimental visualization in Appendix G.2 (Figure 8) for reference. This pattern emerges naturally in the sinogram domain data during actual training, and we have presented an abstract representation of this phenomenon in Figure 5's right subplot for conceptual clarity.

Q2: About ablation experiments

We thank the reviewer for pointing out the advice on ablation. We have now run the experiment “RNO w/o P.A.” (only Radon blocks) on the 85×85 Darcy dataset. Our model enjoys slightly better performance over FNO, exhibiting enhanced capability in capturing local features.

Darcy 85×85 grid, batch size 2, single RTX-4090

As an additional reference, we have conducted an ablation study where we replaced the GPU optimized pytorch algorithm in FNO with a standard algorithm. This experiment provides empirical support for the potential acceleration of our computational efficiency.

Darcy 43×43 grid, batch size 2, single RTX-4090

Tips: While Darcy flow simulations typically employ an 85×85 grid configuration, we conducted our comparative analysis using a reduced 43×43 grid size due to the substantial memory requirements of Non-GPU optimized pytorch algorithm.

[1]Fourier neural operator with learned deformations for pdes on general geometries, JMLR 2023

[2]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023

[3]Scalable transformer for pde surrogate modeling, NeurIPS 2023

[4]Transolver++: an accurate neural solver for pdes on million-scale geometries, ICML 2025

[5]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M

W1/Q1: About practical runtime & memory

We appreciate the reviewer’s insightful feedback on the need for concrete runtime and memory figures to assess RNO’s practicality. In Section 4.3 and Appendix G.5, we provide detailed comparisons of runtime, GPU memory usage, and parameter counts for RNO against FNO on representative PDE benchmarks. These results confirm RNO’s quasi-linear computational complexity, enabled by efficient sinogram-domain operations, with runtimes and memory usage comparable to FNO. Additionally, in Appendix G.5 (Table 9), we provide comprehensive results including memory usage, time, parameters, and errors under varying angular conditions. These empirical findings further support our claim regarding the quasi-linear computational complexity of the proposed method. To further strengthen this evaluation, we will include additional comprehensive comparisons in the camera-ready manuscript, ensuring a clear and robust demonstration of RNO’s efficiency.

Darcy 85×85 grid, batch size 2, single RTX-4090

Our model demonstrates improved performance over WNO across multiple metrics and achieves state-of-the-art (SOTA) results in terms of accuracy. We acknowledge that, at first glance, our approach may appear computationally slower than FNO. However, our ablation studies reveal that FNO’s efficiency relies heavily on PyTorch’s GPU optimizations—when these optimizations are disabled, its performance degrades significantly. This observation suggests that RNO holds promising potential for further computational efficiency improvements.

Darcy 43×43 grid, batch size 2, single RTX-4090

Tips: While Darcy flow simulations typically employ an 85×85 grid configuration, we conducted our comparative analysis using a reduced 43×43 grid size due to the substantial memory requirements of Non-GPU optimized pytorch algorithm.

W2: About our motivation

We sincerely appreciate the reviewers’ insightful comments. RNO achieves a 10–14% improvement on Darcy and Airfoil datasets, marking a significant advancement. More importantly, as shown in Section 4.2, Appendix G.3, and Table 2, RNO’s key strength is its exceptional generalization, outperforming FNO by ~70% across scales and Transolver at high resolutions.

Following the reviewers’ suggestions, our latest experiments show promising results even with less than half the angular information, highlighting a potential direction for sparse data processing. We will include further visualizations and analysis in the camera-ready version.

To better highlight RNO’s advantages and clarify our motivation, we provide comparative results below. We apologize for the markdown formatting limitations and will ensure clearer presentation in the final version.

W3/Q2: Applicability to unstructured or irregular domains

We appreciate the reviewer’s comment on RNO’s applicability to irregular domains. Our paper already evaluates RNO on three non-regular datasets (Section 4.1, Appendix G), demonstrating its ability to handle irregular geometries—unlike traditional methods (e.g., FNO, Table 1). This is enabled by a padding strategy and the Radon transform’s flexibility, which projects data along arbitrary directions via mesh-based integration.To further address this, we will expand Section 4 to better highlight how RNO’s resolution-agnostic θ-grid convolution inherently supports irregular domains.

