Solving Partial Differential Equations via Radon Neural Operator
Authors: Wenbin Lu
Yihan Chen
Junnan Xu
Wei Li
Junwei Zhu
Jianwei Zheng
Venue: NeurIPS 2025 poster ()
Keywords: Neural Operator
Radon Transform
Primary area: deep_learning
TL;DR: A discretization-invariant, angle-aware Radon Neural Operator for PDE solving that achieves state-of-the-art performance.

Abstract: Neural operator is considered a popular data-driven alternative to traditional partial differential equation (PDE) solvers. However, most current solutions, whether fulfilling computations in frequency, Laplacian, and wavelet domains, all deviate far from the intrinsic PDE space. While with meticulous network architecture elaborated, the deviation often leads to biased accuracy. To address the issue, we open a new avenue that pioneers leveraging Radon transform to decompose the input space, finalizing a novel Radon neural operator (RNO) to solve PDEs in infinite-dimensional function space. Distinct from previous solutions, we project the input data into the sinogram domain, shrinking the multi-dimensional transformations to a reduced-dimensional counterpart and fitting compactly with the PDE space. Theoretically, we prove that RNO obeys a property of bilipschitz strongly monotonicity under diffeomorphism, providing deeper insights to guarantee the desired accuracy than typical discrete invariance or continuous-discrete equivalence. Within the sinogram domain, we further evidence that different angles contribute unequally to the overall space, thus engineering a reweighting technique to enable more effective PDE solutions. On that basis, a sinogram-domain convolutional layer is crafted, which operates on a fixed $\theta$-grid that is decoupled from the PDE space, further enjoying a natural guarantee of discrete invariance. Extensive experiments demonstrate that RNO sets new state-of-the-art (SOTA) scores across massive standard benchmarks, with superior generalization performance enjoyed. Code is available at <https://github.com/wenbin-lu/Radon-Neural-Operator>.

Solving Partial Differential Equations via Radon Neural Operator
Wenbin Lu
Yihan Chen
Junnan Xu
Wei Li
Junwei Zhu
Jianwei Zheng
Zhejiang University of Technology. Hangzhou, Zhejiang
Abstract
Neural operator is considered popular data-driven alternative to traditional partial differential equation (PDE) solvers. However, most current solutions, whether fulfilling computations in frequency, Laplacian, and wavelet domains, all deviate far from the intrinsic PDE space. While with meticulous network architecture elaborated, the deviation often leads to biased accuracy. To address the issue, we open a new avenue that pioneers leveraging Radon transform to decompose the input space, finalizing novel Radon neural operator (RNO) solve PDEs in infiniteddimensionall function space. Distinct from previous solutions, we project the input data into the sinogram domain, shrinking the calti-dimeesional transfor- mations to a reduced-die dimeeesioiianii counterpart and fitting compactly with the PDE space. Theoretically, we prove that RNO obeys a property of bilipschitz strongly monotonicity under diffcomorphism. providing deeper insights to guarantee the de- sired accuracy than typical discrete invariance or continuous-disce -iscrreee equivalence. Within the sinogram domain, we further evidence that different angles contribute unequally the overall space, thus engineering reweighting technique enable more effective PDE solutions. On that basis, a sinogram-domain convolutional layer crafted, which operates on a fixed O-grid that decoupled from the PDE space, further enjoying a natural guarantee of discrete invariance. Extensive ex- periments demonstrate that RNO sets new state-of-the-aa (SOTA) scores across massive standard benchmarks, with superior generalization performance enjoyed. Code is available at hhttps://gttuub.comm wwttm//wnbbn-llue Www/Radon-Neuus Newura--uueraaee Operator
1 Introduction
PDE solving is considered an important field of modern mathematics. Although traditional finite element methods (FEM) [5] and finite difference methods (FDM) (35] have been shown to yield no- table outcomes, they require substantial computational resources and careful discretization, a process necessitating specialized expertise. In addition, certain transformations have been mathematically proven to exhibit superior performance in the realm of PDE solving, including Fourier, Wavelet, and Laplace [16] 7 46). Such transformations, though useful to provide some specific merits in other domains, often fail to preserve the intrinsic geometry of the original PDE space.
Recently, the paradigm has shifted to a data-driven pattern. Owing to the inherent capability in discerning behaviors of diverse PDEs directly from data, newly elaborated solvers have achieved remarkable improvements in speed, robustness, and accuracy. These methods, categorized under operator learning framework partollllllllllllyy neural operator -Staabsta explicit mappings from input conditions, e.g., initial/boundary configurations, to PDE solutions in spaces. The most pioneering work was presented in [8. which provides mathematically rigorous alternative to traditional numerical methods by preserving operator continuity in Banach spaces.
function
"ooresponning author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Current NOs predominantly fall into two branches. The first encompasses transformation- based approaches derived from numerical analysis techniques, including Fourier neural operators (FNO) [23] Laplace neural operators (LNO) 3 and Wavelet neural operators (WNO) 37 While effective, these methods inherently inherit the limitations of their underlying numerical schemes they often fail to capture the inherent PDE solution space accurately, enlarging the original dimensionality and presenting scalability challenges for high-dimensional problems. The second endeavors to elaborate sophisticated neural architectures, such as employing transformer [4)] or Mamba [9], yet with the intrinsic property of PDEs disregarded. To overcome the limitations, we propose the Radon neural operator (RNO). In contrast to previous domains, project the input data into the sinogram domain, achieving dimensionality reduction while maintaining strong alignment with the original PDE solution space. Building upon this, rigorously prove that RNO satisfies the bilipschitz strong-monolee property from a diffeomorphism perspective, theoretically providing a deeper equivalence guarantee than traditional discrete invariance 18 or cantiiuuuu-discrete equivalence (2
During NO evolution, the pursuit of holistic features is known as the most critical point. FNO and LNO enjoy strongly the global property due to the employment of convolution op- eration in spectral domain, coupled with the suppression of high-frequenge components, yet suffer from the loss of local- ized information. This poses significant challenges for prob- lems rich in local features, such as shock waves in hyperbolic PDEs or steep gradients in elliptic PDEs. Conversely, convolu- tional neural operator (CNO) 34 focuses on local features but struggles with global feature extraction. For more balanced feature extraction, several attempts have been performed to combine FNO or LNO with differential and integral operators 225 albeit at the cost of compromised generalization capabil- ity. Generally, Radon transform (RT) holds similar properties Fourier transform, yet with less popularity enjoyed. This fact has long plagued the naive use of RT in the context of NO learning.
FNO+integral
INO+differential
FNO
WNO
LNO
RNO
CNO
Transolver
Global
Local
Figure 1:
Comparison of feature holism from different architectures.
In this study, we have dug out that different angles within the sinogram domain contribute unequally to the final feature representation, which motivates us to innovate an reweihhiii technique striving for effective PDE solution. On that basis, we enrich our elaboration by incorporating sinogram- domain convolution within the weighting network to better capture the holistic features. The sinogram convolution operates on fixed 0-grid that is decoupled from the spatial resolution of PDE manifold, guaranteeing discrete invariance as confirmed by our generalization experiments. Furthermore, a physics-attention mechanism is integrated, leveraging the global capability of transformer, into RNO, ultimately leading to the complete architecture. Fig. T provides a visualized comparison of existing architectures on feature learning. Overall, our contributions are summarized as follows.
We pioneer the introduction of Radon transform into neural operators and, for the first time, perform weight analysis in the sinogram domain solve partial differential equations.
Following the Radon forward transformation, we elaborate a convolution operation in the weight network, which assists in the holistic learning of both global and local features.
We rigorously prove that RNO is a bilipschitz operator, which ensures discretization in- variance under diffeomorphism and avoids the introduction of any topological obstructions. Empirically, new state-of -fte-oof-the-ar scores are earned by RNO across most PDE benchmarks.
(cosd,
2 Preliminaries
a
Mathematically, RT is proposed by Johann Radon 32). as an integral transform that maps a function f. defined on Cartesian coordinates, to a function Rf, defined on the plane-wise two- dimensional space of lines, where the value determined by the line integral of the function along that direction, as generally illustrated in Fig. 2 the figure, 8 denotes the perpendicular distance from the line to the origin of the image coordinate system, z represents the projection distance, and a indicates the orientation. The normal vector 7 characterizes the line direction.
[(x,))
Figure 2: Radon transform maps f from (a, y) to Rf in (a, s).

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In the sequel. the Radon transform in its high- -ige-niieniional form is initially formulated, from which the essential dumension-reduction property becomes manifest 77.. For simplicity, we denote S for the unit sphere à (0, in RR, typical point of which is represented as = (W1,".. .n). Then, the plane with unit normal w 5 a distance 8 from the origin can be written as follows:
(Ts,w) == (y R w == =)
Definition 1. The Radon transform Ru- = ū of a function u C. (R") is given as
([s,w)= = J Ju(eww) u dS, where w E S5-1, y = 8 s TR.
The right term is the integral over plane II(swW) w) with regard (n 1) dddmmesioonall surface measure.
We mainly use the two-dimensional variant of the Radon transform, whose mathematical form is as:
Rf((, // f(x, v) 6( cos y + y ysin ip s) dz dy,
(1)
where fz, y) is the imput function, ip the angle of the normal vector the line (typically yp e ())) is the signed distance from the origin to the line, and 6(-) is the Dirac delta function restricting integration the line x cos yp + y sin o = s.
To achieve the concerned inverse transform of RT, we first reveal that a close relationship exists between the Radon transform and the Fourier transform.
Theorem 1. (The connection between Radon and Fourier transforms) Assume u E C (IR"), then
u(r,w)= = | ū(s,w))-imsd = (2m) R. SM-1).
where à = Fu the Fourier transform. See Appendix Appendix detailed proof.
Due to the close connection between the Radon and Fourier transforms, in mathematics it is posited by the projection-slice theorem (also known as the central slice theorem or the Fourier slice theorem in two dimensions) that the results of the following two calculations are deemed equivalent:
A two-dimensional function f(r) is taken, projected onto one-dimensional line (c.g.. through the Radon transform) and subjected to a Fourier transform of the resulting projection.
That same function taken, subjected first to a two-dimensional Fourier transform, and subsequently sliced through its origin along a plane parallel to the projection line.
Since inverting Fourier transform is already known, from Theorem I we likewise obtain RT inversion. Theorem 2. (RT Inversion) The inversion of RT is given as (See Appendix A.2. for proof.)
[
u(x)= 2(2x)" S.-1 ū(r,
giwwx
However, yet with analytical solution existing theoretically for the inverse Radon transform, the practical implementation faces significant challenges. Direct practices often result in reconstructed PDE manifold with pronounced blurring and amplified noise due the inherent instability of this inverse problem. By applying a frequency-dome
filter to the projection data prior to back projection, the filtered back projection (FBP) algorithm stands out natural solution to address the limitations 30). With the theoretical foundation derived from the projection-slice theorem, FBP functionally stabilizes the computation process, effectively suppresses artifacts, and demonstrates particular robustness when handling discrete datasets and noisy acquisition scenarios.
The mathematical form of the FBP algorithm can be expressed as:
f(x) {.¹ (Rf(: h)) ((x, no)) d0,
(2)
in which the convolution kernel h, e.g., h(kk==))))))) = [k], is referred to as Ramp filter in some literature. In this article, the practical number of projection angles is established as a hyperparameter to accommodate various datasets, and it is demonstrated through experiments that, at ultra-low resolution, fewer angles are preferable to a greater number. We provide more details of FBP and the numbers of angle in Appendix A.2.2
3

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Ply Sllce-Siiieeeeeeeeeeeee
Linerr
DeSlice
Linear
Attention
Physic-Attention Block
Raddn Transform
speilicangle weight
Inverse Raden Transform
Radon Block
Figure 3: The full architecture of RNO. Given the input data, non-local features are derived through physics-attention, followed by the operations of weight analysis and sinogram domain convolution with the aid of forward/inverse Radon transform, aiming at enriching the more holistic features.
3
Methodology
3.1 Problem Setting
Our methodology strives for mapping bridging two infinitediimensional spaces from finite collection of given input-output pairs. Let D C IR be bounded and open set, and let X (D; Rdx) be individual Hilbert spaces of square-interrable functions with elements in Rd and Rdv, respectively. Moreover, let Gt X y be a (typically) non-linear mapping. We probe into maps G that arise as the solutions of parametric PDEs. Assume that we are given observations {agj. my)}}=1 where aj is an independent and identically distributed (i.i.d)) sequence from the probability measurement supported on X, and u, = Gf (az) is probably corrupted with noise. The goal is to elaborate an approximation gt by forming a parametric map
(2, as well as
equivalently, Go X + OE
for certain lnite-dimensionall parameter space O, with the optimal choice Øt θ so that G 0i) Ge, R3 Gt. This is natural idea for learning in infinite dimensions as we can intuitively define a loss functional C y x x and pursue a minimizer of the problem
min
Eawy [C (G(a, 0), G' 8EB
which directly parallels the typical miite-dimensional configuration 39.. However, evidencing the existence of minimizers, especially in the problem. We attempt to approach this issue in the test-train setting by using empirical and data- driven approximation to the loss, determining the final 0 and testifying the concerned accuracy. Recall that conceptualize methodology in the infiniteddimensional environment, hence all finite- dimensional approximations enjoy shared parameter set that is consistent in infinite dimensions.
environment, remains a challenging
3.2 Radon Neural Operator
Generally, a typical NO architecture is constructed from three primary components: lifting, iterative kernel integration, and projection 18, which is written as:
Neural Operator Architecture.
Go : Q oz (Lr K2 by-1) o (Lo + KG + o
(3)
where P Rdx + do Q dRd + dd are the lifting and projection mappings, respectively. L2 E Rd xd., denote linear operators (matrices), Kr {2r D; + R + {r++1 D + Rd } represent kernel operators, and b, D dd are bias functions. Acting as maps

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Weight
Radom Tranform
Analysis
Angle Weight
Prediction
Sinogram
Inversion
Inverse Radon Tranffomm
SINOGRAM DOMAIN
Ground Truth
Chanclsl
Figure 4: The flowehart of a Radon block. The input data first enters the sinogram domain through the Radon forward transform, where weight analysis is performed in the sinogram space to assign specific weights to each angle. Afterwards, the signal is projected back to the spatial domain through the FBP algorithm. Mcanwhile, a skip connection is imposed to introduce more original information.
Rd + Rd in each layer, oe is a fixed activation function. Note that the output dimensions dip. dey. the input dimensions d1, ... dr, ,.. and the domains of definition ",, Dr-1 hyperparameters of the architecture.
Kernel Integral Operator IC. Following 18. the kernel integral operator for Eq. is defined by
J (z, y, a(x), a(y): vo(y) dy, Yz D,
(K(a; o)(z) (z)
(4)
in which is usually parameterized by neural network such that Kg ((( + Rdd Ox. Functionally, vi plays the role of kernel integral which we learn from data.
xd. with
Radon Neural Operator. We propose to innovate the kernel integral operator in Eq. with the Radon transform discussed in Section [] Recall that Eq. ( presents the concerned forward transform, by which the data are projected into the sinogram domain. Eq. (2) undergoes the inverse transform, by which the data are returned from the sinogram domain to the original domain. Unlike FNO, which draws inspiration from Green's 's function, the Radon neural operator is motivated by the impulse response. Concretely, it defined that x (I. y, (((x),0u a(v); (p) = 8(x cos yp + ysin yp - -)). Note y is the angle of the normal vector to the line (typically &o E [0, r)). is the signed distance from origin to the line, and 8(.) is the Dirac delta function restricting integration to the line x cos yp + y y 8.
On that basis, Radon neural operator can be formally defined as follows.
(K(())))(z)) R RPp (R)) Vz E D,
(5)
where Pa is learnable parameter used to angle-wise impose different weights within the sinogram domain. Mathematically. data formation derived from various perspectives has been extensively investigated. Specific to the property of the Radon transform, an angle-based basis is considered, which differs from the global spectral bases used in FNO or LNO and the local wavelet used in WNO, providing an opportunity to learn the holistic features within a single transform. We will elaborate on the implementation of the detailed technique in Subsection 3.4 Note most currently well-know NOs can be consistently expressed by the following formula [I] which advocates the joint learning of global-local features, property that our RNO naturally fits.
(vv) o(K(a)()))
=o( / (o, du I (y, dul + Da (v) + Wa(y)+ b(v)).
(6)
y
On that basis, we consider that the non-local features also favor the final solution. Therefore, drawing inspiration from Ref. [43,, we further engineer the physics-attention mechanism that decomposes the discretized domain into series of learnable slices, within which attention is computed on physics- aware tokens. The overall network architecture with general explanation is given in Fig. 3 Physical attention is specifically embodied in the kernel function in Eq. (6) given as follows.
M
M
exp ( WW 2//VC)
w(x, 8K)
RGlobal (y,, x7)
w(y 85)
Wv
(7)
EnT exp (xz)) WKZP/V/ /VC)
j=1 1=
w(E, 8k) dg
5

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Sinogram
0.5
Extract features
Generate weights
50
044
75
0.3
1x16 Conv
16x1 Conv
100
50
100
350
200
250
G
GELU Function
S
Sigmoid Function
Sinogram Weights
Figure 5: The structure sinogram-domain convolution. With convolution operation performed in the sinogram domain, the local information from different angles can be further learned based on the global features of RT. The data sequentially passes through a chain of convolutoon-activati convolution to achieve the weight distributions.
where Q is the computational domain, MM the number of learnable slices, 8, denotes the j-th slice, w(-)s;) represents the soft assignment weight from any point to slice 8y, 24 Sa s()(()))((E) dÊ/ Sg w(E, de is the physics-aware token of slice sj. Wo. WK.
: Wv E RCx the query, key and value projections used in the token-wise attention, and C is the channel dimension.
Parameterizations of Pa- With the Radon transform performed, the input data originally within the spatial domain are now converted to the sinogram domain, in which we further find that the contri- butions of different projection angles the overall spatial representation are inherently onn-unformm Building upon this observation, we conceptualize the angle as a form of basis and introduce learnable parameters dynamically adjust angular weights, thereby achieving more effective solutions.
Quasi-linear Complexity. Given discrete counterpart
# of two-dimensional continuous function, it is firstly restructured, yielding u E RNND where = H x W. The forward Radon transform is then applied to project image along A angles, which necessitates ((AN) computations owing to the rotation and summation the image for each angle. Afterwards, the inverse Radon transform, specifically the FBP algorithm, performs filter operation within the sinogram domain, converting the angular information into the frequency domain. This routine necessitates O(A x max H, W') log max (H, W)), followed by back projection that accounts for a complexity of O(AN). Since that is usually a small value, i.e., A < N, we consider the overall computational cost quasi-linear complexity. The complexity of other well-known models detailed in Appendix E In practice, the runtime complexity the number of angles specified. Intuitively, the larger the angle parameter, the more effective the outcome deemed. However, we have noticed that the larger number often results in a greater risk of overfitting. Practically, ultra-low resolution, more pronounced the error occurs when we large A value. Further details are elaborated in Appendix G.4
RNO is strongly dependent on
3.3 Discretization Invariance under Diffcomorphism
In Ref. [8) no-go theorem is proposed explain the fundamental obstacle separating infinite and finite dimensions, along which the concept of diffeomorphisms and the property discretization invariance are elaborated within the framework of category theory. Moreover, it proves that bilipschitz NO layer can be expressed as a combination strongly monotone neural operator layers with simple isometry. In addition, the strongly monotone NO can be continuously approximated by strongly monotone diffeomorphisms a -inee-dimeniional space.
Following this theorem, in this study we prove that Radon operator actually behaves bilipschitz operator. Since RNO is primarily performed in 2D space, hence in the sequel, the informal proof for the two-dimensional case is provided. More details are referred to Appendix C.I

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By leveraging the linear property of Radon transform, the definition of bilipschitz operator in the context of RT is formally given as follows, in which e and C are some positive constants.
ellfll>> & |Rf I ll
(8)
Problem 8 can be further broken into two parts: the right upper bound and the left lower bound.
Upper Bound: Within (). the norm of Rf is computed with the expression ||
Sst SR [R[(s,)) 0)1² ds d0. Following the Fourier slice theorem, ID Fourier transform of Rf(:,0) given as F((R((-(O)(() = f(o0). On that basis, the Plancherel theorem further leads us to S& (ff(s,))3² = fx If( do, hence the norm of Rf turns into || =
Ss: Sa [f())) do d8, By further switching to polar coordinates, key expression is deduced:
|Rflli= = f |f(()) dg.
The weight ||E||= is manageable since S ||511 de 2n oo, which manifests that we can bound the integral as Ja (((()) de S C₁ [|2 leading to the upper bound with C = 22CT:
|R S 2212
Lower Bound: For the left part of (8) note first that R is injective. That is, if Rs = 0, then f(r0) = O. hence leading to f = 0. Besides, the inverse R-1, roughly given by filtered back
projection f(x) Jg. (HO, Rf)(2 - 0,0) )) dR, bounded in T.2, satisfying RR- gglzz S K flgl ge.
By applying this to Rf, we get [| = || (R))l S KK| which refers to lower bound:
|R L. 2 K [fflle,
with c c= k- Note that for future research, we also provide an n-dimensional proof in Appendix C.1
3.4 Radon Block
As mentioned, our methodology employs a fundamentally different design paradigm within the sinogram domain. The core component, i.e., Radon block, illustrated in Fig. 4 According to Eq. ( the data are first transformed from the original PDE space into the sinogram domain through the Radon forward transform, which achieves dimensionality reduction while preserving the essential information in the original space. From Fig. 4 we observe that while different angular lines are equally distributed, their valid coverages on the PDE manifold are severely distinct from each other. Moreover, different regions of the entire PDE manifold also contribute distinctly to the final representation. Therefore, it is naturally considered that the angular lines benefit from adaptive weight coefficients to favor a better reconstruction.
To put the natural consideration into practice, further elaborate sinogram-domain convolution to perform the weight computation. The practical implementation is presented in Fig. 5 within whose left panel the horizontal and vertical axes respectively represent the spanned angles and the concerned density values. As seen, the actual convolution is performed on an inter-angle scope, adopting multi-line information finalize the weight learning. While multiple lines are jointly considered, the covering span is limited by the convolution kernel. hence with the local features learned. This fact, together with the globality essence of RT and the non-local property of the physic-attention block, ensures more holistic feature learning. Unlike conventional computations that depend closely on the spatial grids, our convolution performed on a f-space that is decoupled from the spatial resolution. Although the sampling interval s varies with resolution, the sinogram convolution maintains strict consistency in the O-direction. This key characteristic enables the model to share convolution kernels across different resolutions, thus further ensuring discrete invariance. Our generalization experiments provide empirical validation of this property.
-inogram-domain We proceed to define the mathematical formulation of sinogram domain convolution. For a sinogram S(0 t), where represents the projection angle and & the position along the detector, the sinogram domain convolution is defined as:
Summ (0, (S K)(0,t) 1) = J S(8', K 8') da'
7