W4/Q3: About sensitivity to noisy

We sincerely appreciate the reviewer’s insightful comments. As presented in Section 2 and Appendix A.2.2, the FBP algorithm inherently incorporates the filtering operation. Additionally, we introduce a learnable weighting mechanism and sinogram domain convolution, which effectively enhances noise suppression in our framework. Additionally, real-world datasets often contain substantial noise. As part of our investigation, we applied our method to the monthly averaged 2 m air temperature forecasting task mentioned in [1], where we achieved promising results (error metrics shown below). This demonstrates our model's robust capability to adapt to and handle noisy data effectively. We plan to include visualizations and more detailed analysis of these results in the final version of our paper.

W5/Q4: Regarding the choice of the number of angles

We sincerely appreciate the reviewer's valuable suggestion regarding the angle selection, which is indeed a crucial parameter in Radon transform. In the main text, we provided a concise description of our angle selection strategy due to space limitations. As detailed in Appendix G.1, we present the number of angles used in our experiments, while Appendix G.4 (Figures 11 and 12) offers a more comprehensive guideline for angle selection. Our experiments indicate that excessively large angles may compromise performance at lower resolutions. Furthermore,Table 9 in Appendix G.5 (ablation studies) demonstrates that while increasing the number of angles generally improves accuracy, it also involves trade-offs in parameter count, memory usage, and computational efficiency. Following this suggestion, we will reorganize the relevant content from the appendix to the main text in the final version to improve readability.

W6: About theoretical proof

We thank the reviewer for raising the important point about the impact of discrete sampling and interpolation errors on RNO’s theoretical guarantees. In Section 3, we prove that RNO satisfies the bilipschitz strong-monotone property under a continuous Radon transform, which ensures theoretical accuracy in infinite-dimensional spaces. We appreciate the reviewer's insightful question. The excellent linear properties of Radon transform establish the connection between the bilipschitz condition and boundedness, which fundamentally remain unchanged in discrete settings. Nevertheless, we agree this aspect deserves careful discussion, and we will provide additional clarification. We will give detailed instructions in the camera ready version. Due to space limitations, we only give a general idea here. The discrete Radon transform (DRT)[2] on an $N \times N$ grid, defined as $R f(s, \theta) = \sum_{(x, y) \in L(s, \theta)} f(x, y)$ , can still satisfy the bilipschitz condition $c ||f||\_{\ell^2}$ $\leq$ $||Rf||\_{\ell^2}$ $\leq$ $C ||f||\_{\ell^2}$ because, as a finite-dimensional linear operator, it is inherently bounded, ensuring an upper bound $||Rf||\_{\ell^2}$ $\leq$ $C ||f||\_{\ell^2}$ with $C$ dependent on $N$ ; moreover, with a sufficiently rich set of lines (e.g., coprime slopes in the Mojette transform), DRT remains injective, and its inverse, though conditioned by grid size, is bounded in $\ell^2$ (i.e., $||R\^{-1} g||\_{\ell^2}$ $\leq$ $K||g||\_{\ell^2}$ ), yielding a lower bound $||Rf||\_{\ell^2}$ $\geq$ $\frac{1}{K}$ $||f||\_{\ell^2}$ with $c = \frac{1}{K}$ , thus upholding the condition under appropriate normalization and grid assumptions.

Q5: About 3D PDEs

We thank the reviewer for their forward-thinking question about extending RNO to 3D PDEs. In Section 2, we present the high-dimensional formulation of Radon transform, and in Appendix C.1.2, we provide the proof of its bilipschitz property across arbitrary dimensions, which theoretically guarantees the feasibility of our approach. The Radon transform's excellent dimension-reduction properties make it particularly suitable for addressing high-dimensional problems.

While our current experiments primarily focus on 2D and time-dependent 3D problems (as most real-world 3D PDE problems are point-cloud based), we would like to highlight that extending our framework to handle such cases is indeed feasible. Following approaches similar to [3], a potential solution would be to incorporate a GNO module for point-cloud processing and 3D PDE applications. Although our present work mainly focuses on introducing the RNO architecture, we believe RNO shows significant potential for solving 3D problems, and we regard this as an important direction for future research.