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Table Performance comparison on standard benchmarks. Relative L2 recorded. smaller value indicates better performance. (Bold: Best performance, Underlined: Second best performance, A Performance increase, V: Performance decrease, ' means that the model does not perform well on that dataset or that the model is not suitable for that benchmark.)
MODEL
MECHANISM
REGULAR GRID
STRUCTURED MESH
Darcy
Navier Stokes
Allen-Cahm
Airfoil
Plasticity
Pipe
DEEPONET
5.88 10
2.97 10
/
3.85
10 2
1.35 10
9.70 x
FNO
Fourier Transform
1,08 10
1.56 10
7.52 10-4
/
WMT
Wavelet Transform
8.20 10
1.54 x 10
1.12
x 10 2
7.50
10 a
7.80 x
7.70 x 10
GALERKIN
Galericin Attention
8.40 x 10
1.40 x 10
1.18
10
1.20
10
9.50 x 10-a
GNOT
Transformer
1.05 x 10-2
1.38 x 10-1
7.60
10-3
3.36 x 10-2
4.70 x 10-3
U-NO ONO TRANSOLVER
1.13 10
1.71
10
4.31 10-2
7,80
10-3
3
10-3
1.00
10-2
Orthogonal Attention
7.60 10
1.20
10
1.71
10
6.10
10-3
4.80
x 10-2
5.20
x 10-3
Physic- Allcction
5.70 10-3
9.00
10
5.32
10-a
5.77
10-3
1.20
10-3
3.30
10-3
RNO (Ours)
Radon Transform
5.10 x 10
8.94
x 10
2
4.61
x
10
AA
4,90
10
3A
1,15
10
120
x 10
Input (Prrus Medium)
Preditioon
Prediction
Transolver
Figure 6: Case study RNO and Transolver. The prediction results and errors provided.
Here, K0) is the convolution kernel, which slides along the 0-axis combine neighboring angular values. We provide more details on sinogram-domain convolution in Appendix D
For the inverse Radon transform, the sinogram domain data projected back to the spatial domain
in accordance with the filtered back projection algorithm, as outlined Section 2 To ensure better preservation of the original information, skip connections
introduced [[2]. As discussed in the analysis of discrete invariance g skip connections play a critical role in preserving the bijectivity and strong monotonicity of neural operators, further guaranteeing that discretization challenges in infiniteddimensional spaces are avoided.
4 Experiments and Analysis
Training Details and Baselines. For fairness, all experiments are consistently conducted on a stan- dardized platform with an NVIDIA GTX 4090 GPU and 2. 10GGHZz Intel(R) Xeon(R) Platinum 8352V CPU. Several well-known PDE solvers are used as the competing baselines, such as Deeponet 27
2N WMT[0, Galerkin 4 GNOT[IT) ONO(44), (-NOO33) and Transolver 43
Standard Benchmarks. To better compare with existing work, performed experiments several publicly available benchmarks, including Plasticity, Airfoil, Pipe with structured mesh and Navier-Stokes, Darcy, Allen-Cahn with regular grid. These benchmark datasets were extensively investigated in seminal works such as FNO 23 geometry-aware FNO (geo-FNOO (22), and WNO 37], and have since gained widespread adoption in the scientific machine learning community. We provide more detailed description of these datasets in Appendix
Implementation Details. All competing methods are trained with l loss and 500 epochs. The ADAM [17] optimizer with initial learning rate of 10-3 is used. For Radon transform, the main hyperparameters lie in the number of Radon blocks and the employed quantity of angles. Note the latter depends on the size of the input data to ensure that sufficient angular information is obtained. We provide more implementation details and hyperparameter configurations Appendix G.T
4.1 Main Results
In Table I. we present comprehensive comparison against existing approaches, in both cases of regular grid and structured mesh. As can be seen, for the regular scene, RNO consistently achieves Mate-oofthhe-ae (SOTA) performance across most PDE benchmarks. Specific to equations Darcy flow and Allen-Cahn the improvements over the second-best competitors reach 10.5% and 13.2%,

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respectively. For irregular meshes, the conventional Radon transform is also applicable through simple zero-padding operation. On the Airfoil benchmark, RNO achieves a 14% performance gain compared to the second-hest approach. However, the padding operation also incurs some redundancies; hence, our proposal lags behind Transolver slightly on the Pipe benchmark.
4.2 Generalization.
Recall that NOs focus on learning mappings between infiniteddimensionall function spaces, pursuing inherent discretization invariance. Generalization capability serves as crucial experimental manifes- tation of this fundamental property. To assess this property of the proposed model, we firstly trained RNO in low-resolution setting, then performed zero-shot inference across higher-resolution settings. Specifically, the Darcy flow equation, originally within resolution of 421 x 421, was downsampled to sizes of 241 x 241, 211 211, 141 141, 85 85, 61 x 61, and 43. The training was conducted on the 43 43 resolution and thereafter assessed across the others.
The concerned results comparing RNO with FNO are provided in Table 2 As seen, RNO achieves satisfactory generalization in all scales, enjoying performance im- provements of roughly 70% over that of the FNO. Fig. and Fig. 8 further visually compares RNO with Transolver. While these two share similar effect at lower resolutions, the performance RNO, at very large scales, ex- ceeds far from that of Transolver. Intuitively, it is demonstrated that RNO possesses strong generaliza- tion capabilities, which hold sig- nificant practical value for super-resolution tasks, image enhancement, data-scarce scenarios, and transfer learning, underscoring its considerable research potential. More detailed explanations offered in Appendix G.3
11.34%
0000k
Relative Eiror
0.07
0.06
-0.0022
Transolver
RNO
0.03
60
100
130
200
200
800
392
400
450
Resolution
Figure 7: Generalization comparison of RNO and Transolver.
Table 2:
Comparative display of generalization performance. Relative 1.2 is recorded. smaller value indicates better performance. (Bold: Best performance)
Model
x
x
141 x 141
211 x 211
421 x 421
FNO
1.16 10-1
1.80
10-1
2.68
10-1
3.16
10-1
3.63 10
RNO (Ours)
3.21 x 10-2
5.04
10-2
6.69
10-2
7.09 x 10-2
7.62 10-2
Relative Improvement
72.42%
71.95%
75.03%
77.56%
79.01%
4.3 Ablation Study
We present ablation studies validate the efficacy of the Radon block. Using the Darcy flow equation as an example, the results of different combinations are given in Table Initially, we replaced the Radon block with alternative transformation methods, i.e., Fourier transform. Experimental results demonstrate that relying solely on global information yields suboptimal performance. We further conducted bidirectional validation by substituting physics-attention mechanism (P.A.) with FNO. The results enjoy a 4 41.67% performance improvement over the baseline FNO, which not only underscores the criticality of holistic information but also confirms that Radon block effectively captures the local features. Moreover, to demonstrate the significance of sinogram convolution, we have replaced it with a simple nonlinear weighting scheme. As evidenced the 4th row of Table the replacement suffers performance reduction of nearly 7%. To further demonstrate the substantial potential of Radon transform, we perform another ablation study by replacing the GPU-accelerated Py PyTch Fourier transform in FNO with the non-accelerated counterpart. The results are given in Table which together with Table shows that the replacement leads significant reduction in efficiency of FNO, rendering it considerably inferior to RNO. This finding underscores RNO

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Input (Porous Medium)
Ground Truth
Error
Error of Transolver
400
421x421
3M
-38
400
4441
211x211
Figure 8: Comparison of generalization between RNO and Transolver. We demonstrate generalization performance two larger resolutions, with RNO significantly outperforming Transolver. The figure shows the error map of RNO, the error map of Transolver, input information and ground truth.
as a promising avenue for future exploration in GPU acceleration. Due to space limitations, the experiments different selections of angle numbers provided in Appendix G.5
Table Ablation studies of Radon block and sinogram domain convolution. All experiments were based on the 85 x 85 Darcy flow dataset and tested when the number of angles was set to 32.
Memory (GB)
Time (s/epoch)
Param (B)
Relative L2 Darcy
Ablations
P.A.+NNO
2.83
40.28
4.02 x 108
8.84 x 10-3
FNO
8.48
2.38
10
1.08
10 2
FNO+Radon Block
2.92
25.58
2.38 x 106
6.30 x 10-3
Only nonlincar weights
3.18
71.98
2.84
10
5.79
10
Ours
37.88
2.83
10
5.34 10
Table 4: Comparison of Training and Inference Time between RNO and FNO without GPU Accelera- tion. (Bold: our method)
Stage
RNO (ours)
FNO (w/o GPU opt.)
Training time (s/epoch) Inference (s)
32.87 3.02
196.79 14.17
5 Conclusions and Future Work
For the efficient solving of PDEs, we propose RNO in this study, which employs Radon transform to reduce the spatial dimensionality of PDEs while preserving their intrinsic information. Theoretically, we prove that the proposed operator possesses a more profound bilipschitz dtrong-monotonicinn which further guarantees discrete invariance under diffeomorphism. Building upon this foundation, we perform weight analysis in the sinogram domain, within which sinogram convolution is newly elaborated and integrated. Distinct from the normal convolution operation, the new proposal is grid-independent, guaranteeing again the discrete invariance. Extensive experiments conducted on multiple benchmarks demonstrate that our RNO achieves state-of-the-art performance. Moreover, the generalization experiments show significant improvements compared to baselines. Future work will focus further investigations in the sinogram domain, enhancing the capture of more detailed features. We also anticipate that RNO will be widely utilized in massive industrial applications.
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Acknowledgments and Disclosure of Funding
This work was supported in part by the National Natural Science Foundation of China under Grant 62276232 and the Key Program of Natural Science Foundation of Zhejiang Province under Grant LZ24F030012.
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Appendix Supplementary Material Solving Partial Differential Equations via Radon Neural Operator
A Related Work
A.1 Neural Operator
Neural operators [8] are regarded as promising way to solve PDEs. Inspired by Green's function solving PDEs, neural operators were pioneered through the incorporation of kernel integral operators, establishing mappings between infinite-dimensional function spaces. The most basic and well-known example is FNO [23]. On that basis, several excellent developments were derived. For instance, the U-FNO 42] integrates the U-Net network with the FNO, while the F-FNO (36] utilizes factorization in the Fourier domain. The Laplace transform, as the complex form of the Fourier transform, is introduced into LNO (3. The multiwavelet-base operator [10] incorporates the principle of multiwavelets. This concept is generalized and leveraged to address arbitrary measures, thereby enabling the development of series of models for operator learning from complex data streams. In addition to these classic mathematical methods, deep learning architectures have been incorporated into neural operators. For example, the famous transformer architecture (40] has given rise to several developments, including FactFormer (21). OFormer [20] GNOT [1] ONO 44 and Transolver [43). Another example, the emerging Mamba architecture
9 has led the development
of the MambaNO [4]].
To provide intuition into the distinctions and connections between RNO and related works, Table 5 offers detailed comparison.
Table 5: Comparison of different neural operators.
Transform
Global Capture
Capture
Geometry Awareness
Discrete
Computational
FFT (PNO)
Excellent (glohal frequencies)
Poor locality)
periodicity
(spectral conv)
(FPT linear)
Wavelet (WNO)
Limited (coarse scales)
Excellent (multi-scale)
Limited (fixed basis)
Yes fmulli-levell
Moderate (älter bank)
Radon (RNO)
Strong (line integrals)
Tunable (8-conv)
Strong (line geometry) Yes (0 (0-grid decoupled) Muderate (O(AN)
Key: v = ad advattage moderate, x weak or missing.
A.2
Supplement to Radon Transform
A.2.1
Proof Supplement
(Radon and Fourier transforms) Assume that E C (R"), then
u(r, w) == / ū(s, irs
irdds ds = (2m) e SR-
where û = Fu is the Fourier transform.
Proof. Take bi,... bn- to be an orthonormal basis of [T(0, Then
{
ū(s,w)
dy
8--1
and so
ü(s we ire
/
Σ j=1
n-1
yr by sw c iirr dyds
By changing variables and rewriting I == ΣØ=1 11
Bibb + sw, then we have
/
/
ū(8 u(s,w)e - ing
u(x))e-in(e -n(())))))))))))))))))()))) dr = (2x) (2x))//((((())
R
pm
21

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(Inverting the Radon Transform) We have
(x) 2(2xT² J L a(n wrr r
Sn-1 Proof. The process is sequentially deduced follows.
J
ür n- e irw.z dS dr = (2T)
((rw)r" 1 frwwr dS dr
gn-1
= -2(xx) m/2
u(rw))"-le (inw dS dr
Sn
2(2
((y)e dy RR
=((2))
A.2.2 Supplement to FBP Algorithm
We have generally introduced the mathematical form of the FBP algorithm in the main texts. Here we elaborate on the implementation process of the FBP algorithm, which can be broken down into two main steps:
Filtering: The projection data, or sinogram, is processed with a high-pass filter, typically a Ramp filter, in the frequency domain. This step is designed to enhance high-frequent ponents, such as edges, thereby counteracting the blurring effect caused by back projection. The filter, which is often modified with window function for practical purposes, ensures that the data are refined prior to reconstruction.
Back Projection: The filtered projection data are subsequently back bock-projected, meaning the data are redistributed into the PDE space along the trajectories from which the projections were obtained. This process entails the accumulation values within the pixels to recon- struct the original function, whereby information from all views is integrated to produce the final generation.
B Discretization Invariance under Diffeomorphism
To commence with, the concept is initially defined regarding what it means for nonlinear function F X - X an infinite-dimension Hilbert space, X, to be estimated by operators in finite- dimensional subspaces V X.
Definition 2. (Ev approximators and weak approxinators)
(i) Let r 0, F C (X; be a family of operators or functions, and ē (ev)ves,(X) be series such that Ev + 0 as V + X. We claim that function
Ax: F X (BBv(O,r):)), F (Fv)veso(x)
an E-aprrxximation operation for functions F in the ball Bx (0. r) assuming values in families Fy c (V:V) if Ax maps a function F: X - X, where F E F. to a series of functions (Fv)veso(X) where Fy E Jv, so that the following valid: For all F: X X satisfying (1 M, we get
sup (v.r)
FF((z) - Pv(F(x))llx S Mcv,
where pv: X - X is the orthogonal projection onto V. that is, Ran(Pv) V.
(ii) We further say that A: ("'X;;) + Xveso(x) C(V:V), C(V:))
F (Fv)veso(X) weak approximation operation for the function family F C C* (X; for any F F and 0 enjoys that
lim
sup
[v(2)) PF((F)))llx +.
(0.r)
22

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Definition 3. (Strongly Monotone) We clain that (nonlinear) operator F: X - X on Hilbert space X, is strongly monotone if there exists constant >> 0 such that
(F( ((1a))F(F22,-2) 21 -22
al|æ1 x211x, for all x,, 22 X.
Definition 4. (Bilipschirz) It can be deemed that F is bilipschitz if there exist constants e >> 0 and C' < 3o such that for all 21, T2 X,
c c| z2ll (F(()) F(22)ll C/|a11 =x2ll-
B.1 No-go theorem for discretization of diffeomorphisms on Hilbert spaces
Definition 5. (Category of Hilbert Space Diffeomorphisms) We let D the category of Hilbert diffeomorphisms with objects Op that are packs F) of a Hilbert space X and (possibly non-linear) C -diffeomorphism F: X X and the gathering of morphisms (or arrows that 'map' objects to others) A that are either
(induced isomorphisms) Maps ag that are ruled for a linear isomorphism 6: X1 - X2 of Hilbert spaces X1 and X2 that maps the objects (X1, P1) OD to the one (e(X1), @o P1
o b-)) E OD. or 2. (induced restrictions) Maps ax,,.., that are ruled for a Hilbert space X1, its closed subspace X2 C. X1, and an object (X1. Fi) E OD such that Fi(22) = X2. Then ax1.X, maps to the object (X1, Fi) E OD to the one (X2, Filx,) E OD
Definition (Cutegory of Approximation Sequences) We let B be the category of approximation sequences, owning objects Og that are of the form (X,S,(X), (Fv)vesu(x)), in which X is Hilbert space,
So(X) S(X) (V IV CVC. X is finite dimensional linear subspace),
are partly ordered lattices, UVES.(x) V X, and Fv: V V are C² C-diiffeomorphisss of spaces V E So(X).
The set of morphisms As comprises either
Schedules As that are defined for linear isomorphism Xx + X2 of Hilbert spaces Xı and X2, and lattices So(Xi) and So xx) = (o(V) V E So So(X))), that schedule the objects (X1,5(X1) (Fv))esxx)))) to (X2, S(x2), (Ø F Fa-1(W) o--1 )wes(x))), or
2. Schedule Ax,,x, that ruled for Hilbert space X1, its closed subspace X₂ X1, and an object (X1,So(X (F) (Fv) so that F(X,) = X, and So(X,) = {V E (X,, | V C [x2] are a partly ordered lattice. Afterwards, Ax,,x, projects the object (X2, So(x)) (F)) (Fv) (Fv))es(Xx.)) to the one (X2, (o(2),, (Fv) (Fv)))s((x.)).
In the sequel, the notion of an approximation discretization functor is given. Practically, an approximation functor an operator that projects function from an infnite-dimensional space X to another function Fy operating within Fv are closely aligned (in reasonable sense) with the function F.
subspaces V of X, that the functions
Definition 7. (Approximation Functor) We engineer the approximation functor, denoted A: D B. as the functor that maps each (X, F)) E Op to some (X, (X),, (Fv) E OB such that the Hilbert space stays as the same. The approximation functor maps all morphisms as to Ag and morphisms X2 to Xa, and enjoys the following properties
(A) For all r > O and all (XFF)EC EO
lim V-X
sup
[[vv(x) F(x)Ixx 0.
(0.)))
In separable Hilbert spaces, this indicutes that when the fiiite-dimensional subspaces V C X expand to fill the entire Hilbert space X, then the approximations Fy converge coincidentally all bounded subsets to F.
23

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Definition 8. We argne that the approximation functor A is continuous if the following state- ment holds: Let F), (XX,FD) E OD be such that the Hilbert space X the same for all the objects and let (X,)) (F, A(X, Fū1) be approximating sequences of (X,F5)) K6,1) Furthermore,
A(X, F) be approximating flows of (X, F) and
(X),, (Fgy) assume that T > 0 and
lim sup zE.x(
(x)(()l =0.
Then, for all V E Su(X) the approximations F() of F(6 and Fv of F fulfill
lim 1000
sup zEVnBv(0.r)
FF (x) Fv(x)lxx
The theorem given below establishes a negative result, especially that continuous approximating functors for diffeomorphisms may not exist.
Theorem 3. (No-go theorem for discretization of general diffeomorphisms) There lives no functor D B that meets the property (A) of an approximation functor and is continuous.
B.2 Strongly monotone diffeomorphisms and approximation
In this subsection, it is evidenced that the obstruction to continuous approximation naturally eliminated when the diffeomorphisms under consideration are supposed to be strongly monotone.
Lemma 1. Let V X be finite- dineesiamms subspace of X, and let Pv: X + X be the orthonormal projection onto V. Let F: - X be a strongly monotone C' -dffeeomorphim. On that basis, Pv Fl V + V is strongly monotone, and C. -iffeemorppism.
In fact, strongly monotone projections can be continuously discretized in weak sense. In the sequel, we concentrate on bounded linear operators and Nemytskii operators.
Lemma 2. Let A: X + X be a linear bounded operator and meet (Au, u) 2 collull & for certain co 0. Then, A: X + X is strongly monotone.
Next, supposing that X = (D; RR), we define Nemytskii operator by
Fd (u) =00 u,
in which R. + continuous. Moreover, we can get:
Proposition 1. Suppose that satisfies [a[()] S Cs1sl + O2 and the derivative of - ass) is defined a.e and satisfies the condition a' (8)) > 0. Then, F (D; RR) + (,2 R) strongly monotonous.
We can now give the sufficient conditions for the layers of an NO to be strongly monotone:
Lemma 3. All strongly monotone layers of NOs (F) are diffeomorphisms.
Theorem 4. Let Amn be the discretization functor that projects F to Pv Fly for each finite subspace V C X. Let Dsmn and B Byna categories in which F: - X and Fv: - V are strongly monotone C'-functions in the form of an NO. Then, the functor Aun Damn + Bamn satisfies condition (A), and is continuous in the sense of Definition 7
straight condition to guarantee strong monotonicity of an NO layer given as follows:
be layer of NO that holds the form F(u) = u+ + TTG(TT where T3 : X + -. 1,2 are compact operators and G X + X is a C smooth projection. Suppose that Fréchel derivative DG of at x meets the following for all x E X,
Lemma 4. Let F X
DD 11-4X S
Then, F: X + X is deemed strongly monotone.
24

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B.3 Bilipsshiiz NOs are conditionally strongly monotone diffeomorphisms
The following theorem states that we may always decompose a bilipschitz NO into the composition of strongly monotone NO layers H, and a reflection operator Au-
Theorem 5. Assume X be a Hilbert space. There is e E X, ||e|xx = 1 so that the following is real: 0 and there are linear
Let F: X - X be layer of bilipschitz NO. Then for all invertible projection Ag: X + X, that is either the identity map or reflection function and strongly monotone operalors Hg that also layers of NOs such that
Hy X + X, Hx(r) = + Ba(x), k = 1,2,.....
where Bg X + X a compact mapping and meets Lip(Ba) <cau and
F(x)= Hoo H2 H1 e Ao(æ), for all a E Bx(0,r)). Furthermore, if XX,, then J = O
(9)
We have noticed that operators of the term identity plus compact form are crucial for continuous discretization. This insight inspires the employment of residual networks as approximators within the framework of finite-rank NOs. In the sequel, suppose that X is separable Hilbert space, with an orthogonal basis = {4 SpnNneN. For E N, we define Ey : -x IR and DN RR + X by
E (1)x, ........ (U.
Dya Σ On ASN
is noted that Pvx N E. in which Pvx X + X is the mapping onto Vv == span/pn Using EN, DN. define the category of residual networks in the separable Hilbert space, with T, N E N and activation function a, as
R (X) (G G Of (lax + DN NN, Ex),
NN: : + N are neural networks with activation function a (f = 1,..., T)}
The next theorem proves universality outcome for each of the layers G. allowing to achieve general universality result for the whole network.
Theorem 6. Let R > O, and let F: X + X be a layer of a bilipschitz NO, as Definition f Let a be the ReLU activation function defined by a (x) == ma x)3. Then, for any E (0, 1), there are T, E N and G E oP.,N. that enjoys the form
G= + DN NNT Ex) . (Ix DN NN, En),
such that each projection (Ix + DN o NN, EN) is strongly monotone -'diiffomoorphisms on some ball and
[F(x) G A(oelxx Se,
x (0.R)
in which A: X ++ X is a linear invertible projection that is either the identity map or reflection function
2(x xe with some unit vector E X. Moreover, G A: Bx (0, R) Go A(Bx (0, R)) invertible, and there is certain NO T:: Go AO R)) ~ A(Bx (0, R)) such that
(G ABBxl.....))--- A-1 o.
C Proof of Discrete Invariance under Diffeomorphism
C.1 Proof that the Radon Transform is Bilipschitz
C.1.1 Proof that the Radon Transform in 2D is BiLipschitz
The Radon transform in two dimensions, denoted R : L2 + L2(R x S¹), maps square- integrable function f e (IR to its line integrals over all lines the plane. For parameter e R (the signed distance from the origin) and a direction 0 e S (the unit circle), the Radon transform is defined as:
R)
((BESS du(x),
25