[1]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M

[2]Discrete radon transform, IEEE Trans. Acoust. Speech Signal Process 2003

[3]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023

W1: About the limitations of our work

We sincerely thank the reviewer for highlighting the importance of discussing limitations to enhance scientific transparency and credibility. We acknowledge that explicitly addressing the limitations of the RNO in the main paper would strengthen its clarity. While we briefly mention challenges, such as the performance drop on the Pipe benchmark (Section 4.2), we agree that a dedicated section would provide a clearer perspective.

Specifically, our current work has two main limitations: (1) Unlike the Fourier transform, which benefits from well-optimized algorithms and seamless PyTorch integration, the Radon transform still has room for improvement in parallel computing and GPU utilization, where further optimizations could enhance computational efficiency; and (2) while our model primarily focuses on 2D or time-dependent 3D problems, we have not yet extensively explored general 3D PDEs (particularly those involving point cloud data). However, in the following response, we provide a feasible and theoretically supported solution to address such cases. To address this, we propose adding a "Limitations" section in the latest revision.

W2/Q1: About operational efficiency and memory

We greatly appreciate the reviewer’s suggestion. We can see in Table 9 of Appendix G.9 that when the number of angles A increases from 64 to 512, the running time increases approximately linearly, which verifies our assertion of "quasi-linear complexity". We have also added relevant experiments, and the relevant data are as follows.

Darcy 85×85 grid, batch size 2, single RTX-4090

The comparative results between our model and other baselines are summarized above. We would like to highlight that RNO achieves superior performance over WNO across multiple key metrics. Furthermore, we conducted an ablation study by disabling the GPU-accelerated PyTorch Fourier transform, which reveals a significant efficiency drop in FNO. This observation underscores RNO’s strong potential in improving computational efficiency.

Darcy 43×43 grid, batch size 2, single RTX-4090

Tips: While Darcy flow simulations typically employ an 85×85 grid configuration, we conducted our comparative analysis using a reduced 43×43 grid size due to the substantial memory requirements of Non-GPU optimized pytorch algorithm.

W3: About irregular grids

Thank you for your valuable feedback. We would like to point out that we already incorporate non-regular datasets in our evaluation (three out of the six test datasets in this paper are non-regular). We need to clarify that conventional transformation methods (such as FNO, as referenced in Table 1) are fundamentally unable to process non-uniform grids. In our approach, we address this limitation through a carefully designed padding strategy for irregular grids, which has demonstrated state-of-the-art performance on several non-uniform benchmark datasets.

As visualized in Figure 14 of Appendix G.6 (illustrating the pipe dataset case), we observe that this particular dataset requires relatively extensive padding in empty regions. Our analysis suggests this padding requirement may be a primary factor contributing to the observed performance degradation. We will ensure to provide a more comprehensive discussion of this aspect in the final version of our paper.

Q2: About angle quantity selection

We thank the reviewers for raising this insightful question about the selection of the number of angles in our experiments. First, regarding the angle number selection guide you mentioned. The number of angles selected for our experiments is presented in Appendix G.1. A more comprehensive guideline for angle selection is presented in Appendix G.4 (see Figures 11 and 12). Our experiments revealed that excessively large angles may lead to suboptimal performance at lower resolutions. As shown in Table 9 of the ablation studies (Appendix G.5), while increasing the number of angles generally improves accuracy, this comes with trade-offs in parameter count, memory usage, and computational efficiency.

Secondly, regarding the adaptive method. We have experimented with adaptive methods to optimize the number of angles, but found they did not produce satisfactory results. Therefore, we suggest setting the number of angles based on both problem scale and computational resources, as our experiments show that simply increasing the number of angles does not guarantee better performance.

Q3: Discussion of 3D PDEs

We greatly appreciate the reviewers’ question regarding RNO’s generalizability to 3D PDEs. In Appendix C.1.2, we provide a rigorous proof of the bilipschitz condition for arbitrary dimensions, which theoretically establishes the feasibility of our approach. This theoretical foundation ensures the soundness of our proposed method across different dimensional spaces. The dimensionality reduction property of the Radon transform (Section 3) is particularly advantageous for 3D PDEs, as it reduces the computational burden compared to methods like FNO, which may face scalability challenges in higher dimensions (Section 2).We would like to highlight that the GNO module integration strategy presented in [1] for 3D problem solving could be directly adapted to our framework. Our methodology inherently supports this extension, and we are fully capable of addressing 3D PDE cases through similar architectural modifications.