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where du denotes the Lebesgue measure on the line {I IR? z. s). The space L² x 5 is
equipped with the norm:
1 (1(22))))))>= = S / (9((,,)) ds
where d6 is the standard measure on S
A linear operator R is bilipschitz if there exist positive constants and C' such that for all fi, fa (RR)).
fi f2ll2?(K2) & |8 Rf2ll & Cl f2llz>(w>)
Given the linearity of R, this is equivalent to proving:
Upper bound: There exists C > O such that || S Cllfll for all f E (²R))),
Lower bound: There exists
such that [ 2 (l)) for all f E (RR)..
We proceed by establishing both bounds separately.
Upper Bound Proof
We show that there exists a constant C > 0 such that:
[R]llL?(RXST) & (l]])l for all & 1.? (R²)..
Proof:
Firstly, we express the 2 norm of Rf: The norm squared is:
|R ( - J / [R((,,0) 0)1² ds do.
We then apply the Fourier Slice Theorem:
For a fixed direction e S consider the mie-immnsioonal Fourier transform Rff., 0) with respect to s:
F((R((,01(()) = Rf () tas ds.
By the Fourier slice theorem, this equals the two-dimensional Fourier transform of f:
A[]J(,0)](() i(00),
where i(L) = Jga f(x)e- dx for & e I22
Now, we invoke Plancherel's Theorem:
Plancherel's theorem for the one-dimensional Fourier transform states:
[ [Rf(s,0)) (F(RR((())())) do.
Substituting the Fourier slice theorem result
Rs(s 0), ds (f(00)22 do.
26

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Thus, the full norm becomes:
S | [((o0)] do dA.
|R (R
To better prove this, we switch to Polar Coordinates:
Parameterize 0 (cos Φb sin Φ) with @ (0,2 so o0 = a Φb, sin Φ). The integral becomes:
| = / [
2t 00
[f(o(cos o, sin o)) do dó.
Since f(00) f(-o(-0) and St is symmetric, the integral can be split into:
[| [| =2 [//0////////10///////00
2x
sin ") (
do do.
In R?, use polar coordinates (r cos o, sin Ø), where ||E|| and de dr do. The L norm
of f is:
11 Jo 2 [
sin
Compare this to our expression:
2x 1f(occos sin o)) do do.
|| || =
Substituting a = T, the integrand lacks the Jacobian factor r:
|Rf [ = / R2
[f(() de.
adjusting for the measure transformation. Consider the integral:
(f(0) 1Ell dg.
The weight ||||| is locally integrable near the origin:
|I|| de [ -- 2xr dr 2x 1' lrr 2 < 00,
and decays as |s| + oo. Since (RR)),, Cauchy-Schwee inequality or a direct estimate shows this integral is finite when / has sufficient decay, but in L², we rely on known results that this operator (the Fourier multiplier 1//) is bounded in certain weighted spaces, adjusted here via:
R |R (RXS)))
Cillf = Cill [[
where C accounts for the constant from the weight's integrability (approximately 2 2n with proper normalization).
In conclusion:
27

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[R (allffllee
Let C = VC1, which establishes the upper bound.
Lower Bound Proof
We show that there exists constant > 0 such that:
|| 2 <2) for all f C (R²).
Proof:
Firstly, we introduce the injectivity of
The Radon transform is injective. If Rf = 0, then for all 0 E St, Rf 0) - 0, so:
[[Rf(;0)](o) f(o0) 0 for all R, 0 St.
Since {a0: a R 8 RR? 0 almost everywhere, implying 0 by the Plancherel theorem.
We then give discussions on the boundedness of the Inverse:
The inverse Radon transform R-1, Im(R) -> (TR) is well-defined the image of R. In two dimensions, R typically expressed via filtered backprojection:
1 (HO,Rf) 0,0) d0,
f(x) = SS
where H is the Hilbert transform. Standard results (e.g., Natterer, 2001,299 show that R-1 is bounded 2
||t gllza(0e) S <lllll) for all E Im(R),
with some constant K > 0. Next, we apply Rf:
Since f R (Rf).
M = (RfIl00)))) K||B////>(xxs1>>
Rearranging:
[ 2 K [1
Let c K which proves the lower bound.
In conclusion, The Radon transform R: T.2(R22) ( x 81) satisfies:
Upper bound [[(f((==((x5T)) S C'].f(l>((R)))
Lower bound [[(f( 2 cll c|]
with positive constants e and C. Thus, R is bi-Lipschitz in the 122 sense, being both bounded and having a bounded inverse on its range.
28

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C.1.2 Proof that the Radon Transform is Bilipschitz
The Radon transform R : (R") + L x Sn-1) maps function f E L (RR) to its integrals over all hyperplanes in R Specifically, for s E R and / 5 (the unit sphere in RR), is defined as:
Rf(s, = |
f(x) du(æ))
(0ER":0-0=s)
where du is the Lebesgue measure the hyperplane {ar s). We aim prove that R is bi-Lipschitz, meaning there exist constants C > O such that for all f1. f2 E (RR),
cllf - felleaRR) & - (2x & Cil f2l1
Since is linear, it suffices to show:
for Upper bound: || S Cllfl for all f E (RR),,
Lower bound: || ( 2 cll for all f (²())).
We will establish these bounds separately.
Upper Bound Proof
We prove there exists constant C 0 such that for all (RR),,
[f 1) S ClIfll
Proof:
Firstly, we define the L norm of Rf:
The L norm Rff is given by:
(RxS=-1) = SS.. / [R ds dB,
Rf
ga-1
where d0 is the surface measure on 5n-1
We then apply the Fourier Slice Theorem:
By fixing I E Sn-1, the one-dimensional Fourier transform of Rf(-:0) with respect to is:
ds === f(af),
where f(E) = JR- f(z) i dr is the n-dimensional Fourier transform of f.
Now,we use Plancherel's Theorem:
By Plancherel's theorem in one dimension,
/ [
[FIRRF(:00)())) ( do = J [((00)2 do.
Thus,
| gn-1 / |((00)) do do.
Next, we switch to Polar Coordinates:
Parameterize Ç a0 with a E R and 0 E Sn-1, Since f(o0) f(-a0) for real-valued f (adjusting for symmetry), consider:
29

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J [1 L |f(00) do do / [((o)] do =2 % f((o)) do, -o=ao-2000
assuming is real (for complex the factor remains bounded). Hence,
| = gm /0 |f((o)]2 do d8.
In polar coordinates, Ç o8, IlEll |ø), and the Jacobian de -n=1 do dl for O. Thus,
f ((())) de I S (1(00))20=11 da d0.
Rewrite the expression for Rf:
Jo 1B(00)1 do d0 - S0 (- 1f(00))20011 (160))2a-11.00 .''-" do f |f(()) dg.
So,
/ [f(()) de.
|| =
The weight ||_||==> must controlled: n 1, |||||=11 1, and the integral is fa lf(() de |lf [l For n 2 2. |ıE||= is locally integrable near & = (since n -1 implies integrability) and decays infinity. There exists constant C,, sup; Ss-- ||= d6 < depending on n, such that:
J (1(9) de Cn Cn 1f(E)) C. C..1
using Plancherel's theorem again: || [[f]l I1
In conclusion:
|| 1) S 20,.1ffl2a((=)
|| 1) S (20u11fllexre.
Set C = 22,,, proving the upper bound.
Lower Bound Proof
We prove there exists a constant e 0 such that for all f (R"),.
Rffll 2 c|lf
Proof:
Firstly, we introduce the injectivity of Rt:
The Radon transform is injective T.2(R)):: Rf 0, then f(a0) 0 for all a R, 0 p implying f 0 almost everywhere, so f
Then, we talk about boundedness the Inverse:
The inverse Radon transform R-1 Im R - I (R.) exists and is bounded in I under certain conditions: Odd n: For n odd, R-1 is given by the filtered backprojection formula, involving derivatives of order n 1, and is bounded in 2 There exists K > 0 such that:
30

---

|| (lLEAR) S [1lllll1a(xxs t),
for y E Im(R). Since / Rt (R,),
[[llZ>(R>) = -I &
Rearranging,
K [
Set c = 1/K.
For n even, R R involves fractional derivatives (e.g., Hilbert transform for n = 2), but remains bounded L with an appropriate constant K, ensuring the same inequality holds.
In conclusion, We have shown:
Upper bound [[ 1) S ((
Lower bound [[f]] cll [L)()>)
Thus, R bilipschitz in L?, being both bounded and non-degenerate.
D Supplement to Sinogram-Domain Convolution
In the sinogram-domain space obtained after the Radon transform, the sinogram-domain convolution discretely invariant. Specifically, for data of different resolutions, the siongram- -ommann convolution of their Radon transforms yields consistent results in the angular direction, independent of the spatial resolution of the input images. This implies that convolution kernel trained on resolution can be applied to another without retraining, enabling super-resolution capabilities.
Consider two resolutions:
Low-resolution image:
Resolution Mi N1, with position sampling iA81, i 0, 11....,,, -1, where Pi depends on Mj and N1;
High-resolution image: Resolution M2 x N2 (where M2 > M > N1). with position
sampling 8j =jAss.j 1,..,, P2 where P2 > P1 and Ass Asi.
Assumption:
The angular sampling 0₁ 2nk remains consistent across both resolutions, i.e., No is
fixed.
D.1 Definition of Sinogram-Domain Convolution
In Equation (9) of the main text, we have presented the sinogram domain convolution formula. To maintain notational consistency, we will employ the same symbols introduced previously while providing more detailed explanation.
Soonv [k;, Σ
Sim, t][[k m]
(10)
where:S[m. = S(Am, t):K[ - m] = K(fp- m) le m computed modulo N, to account for the periodicity of 0.
Here, K(0) is the convolution kernel, assumed depend only on the angular difference 0 0', and is discretized K[k], 0, 1....,N. 1.
31

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D.2 Sinogram-Domain Convolution at Different Resolutions
Let the sinograms for the low-resolution and high-resolution images be:
Low-resolution: (0x, 44), where i; = i High-resolution: Snigh tj), where łj j2t2.
The discrete convolutions are:
Low-resolution:
No
Scony, low [ki, [4]
=
Σ
Swo (m, Ki]KK - m))
(11)
m=0
High-resolution:
Ng -1
Sconv, high [ke ty]
Snigh Kn,ttKKKKk [k- ml
(12)
Observation: The convolution operation depends only the angular index k. the kernel K [k], and the sinogram values S S(m, at the respective t-values. The structure of the convolution is identical across resolutions, with differences only in the t-sampling.
D.3 Proof of Consistency
To prove discrete invariance, we need to show that the convolution results in the O-direction are consistent across resolutions, independent of the I-sampling.
For fixed f, the discrete convolution Scome [k DMT-' Sim, /) K mi depends on:
The angular sampling lg. which is fixed at 0g = 2xk No
The kernel [{],, which is identical for both resolutions;
The sinogram values S(m, t], which vary depending on the specific t-value.
In the low-rcsolution case, = t, = itt, so the convolution uses Slow [m, tz]. In the high-rcsolution case, = t, = j Atz so it uses Snigh (m, t-]- While i; and t, represent different sampling densities, the convolution operation itself is defined solely over the f-direction. The summation over m (i.e., the angular indices) is identical in both cases:
Same number of terms (Ng);
Same kernel values K[k- m];
Same angular indices m.
The difference in (-sampling (i.e., At, vs. At2) affects the number of convolution outputs along the f-axis (Pi outputs for low-resolution, P2 for high-resolution). but for any specific t-value, the convolution process in the d-direction remains unchanged. t, = ty (i.e., they correspond nearly the same physical detector position, adjusted for sampling), then Sjow/ [mz, t.] ~ Snigh m, ty], and the convolution results Sconv, law [ks t.] Senvy high k, tyl. The operation's 's form ensures consistency the f-direction across resolutions.
E
Time Complexity Analysis
Given a discrete counterpart E HX of 2D continuous function, we can reform u to E NCC with N H W. aforementioned the main texts, the variable represents the number of projection angles.
32

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Table 6: Comparison of runtime complexity
Models
Complexity
GNO (kernel)
O(N(N - 1)) ((N2) O(N log N) O(N) O(AN)
Transformer
FNO (FFT)
CNO
RNO
Graph neural operator (GNO)
The integral formulation of GNO is as follows:
(K(v))(a)= = Socc. K(x,y) w(y)dy
Σ K(x,y) v(y))
BEU(a)
where is kernel (typically parametrized by deep network) and dg E IK are suitable quadrature weights. Yet GNO is capable of expressing local integral operators by choosing appropriately small neighborhood (x) S D, computing the kernel and performing aggregation within each neighborhood (x) is computationally expensive and memory-demanding for general applications k: x D + R". For each point, it is required to access all of the neighboring elements. As reported
in 24 the runtime complexity of GNO is then ONN - 1)).
Transformer
The integration of Transformer-based NO is given as:
(K(()))()) = / ((v(z),
(K()))(z) ~0 Σ v(()))))
W.
exp
Vdu
(K(())(z))
W. Wve(K))
(13)
Wer(s)) (aa(e))
1== exp
in which K is perceived employing softmax function onto three transformed vectors with length By. Evidently, Eq. (13) closely aligns with the commonly applied self-attention mechanism in vanilla transformers, where the matrices W Ws, W, E Rdd Xd. concern the learned transformations of queries, keys, and values, respectively. From Eq. (13), it obvious that two nested loops of length ny must be traversed. On that basis, the runtime complexity of transformer-based NOs is O(N2).
FNO FNO substitutes the global convolution in the time domain with multiplication operations in the frequency counterpart; hence, its kernel integral is written as:
(K(())(z) = =f K(x, v)v(y)dy
/ K(r y)u(y)dy
Ja
=F (Fv)) (x)
in which Rs designs N multiplications, parameterized by Φ. therefore its runtime complexity is O(N). Moreover, the time complexity of the Fast Fourier Transform is O(N log N), leading to an overall complexity of O(N log N).
33

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CNO Typically, the local convolution operation is elaborated by
(I(()))()== / x(x - by)
Kazu( 2as)
where E Bu, x is discrete kernel with size & N, Zij are the resultant grid points. Evaluating each point involves k2² multiplications. So the total time complexity O(N12) O(N).
PDE Benchmarks
We tested our data benchmarks, including three categories:
Solid material [0]:lasticity
Navier-Stokes equations for fluid 28 :Airfoil, Pipe, Navier-Stokes
Darcy's 's law 14
Darcy
Deaction-Diffusiin 38 AAlle--Cahn
Table 7: Summary of experiment benchmarks, where the first six datasets are from FNO and geo-FNO. Mesh records the size of discretized meshes. Dataset is organized the number of samples in training and test sets.
Geometry
Benchmarks
Dim
Mesh
Input
Output
Dataset
Plasticity
2D+Time
3,131
External Force
Mesh Displacement
(900,
Structured Mesh
Airfoil
2D
11,271
Structure Structure
Mach Number
(1000, 200)
Pipe
2D
16.641
Fluid Velocity
(1000, 200)
Navier-Stokes
1D+-Time
4,096
Past Velocity
Future Velocity
(1000. 200)
Regular Grid
Darcy
2D
7,225
Porous Medium
Fluid Pressure
(1000, 200)
Allen-Cahn
129xx129
Initial Phase Field
Evolved Phase Field
(1000, 200)
Here are the details of each benchmark.
F11
Solid Material
The fundamental equation governing the behavior of solid materials expressed as:
82
=0
(14)
0t
where p² E IR represents the density of the solid, and V signifies the nabla operator. The variable function of time while corresponds the
denotes the displacement vector of the material stress tensor. The model Plasticity [22] is governed by the equation, as presented in (14..
Plasticity. This benchmark addresses the plastic forging process, where plastic material undergoes impact from an arbitrarily shaped die applied from above. The input consists of the die's geometry, represented structured mesh. The output comprises the deformation of each mesh point over the subsequent 20 time steps. The structured mesh has a resolution of 101 x 31.
F.2 Navier-Stoker Equation
The differential form of the fluid dynamics equations given by:
Op
at (pU)
(15)
OU
+ VU = f -/ (Tijee;))
(16)
01
34

---

a (e (e+ 3 U2) + U v et ( - -
=f
-
(U Tyee)e)) + AT,
(17)
Ot
where Equations (15 16 and 17 represent the conservation of mass, momentum, and energy, respectively. Here, denotes the fluid density, U the velocity vector, / represents the external force, and is the internal energy. The stress tensor in the fluid is denoted by T, with 6; as the basis vector, and Tiee adheres to the Finstein summation convention. All variables are functions of space and time, and the term À AT accounts for heat conduction. For a Newtonian fluid, the stress tensor I" depends on the pressure viscosity coefficient V, and velocity vector U. Consequently, Equation (T6) for Newtonian fluid can be reformulated as:
OU + + VU = - VP-VVV
(18)
Additionally, Equation (17) can be derived similarly, but its complexity precludes inclusion here; (28) for details. These equations for Newtonian fluids are commonly referred to as the Navier-Stokes equations. Below, we elaborate on the partial differential equations (PDEs) underlying our fluid benchmarks.
Navier-Stokes. The Navier-Stoker dataset, sourced from 123]. models incompressible, viscous flow on unit torus, where the fluid density p in Equation (T5) remains constant. In this context, the energy conservation in Equation (17) is decoupled from mass and momentum conservation. Thus, the fluid dynamics are governed by Equations (15) and [8]:
-U=
dw Ōt
+U Vw ~
(19)
w 1= Wo:
where U = (u, v) is the 2D velocity vector, w V x
U
0v du represents the vorticity, By dz
and wy E IR is the initial vorticity at 0. The dataset uses viscosity 10- and a 2D field resolution of 64 x 64. Each sample includes 20 consecutive frames, with the task of predicting the next frames based on the previous 10.
Pipe. The Pipe dataset, from 22 examines incompressible flow within a pipe. The governing equations derived from Equations 15 and (8]
-
aU
(20)
VU -f
The dataset is generated on a structured mesh with resolution of 129 x 129, For experiments, the mesh structure serves as the input, and the output the horizontal fluid velocity within the pipe.
Airfoil. The Airfoil dataset, also from [22] investigates transonic flow over an airfoil. Given the negligible viscosity of air, the viscous term V is omitted from the Navier-Stokes equations. The governing equations are thus:
0pl
+ V (p'U)
Øt
0p'U + UU + +pl) =
(21)
OF + v ((E + p)U) = =0
Øt
where of denotes the fluid density, and F. represents the total energy. The data is generated structured mesh with a resolution of 200 x 50. The mesh point locations are used as inputs, and the Mach number each mesh point is the output.
E33
Darcy Flow
Darcy, Darcy's 's law governs the flow of fluids through porous media, such as water permeating sand. We utilize the Darcy dataset introduced by (23), which models 2D Darcy flow equations within a unit
35

---

square, expressed as:
-V (aVu) (aVu)= ul 0,
(22)
where a E IR+ represents the diffusion coefficient, and / denotes the external force, fixed this dataset. The input for this dataset is the diffusion coefficient a, with the output being the solution Data samples are structured on regular grid with resolution of x 85.
F,4 Allen-Cahn equation
The Allen-Cahn equation, a reaction-diffusion model, is widely applied study chemical reactions and phase separation in multi-component alloys. We used the dataset proposed by [37] In two-dimensional space, the Allen-Cahn equation is formulated as:
Allen-Cahn Equation.
0,u(x, y, t) edu(22, y, u(x, v.t) u(z,8,t)² u( = 0, t) = u(r = 3. 3. t). u u(x, y = 6) u(x, 3, t):
I, E (0,3), E [0, 20).
y E (0,3), t E [0, 20).
(23)
x E (0,3), E [0, [0]
( V. 0) = uo (2 (z,y). y):
y E (0,
where e R is a positive constant controlling the diffusion magnitude. The problem is defined with periodic boundary conditions, setting = 10-3, The initial condition is generated from Gaussian Random Field using the kernel:
K(x,y) = (a-1) (x² +
(24)
with parameters and 1. The objective learn the operator un y) + u(x y,
Here, the solution is computed at = 20 on grid with resolution of 129 x 129.
G More Experiments and Analysis
G.1 Implementation Details
As shown in Table g all the baselines are trained and tested under the same training strategy. For the Allen-Cahn equation, plasticity. and airfoil problems, we set the number Radon transform angles to 64, while for pipe flow, the Navier-Stokes equations, and Darcy flow, we use 32 angles. Given the physical field u and the model-predicted field ú, the relative 1,2 norm of the model prediction can be calculated as follows:
||u all
Relative 1.2
(25)
|/
Table Training and model configurations of RNO. Here L. and L. represent the loss volume and surface fields respectively. As for Darcy, we adopt an additional spatial gradient regularization term Lg following ONO 4k
Training Configuration (Shared in all haselines)
Model Configuration
Benchmarks
Loss
Epochs Initial LR
Optimizer
Layers L Hcads Channels C Angles Blocks
Allen-cahn
128
64
Plasticity
128
Airfoil
Relative
Adam
128
500
10
8
8
Pipe
L22
26
128
Navier-Stokes
256
32
Darcy
Cua + O.1.2x
128
32
G.2 Experiment Visualization
We have already given relatively complete description of Radon block in Section 3f4 the main text. We then show the self-learning adjustment of the sinogram domain weights Figure 9 in which the weight change of the first Radon block is provided. As seen, while the training steps proceed deeper and deeper, the resultant weighs would adaptively gather into some specific angles with enriched features.
36

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eight adjustment
Figure 9: Self-learning adjustment of the sinogram domain weight. From top to bottom, from left right, it represents the adaptive change of the angle domain weight as the training time increases.
G.3
Generalization
The discussion of generalization is an important issue neural operators, which reflects the learning of the mapping relationship between neural operators and infiite-dimensiona function spaces. We have introduced this in Section 4.2] of the main text. The comparison in Table 9 reveals RNO's superior generalization performance over Transolver and FNO when handling larger resolutions. We futher show results of the generalization experiment on the Darcy equation dataset in Figure 10
Table 9: Comparative display of generalization performance. Relative 1.2 is recorded. A smaller value indicates better performance. (Bold: Best performance)
Model
61 x 61
85 x 85
141
141
211 x 211
421 x 421
FNO
1.16 x 10-1
1.80
10
2.68 x 10
3.16 x 10
3.63 10-1
Transolver
3.28 x 10-2
5.11 10-2
6.78 x 10-2 2
7.46 x 10
8.64 10-2
RNO (Ours)
3.21 x 10 5.04 x 10 2
6.69 10 2
7.09 x 10
7.62 x 10 2
61x61
85x85
141x141
211x211
421x421
Prediction
Ground Truth
Relative Error
Figure 10: The effect of model generalization is shown. From top bottom, they the prediction map, the real map, and the error map. showing the generalization of RNO on the Darcy dataset. The model was trained low resolution of 43 x 43 and tested five other high resolutions.
37