Q4: Discussion about inverse transform

We sincerely appreciate your valuable suggestions regarding the inverse transform selection. We have indeed explored multiple approaches in this regard, including the framework proposed in [2] and alternative implementations using UNet-style backprojection networks to replace the fixed FBP algorithm. Our extensive experimental comparisons demonstrate that the FBP algorithm achieves superior performance in PDE-related tasks, exhibiting advantages in both reconstruction accuracy and computational efficiency, while better preserving strict discrete invariance properties. That being said, we fully agree that learnable inverse transforms represent a promising research direction worthy of further exploration. Using Darcy flow as a representative case study, we present our experimental findings to demonstrate the method's performance.

Q5: About noise-sensitive and real-world datasets

We appreciate the reviewer's valuable perspective, which mirrors our own thought process during this research. As discussed in Section 2 and Appendix A.2.2, the FBP algorithm inherently includes the filtering operation, which can contribute to noise reduction to some extent. The RNO architecture leverages the sinogram-domain convolution and a reweighting technique (Section 3.4) to prioritize relevant angular contributions in the sinogram domain, which inherently provides some resilience to minor perturbations by focusing on dominant features. Additionally, the physics-attention mechanism (Section 3.2) enhances feature extraction by emphasizing physically meaningful patterns, potentially mitigating the impact of noise.

Moreover, real-world datasets often contain significant noise. Following the methodology in [3] for forecasting the monthly averaged 2 m air temperature, our experiments achieved competitive results (error metrics provided below), demonstrating our model’s robustness in handling noisy data. We will consider incorporating additional visualizations and analysis in the final version to further illustrate these findings.

[1]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023

[2]Radon Inversion via Deep Learning, IEEE Trans. Med. Imaging 2020

[3]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M

Q1/Limitations: About Radon vs. FFT/wavelets

We thank the reviewers for their valuable comments. In the manuscript, we note that the Radon transform projects input data into the sinogram domain, achieving dimensionality reduction while maintaining alignment with the PDE solution space. Unlike the Fourier Neural Operator (FNO) and Wavelet Neural Operator (WNO), which primarily capture global frequency-based features but may lose localized information, the Radon transform integrates data along lines at various angles, naturally encoding both global patterns (via projections across the entire domain) and local details (via angle-specific convolution operation).

Intuitively, the Radon transform can be thought of as “slicing” the PDE solution space from multiple perspectives, akin to taking "X-ray" images of an object from different angles. Each projection captures a global view along a specific direction, while the collection of projections across angles preserves fine-grained local variations, such as sharp gradients or discontinuities in PDEs (e.g., shock waves in hyperbolic PDEs mentioned in Section 2). The sinogram-domain convolution (Figure 5) further processes these projections to balance global and local feature extraction, unlike FFT, which emphasizes global frequency modes, or wavelets, which focus on localized frequency components but struggle with global coherence. In the revised manuscript, we will add a dedicated subsection with this intuitive analogy and visual aids to clarify how RNO’s Radon-based approach outperforms FNO and WNO in capturing both feature types, enhancing the manuscript’s accessibility.

To directly address the reviewer’s suggestion, we have added the following concise comparison table in the revised manuscript. It summarizes the key advantages and disadvantages of Radon, FFT, and Wavelet transforms specifically for neural operator learning.

Key: ✅ = advantage, ⚠️ = moderate, ❌ = weak or missing.

We hope this table provides the requested intuitive side-by-side view and clarifies why Radon offers a distinct niche between purely global FFT and purely local wavelets in the context of neural operators.

W2: About running time

We sincerely appreciate your valuable feedback and constructive suggestions. Regarding the discussion on computational efficiency, we have included preliminary time comparison results in Section 4.3 of the main text and Appendix G. We highly value your comments and will further supplement and refine the relevant analysis in the final version. For your reference, we would like to briefly share some additional comparative data.

Darcy 85×85 grid, batch size 2, single RTX-4090

We would like to highlight that our model demonstrates superior performance over WNO across multiple metrics, while achieving state-of-the-art results in terms of accuracy. It is worth noting that although our approach may appear computationally more intensive than FNO at first glance, the ablation studies presented in W4 reveal a noteworthy observation: when FNO's PyTorch GPU acceleration is disabled, its computational efficiency decreases significantly. This finding suggests the potential of RNO in advancing computational efficiency in this research direction.