---

G.4 Parameter Setting of Radon Transform
It is generally considered that an increase in the number of angles in the Radon transform enhances the accuracy data prediction and reconstruction; however, the opposite is sometimes observed. As demonstrated in Figures| I1 and 12 it has been found through experiments that, at ultra-low resolution, an increased number of angles does not improve results; conversely, greater errors are produced. This finding further informs our choice of the number of angles and, as discussed in Section [.2] lower angle count reduces computational complexity. This finding provides valuable insights for future applications of the Radon transform.
G.5 Supplementary Ablation Experiments
Note that the positive roles of replacing other transformations with RT have been evidenced in previous discussions. Hereafter, we further evaluate the behaviors of Radon block when imposed on the other architectures. Practuccc....a.ccalll...... we performed experiments by integrating Radon block with several global methods, including self-attention (S.A.), and orthogonal attention (O.A.). Similar to our method, these attempts all take advantage of global feature extraction and local feature learning. Similar to our method, these attempts all take advantage of global fea- ture extraction and local feature learning. We observe that previous works have addressed similar local problems (25) As demonstrated in our main results table, the proposed FNO+Radon block achieves superior performance compared to approaches using either Differential Kernel Local Integral Kernel operators.
Table 10: Ablation studies by integrating Radon block with other off-the-shelf modules All ex- periments were based on the 85 x 85 Darcy flow dataset and tested when the number of angles was set to 32.
Ablations
Memory (GB)
Time
Param
Relative L2 Darcy
(sieprch)
S.A.+Radon Block
19.564
508.83
x 10 10-2
O.A.n Block
15.038
110.36
2.03 10 7,80 10
ours
3.191
43.21
2.83 x 10 5.31 10
As shown in Table [ we conduct systematic experiments by varying the number of model layers. Taking the Darcy dataset as an example, our results demonstrate that increasing the layer depth does not yield performance improvements, but rather leads to a significant growth in both parameter count and computational time. Similar observations hold for other datasets such as Navier-Stokes, where deeper architectures tend to suffer from overfitting. Based these empirical findings, we deliberately avoid using larger layer counts in our main experiments.
We conduct extensive experiments to evaluate the impact of angle parameter settings, with results summarized in Table 12] Our findings reveal that while higher angular resolution leads to improved accuracy, this comes at the cost of increased computational time- a trade-off that aligns with our theoretical complexity analysis. Notably, the angle parameter selection has no measurable effect on the total number of model parameters.
Table 11: Ablation studies by varying the number of model layers from 16 to 40.
Memory (GB)
Time (slepoch)
Param (B)
Relative 1.2 Darcy
Settings
layer=40
11.136
110.03
1.39 x 107
4.89 x 10
layer=32
9,070
91.45
1.12 x 107
6.78 x 10-3
layer=24
7.004
73.25
8.38 x 106
5.73 x 10 3
1ayer=16
4.940
55.64
5.60
109
5.20
10-3
G.6 Showcases
To facilitate improved visualization of the experimental results, visual representations of datasets, along with prediction images and error maps, are presented. As follows, Figs. [4] 13 show the 6 benchmarks. We provide comprehensive visualization of our experimental results, including input data, ground truth, predicted outputs, and error maps.
38

---

Table 12: Ablation studies by varying the number of angles for Radon transform from 64 to 512.
Memory (GB)
Time (slepoch)
Param (B)
Relative L2 Darcy
Settings
Angles=512
4.400
283.88
2.83 x 10
4.31 x 10-3
Angles=256
3,562
145.63
2.83
10
4.65 x 10
Angles=128
3.152
85.11
2.83 x 106
4.87 x 10-3
Angles=64
2,962
54.19
2.83 x 109
4.90 x 10-3
H Limitations
Our work currently faces two main limitations. Unlike the Fourier transform, which benefits from well-optimized algorithms and seamless PyTorch integration, the Radon transform exhibits potential for further improvement in parallel computing and GPU utilization. To substantiate this claim, we have conducted ablation study in Section 4.3 Additionally, whereas model is designed for 2D or time-dependent 3D problems, its application to general 3D PDEs, particularly those involving point cloud data, remains unexplored. Regarding this issue, we provide rigorous proof of the billipschitz condition for arbitrary dimensions in Appendix C.I.2 which theoretically guarantees the feasibility of our approach. This theoretical foundation ensures the soundness of proposed method across spaces of different dimensionalities. The dimensionality reduction property of the Radon transform is particularly advantageous for 3D PDEs, alleviates the computational burden compared to methods like FNO, which may face scalability challenges in higher dimensions We emphasize that the GNO module integration strategy proposed in 24 for solving 3D problems can be directly adapted to our framework. Our method inherently supports such an extension, and we are well well- positioned address 3D PDE cases through similar architectural modifications.
I Broader Impacts
Partial differential equations represent the most fundamental mathematical tool in modern physics and engineering, making their numerical simulation research area of both theoretical and practical significance. While mathematically inspired, our work innovatively incorporates operator learning architectures, achieving superior performance in terms of solution accuracy and generalization capability. This advancement demonstrates substantial potential for both industrial applications and theoretical research in PDE solving. Meanwhile, neural operators have proven to be remarkably versatile. They integrate seamlessly with various algorithms like transformers [43] Mamba (7].
and diffusion models 48 [3], and demonstrate broad applicability in domains such as remote sensing [5] BT, medical images 15 [9] and super-resolution 41). We hope our work offers fresh perspectives for this dynamic field.
39

---

Ground Truth
n_angles=512
Relative Error :0.0627
Sinogram
Shaggams
Error
Figure setting of the number Radon transform angles ultra-low resolution. From left to right, the true image, the angle domain image, and the transformed image and eтror are shown. The image above shows the number of angles 512.
Ground Truth
n_angles=64
Relative :0,0607
Sinogram
Seague
Error
The setting of the number of Radon transform angles at ultra-low resolution. From left to right, the true image, the angle domain image, and the transformed image and error are shown. The image above shows the number of angles 64.
Figure 12:
40

---

Ground Truth
Prediction
Error
Plasticity
Navier-Stokes
Figure 13:
Experimental visualization of Plasticity, Navier-Stokes equations, including the ground truth plots, predicted plots, and error plots.
Input
Ground Truth
Prediction
Error
Darcy
X
Allen-cahn
WW
b
W
W
Pipe
Airfoil
Experimental visualization of Darcy, Allen-cahn, Pipe, Airfoil equations, including the input plots, ground truth plots, predicted plots, and error plots.
Figure 14:
41