W3: About sinogram domain weights

We thank the reviewer for their helpful feedback on our presentation. In the manuscript, the weight in the sinogram space refers to the output of the reweighting technique applied to the sinogram-domain data after the forward Radon transform. We would like to clarify that this represents an intentionally designed mechanism in our approach. The core concept follows naturally from the Radon transform's ability to extract multi-angle information. By incorporating our weighting network, we enable more refined analysis of PDE-relevant features. Specifically, the Radon transform projects the input PDE data $u(x)$ into the sinogram domain as $\mathcal{R}u(s, \theta)$, where $s$ is the distance from the origin and $\theta$ is the projection angle (Definition 1). The reweighting technique assigns a weight $w(s, \theta)$ to each sinogram entry to account for the unequal contributions of different angles to the feature representation, as noted in Section 3. Mathematically, for a sinogram $\mathcal{R}u(s, \theta)$, the weighted sinogram is computed as: $\tilde{\mathcal{R}}u(s, \theta) = w(s, \theta) \cdot \mathcal{R}u(s, \theta),$where $w(s, \theta)$ is learned via a neural network to emphasize angles with greater relevance to the PDE solution. This weighted sinogram is then processed by the sinogram-domain convolution (Figure 5) to extract holistic features. In the revised manuscript, we will clarify this in the caption of Figure 4 and include the above equation in a new subsection to provide a precise mathematical description, ensuring clarity on the role of weights in the sinogram space. Thank you for this suggestion, which will improve the manuscript’s technical precision.

W4: Discussion on algorithm efficiency

We note that FNO's computational efficiency benefits significantly from PyTorch's GPU-optimized implementations. In contrast, Radon Neural Operator, as we firstly proposed in this field, has not be explored in any of these regards. To address the reviewer’s concern, we provide detailed evaluations between FNO and our proposed RNO. Notably, we conducted a specialized ablation study where we replaced the gpu optimized pytorch algorithm with conventional Fourier transform computation. This controlled experiment serves to better demonstrate our method's capabilities under different computational paradigms.

Darcy 43×43 grid, batch size 2, single RTX-4090

Tips: While Darcy flow simulations typically employ an 85×85 grid configuration, we conducted our comparative analysis using a reduced 43×43 grid size due to the substantial memory requirements of Non-GPU optimized pytorch algorithm.

W5: About limitations

We sincerely appreciate the reviewer’s valuable feedback regarding the importance of discussing limitations to improve scientific transparency and credibility. We fully agree that explicitly addressing the limitations of the RNO in the main paper would enhance its clarity. While we briefly touch upon certain challenges—such as the performance drop on the Pipe benchmark (Section 4.2)—we acknowledge that a dedicated section would offer a more comprehensive perspective.

Specifically, our current work has two key limitations: (1) Compared to the highly optimized Fourier transform with seamless PyTorch integration, the Radon transform still requires further improvements in parallel computing and GPU utilization to maximize efficiency; and (2) while our model excels in 2D and time-dependent 3D problems, its extension to general 3D PDEs (particularly those involving point cloud data) remains an open challenge. To address the latter, we propose incorporating the GNO module [1], which offers a theoretically grounded and empirically viable solution for such cases. In response to this suggestion, we plan to incorporate a "Limitations" section in the revised version of our paper.

W6: typo: $\phi$ and $\varphi$ in eq. 1.

Thank you very much for your correction, we have fixed this.

Q1: What is the "detector " in line 247?

Since Radon transform is often used in CT tasks, we continue to use related expressions[2]. In the mathematical framework of the Radon transform, a detector can be abstractly understood as a point or line that records projection data. The Radon transform maps a two-dimensional function into a set of one-dimensional projections along specific directions. Each "detector" corresponds to a projection value at a particular angle and position.

Mathematically, the Radon transform $R(f)(\theta, s)$ represents the line integral of a function $f(x, y)$ along a line defined by angle $\theta$ and distance $s$ : $R(f)(\theta, s) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \delta(x \cos \theta + y \sin \theta - s) dxdy$Here, the "detector" can be thought of as a virtual point that receives these integral values, representing the projection data for a specific direction.

[1]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023

[2]The Radon transform-theory and implementation, 1996