Reviews:
[{"review_id": "f8rfSbxQSP", "reviewer": "Reviewer_XT7t", "summary": "This paper introduces the Radon Neural Operator (RNO), a novel neural operator that projects PDE input data into the sinogram domain using the Radon transform, followed by learning with angle-weighted sinogram-domain convolutions and a physics-informed attention mechanism. The proposed method achieves SOTA performance across multiple PDE benchmarks and demonstrates strong generalization to higher resolutions. A key theoretical contribution is the bilipschitz proof of the Radon operator, establishing discretization invariance under diffeomorphism, addressing a fundamental obstacle in operator learning.", "questions": "- How computationally intensive is the their pipeline compared to other models such as Fourier or Wavelet transforms? Can the authors provide runtime or memory usage comparisons?\n- How was the number of angles chosen in each experiment? Is there a principled or adaptive way to select this?\n- Can RNO model generalize to 3D PDEs, or are there limitations to extending this framework to higher dimensions?\n- Have the authors considered using learnable inverse transforms, rather than the fixed FBP, to improve robustness for example in the presence of noise?\n- Is the model sensitive to data noise, especially since inverse Radon transforms can amplify artifacts in noisy inputs?", "limitations": "While the paper is strong, it omits a critical reflection on limitations. For instance:\n- computational cost: Although the paper claims quasi-linear complexity, it does not benchmark runtime or memory usage against baselines like FNO. Applying Radon transforms could become expensive in high dimensions or with large angle counts.\n- sensitivity to angular hyper-parameter: The performance of RNO is dependent on the number and selection of projection angles, yet there is no clear guidance or adaptive mechanism for tuning this.\n- limited discussion of failure modes: The model underperforms on the Pipe dataset due to padding, which hints at limitations on complex or irregular geometries, but this is not analyzed in depth.\n- despite claiming industrial applicability, there is no proof of deployment or even a real-world dataset test case.\n\nI would encourage the authors to include a clear and thoughtful discussion of limitations in the camera-ready version. Note that acknowledging limitations is not a weakness, it reflects intellectual honesty and strengthens the trust in the work.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n- Novel use of Radon transform in operator learning, offering a new path distinct from the more common Fourier or wavelet-based approaches.\n- Theoretical depth: Clear and rigorous bilipschitz proof linking Radon transform to discretization invariance.\n- Effective implementation: Sinogram-domain convolution and learnable angle-weighting are both intuitive and empirically validated.\n- Nice empirical results: Outperforms strong baselines such as FNO, WNO, and Transolver across a variety of PDE types and geometries.\n- Generalization evaluation: Shows impressive resolution transfer, with 70–79% relative L2 error reduction compared to FNO.\n- Ablation studies are informative (e.g., quantifying benefits of sinogram conv vs. simple weights).\n\nWeakness:\n- No explicit limitations discussion: The paper lacks a section acknowledging limitations, which would improve scientific transparency and credibility.\n- Limited runtime benchmarks: The authors provide partial runtime metrics (e.g., per-epoch time and memory in Table 3 for instance), but do not compare end-to-end training or inference time versus other neural operator baselines.  Also, it would have been valuable to report scaling behavior across problem size, resolution, or angle count, which would better support their efficiency claims.\n- Limited failure analysis: The performance drop on the Pipe benchmark is mentioned but not analyzed in depth, leaving ambiguity about limitations on irregular meshes or real-world geometries.", "was_revised": false, "final_justification": "", "rebuttal": "> **W1: About the limitations of our work**\n\nWe 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.\n\nSpecifically, 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.\n> **W2/Q1: About operational efficiency and memory**\n\nWe 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.\n\n*Darcy 85×85 grid, batch size 2, single RTX-4090*\n| Model | Train (s/epoch) | Inference (s) | Memory (GB)|Relative L₂|\n| ----- | ----------- | -------------- | ----------- |--------- |\n| FNO         | 8.48  | 0.35           | 1.418       |0.0108     |\n| WNO         | 51.3  | 3.34           | 2.803       |0.0084     |\n| RNO         | 36.7  | 2.21           | 3.994       |0.0051     |\n\nThe 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.\n\n*Darcy 43×43 grid, batch size 2, single RTX-4090*\n\nTips: 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.\n| Stage    | **RNO (ours)** | **FNO(without GPU optimization)**|\n| --------------------------- | ------- | -------------- |\n| **Training time per (s/epoch)** |32.87   | 196.79          |\n| **Inference (s)**    |3.02    |  14.17         |\n\n> **W3: About irregular grids**\n\nThank 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.\n\nAs 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.\n\n> **Q2: About angle quantity selection**\n\nWe 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. \n\nSecondly, 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.\n\n> **Q3: Discussion of 3D PDEs**\n\nWe 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.\n\n> **Q4: Discussion about inverse transform**\n\nWe 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.\n| Method    | Relative L₂ | Time (s/epoch)|\n| --------- | ----- |------|\n| iRadonMAP |❌0.126  |127.5|\n| Unet      |❌0.271  |286.4|\n| FBP(Ours) |✅0.0051 |36.7  |\n\n> **Q5: About noise-sensitive and real-world datasets**\n\nWe 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. \n\nMoreover, 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.\n\n| Method    | Error |\n| --------- | ----- |\n| WNO       |0.0155  🥈|\n| RNO(Ours) |0.0121 🥇 |\n\n\n[1]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023\n\n[2]Radon Inversion via Deep Learning, IEEE Trans. Med. Imaging 2020\n\n[3]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M"}, {"review_id": "SPZAB50KC9", "reviewer": "Reviewer_vnqR", "summary": "This paper introduces Radon transform into neural operators and perform weight analysis in the sinogram domain to solve partial differential equations. Following the Radon forward transformation, a convolution operation in the weight network is designed, which assists in the holistic learning of both global and local features. Theoretically it  is shown that RNO is a bilipschitz operator, which ensures discretization invariance under diffeomorphism and avoids the introduction of any topological obstructions. Empirically, new state-of-the-art scores are earned by RNO across most PDE benchmarks.", "questions": "What is the \"detector \" in line 247?", "limitations": "It is better to give a table to describe the advantages and disadvantages of Radon transform in comparison to FFT and wavelet etc. in the context of neural operator learning.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The introduction of Radon transform into neural operators to solve partial differential equations is novel.\n2.  RNO is a bilipschitz operator, which ensures discretization invariance under diffeomorphism and avoids the introduction of any topological obstructions.\n3. Extensive experimental results are provided.\n\nWeaknesses:\n1. Please give a more intuitive description why Radon transform can assist in the holistic learning of both global and local features, as opposed to FFT and wavelet.\n2. The timing for training and inference is not explicitly provided.\n3. In fig.4, what is the weight in the sinogram space? Better use math to describe. Is it the output of of Radon transform?\n4. It seems that this method is much slower than FNO. \n5. Please give a description of the limitation of this method. \n6. typo: $\\phi$ and $\\varphi$ in eq. 1.", "was_revised": true, "final_justification": "The proposed method is novel and extensive experimental results support its effectiveness. Most questions concerning presentations and efficiency are clarified by the rebuttal.", "rebuttal": "> **Q1/Limitations: About Radon vs. FFT/wavelets**\n\nWe 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).\n\nIntuitively, 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.\n\n\nTo 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.\n\n| Transform         | Global Feature Capture           | Local Feature Capture     | Geometry Awareness       | Discrete Invariance      | Computational Cost         | Typical Bottlenecks in NO Learning              |\n| ----------------- | -------------------------------- | ------------------------- | ------------------------ | ------------------------ | -------------------------- | ----------------------------------------------- |\n| **FFT** (FNO)     | ✅ Excellent (global frequencies) | ❌ Poor (no locality)      | ❌ Assumes periodicity    | ✅ Yes (spectral conv)    | **Low** (FFT + linear)     | Oversmoothing of shocks, Gibbs artifacts        |\n| **Wavelet** (WNO) | ⚠️ Limited (coarse scales)       | ✅ Excellent (multi-scale) | ⚠️ Limited (fixed basis) | ✅ Yes (multi-level)      | **Moderate** (filter bank) | Choice of mother wavelet, boundary handling     |\n| **Radon** (RNO)   | ✅ Strong (line integrals)        | ✅ Tunable (θ-convolution) | ✅ Strong (line geometry) | ✅ Yes (θ-grid decoupled) | **Moderate** (O(AN))       | Angle count trade-off |\n\nKey: ✅ = advantage, ⚠️ = moderate, ❌ = weak or missing.\n\nWe 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.\n\n> **W2: About running time**\n\nWe 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.\n\n*Darcy 85×85 grid, batch size 2, single RTX-4090*\n| Model | Train (s/epoch) | Inference (s)   |Relative L₂|\n| ----- | ------- | -------- |----------- |\n| FNO         | 8.48  | 0.35           | 0.0108     |\n| WNO         | 51.3  | 3.34           | 0.0084     |\n| RNO         | 36.7  | 2.21           | 0.0051     |\n| Transolver  | 28.9  | 1.12           | 0.0057     |\n\nWe 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.\n\n> **W3: About sinogram domain weights**\n\n 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**.\nMathematically, for a sinogram $ \\mathcal{R}u(s, \\theta)$, the weighted sinogram is computed as:\n$$ \\tilde{\\mathcal{R}}u(s, \\theta) = w(s, \\theta) \\cdot \\mathcal{R}u(s, \\theta), $$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.\n\n> **W4: Discussion on algorithm efficiency**\n\nWe 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. \n 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.\n\n*Darcy 43×43 grid, batch size 2, single RTX-4090*\n\nTips: 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.\n| Stage    | **RNO (ours)** | **FNO(without GPU optimization)**|\n| ------------ | ------- | --------- |\n| **Training time per (s/epoch)** |32.87   | 196.79          |\n| **Inference (s)**    |3.02    |  14.17         |\n\n> **W5: About limitations**\n\nWe 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. \n\nSpecifically, 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.\n\n> **W6: typo: $\\phi$ and $\\varphi$ in eq. 1.**\n\nThank you very much for your correction, we have fixed this. \n\n> **Q1: What is the \"detector \" in line 247?**\n\nSince 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.\n\nMathematically, 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  $s$ :\n$$R(f)(\\theta, s) = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} f(x, y) \\delta(x \\cos \\theta + y \\sin \\theta - s)  dxdy$$Here, the \"detector\" can be thought of as a virtual point that receives these integral values, representing the projection data for a specific direction.\n\n\n[1]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023\n\n[2]The Radon transform-theory and implementation, 1996"}, {"review_id": "iF0ZsA2rSQ", "reviewer": "Reviewer_JP5G", "summary": "This paper introduces the Radon Neural Operator (RNO), which learns to solve parameterized PDEs by working in the sinogram domain. The authors take input functions, apply a Radon transform to get projections, run a learnable weighting and convolution on those projections, and then reconstruct the solution via filtered back-projection. They back this up with a bilipschitz-under-diffeomorphism argument to claim stronger discretization invariance, and show experiments on various PDEs (Darcy flow, Navier–Stokes, Allen–Cahn, Airfoil, Plasticity, Pipe flow). Results suggest RNO often outperforms FNO and related baselines, especially in zero-shot generalization from coarse to fine grids", "questions": "- Can you share wall-clock runtimes and GPU memory usage for RNO vs. FNO (and any other strong baselines) on representative problems, especially at high resolutions?\n- Have you tried RNO on non-Cartesian or unstructured meshes? If not, can you speculate on how one might extend the sinogram approach to arbitrary domains?\n- How sensitive is RNO to noisy or perturbed inputs? Would adding learned filters or regularization in the sinogram domain improve robustness?\n- What guidelines can you offer for choosing the number of projection angles? Do you observe overfitting or diminishing returns as angle count increases?\n- Do you see a straightforward path to 3D PDEs via 3D Radon transforms? What challenges—computational or theoretical—do you anticipate for higher-dimensional extensions?", "limitations": "N/A", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### **Strengths**\n- The idea of using the Radon transform for operator learning feels fresh. It leverages well-known imaging theory in a new context.\n- The theoretical section goes beyond a simple invariance remark: showing bilipschitz behavior under diffeomorphisms is a solid contribution.\n- Empirical results cover a nice variety of PDEs and clearly show gains in resolution generalization. Seeing consistent error drops when moving from low to high resolution is convincing.\n- The ablation studies isolate the Radon block’s benefits, helping to understand why the approach works.\nZero-shot upscaling experiments (e.g., going from 43×43 inputs to 421×421 outputs) show that RNO handles discretization shifts better than some prior methods.\n\n\n\n### **Weaknesses**\n- It’s not obvious how expensive this is in practice. While the paper claims quasi-linear cost, I’d like concrete runtime and memory figures compared to FNO or other baselines on large grids. Without that, it’s hard to judge practicality.\n- Actually most metrics in Table 1 seem quite close to TRANSOLVER, within 10% and sometimes TRANSOLVER is better. I would not consider them to be materially different. The real advantage of RNO was not deeply demonstrated. What is it that RNO would be most special about and most suited for? Why do we need to work with it given the hundreds of neural operators out there already? The authors should devise experiments and identify suitable cases to uncover that. As of now, the motivation is not strong.\nResults and analyses should be strengthened. More in-depth analysis is called for.\n- The link to unstructured or irregular domains is unclear. The current setup seems tied to Cartesian grids—how would you apply RNO if the domain is irregular or mesh-based?\n- Radon inversion can amplify noise. There’s no study on robustness to noisy inputs or discussion on how to regularize or filter in the sinogram domain. This feels like a gap if someone wants to apply RNO to real-world data.\n- The choice of projection angles is glossed over. How many angles are needed? Could too many or too few hurt performance or cause overfitting? A systematic angle-count study would help.\n- The theory assumes ideal continuous Radon/FBP; in practice, discrete sampling and interpolation errors matter. The paper could discuss how discretization affects the bilipschitz guarantees.", "was_revised": true, "final_justification": "The authors addressed my questions and proposed a good plan to make changes.", "rebuttal": "> **W1/Q1: About practical runtime & memory** \n\nWe 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.\n\n*Darcy 85×85 grid, batch size 2, single RTX-4090*\n\n| Model | Train s/epoch | Inference (s) | Memory (GB)|Relative L₂|\n| ---- | --- | --- | --- |--- |\n| FNO | 8.48  | 0.35  | 1.418 |0.0108 |\n| WNO | 51.3  | 3.34  | 2.803 |0.0084 |\n| RNO| 36.7  | 2.21 | 3.994 |0.0051 |\n| Transolver  | 28.9  | 1.12 | 3.983 |0.0057 |\n\nOur 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.\n\n*Darcy 43×43 grid, batch size 2, single RTX-4090*\n\nTips: 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.\n\n| Stage| **RNO (ours)** | **FNO(without GPU optimization)**|\n| ---- | ---- | ---- |\n| **Training time per (s/epoch)** |32.87 | 196.79 |\n| **Inference (s)**    |3.02 |  14.17 |\n\n> **W2: About our motivation**\n\nWe 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.\n\nFollowing 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.\n\nTo 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.\n| Perspective  | **Radon Transform (RNO)**| **Transolver** | **FNO** | **WNO**|\n| --- | --- | --- | --- | ---- |\n| **Global–Local Fusion** | ✅ Single transform yields global (line integral) & local (angle-wise) features. | ⚠️ Needs separate global attention. | ❌ Global only.  | ❌ Local only. |\n| **Geometry Awareness** | ✅ Integral along rays captures transport directions. | ⚠️ Mesh-based geometry via attention. | ❌ Periodic boundary artifacts.  | ❌ Axis-aligned wavelets. |\n| **Discrete Invariance**| ✅ Angle grid decoupled from spatial resolution. | ⚠️ Attention weights re-learn per mesh. | ❌ Frequency grid tied to resolution. | ❌ Scale grid tied to resolution.|\n| **Computational**  | ✅ Quasi-linear O(AN) with small A | ⚠️ Attention overhead. | ✅ O(N log N).| ✅ O(N log N). |\n\n> **W3/Q2: Applicability to unstructured or irregular domains** \n\nWe 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.\n\n> **W4/Q3: About sensitivity to noisy**\n\nWe 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.\n| Method    | Error |\n| ----- | --- |\n| WNO |0.0155  🥈|\n| RNO(Ours) |0.0121 🥇 |\n\n> **W5/Q4: Regarding the choice of the number of angles**\n\nWe 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.\n\n> **W6: About theoretical proof**\n\nWe 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.\n\n> **Q5: About 3D PDEs**\n\nWe 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.\n\nWhile 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.\n\n[1]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M\n\n[2]Discrete radon transform, IEEE Trans. Acoust. Speech Signal Process 2003\n\n[3]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023"}, {"review_id": "GkrNAF1MnW", "reviewer": "Reviewer_Qcdm", "summary": "The authors propose a neural operator architecture, Radon Neural Operator (RNO), which performs operator learning in the sinogram domain based on the Radon transform. The architecture first applies the Radon transform to project function features from the spatial domain into the sinogram domain. Then, convolution operations are performed in the sinogram domain to extract local features and reweight the basis features corresponding to different projection angles. Finally, an inverse Radon transform is implemented via filtered back projection to map the features back to the spatial domain. Additionally, a Physics Attention Block is introduced to capture global features. Experiments on both regular grids and structured meshes demonstrate that the proposed RNO achieves excellent performance in terms of accuracy, efficiency, and generalization ability. The authors also provide a series of theoretical analyses, which strongly support the effectiveness of the proposed model and lay a solid foundation for future improvements.", "questions": "1. After the function features are projected into the sinogram domain, what is the shape of the data and what does the convolution process look like? In the left subfigure of Figure 5, what does the red dashed arrow represent? In the right subfigure of Figure 5, why do the green and blue blocks exhibit a sine wave pattern?\n\n2. In the ablation study on model structure (Table 3), the \"Only nonlinear weights\" experiment is used to evaluate the importance of the sinogram convolution. However, why is there no experiment result for a model composed only of the Radon Block (i.e., removing the Physics Attention (P.A.) block from RNO)? Such an experiment would be crucial to fully demonstrate the importance of the P.A. block in the RNO architecture.", "limitations": "It would be beneficial to validate the effectiveness of RNO on real-world datasets.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strength\n1. The authors innovatively introduce the Radon Transform into neural operator models and propose a novel mechanism for reweighting basis functions corresponding to different projection angles in the sinogram domain. This enables the model to focus on high-frequency feature learning, addressing a key limitation of neural operators based on Fourier and Wavelet Transforms, which often overlook important local or high-frequency components.\n\n2. The authors theoretically prove that the proposed RNO model satisfies the bilipschitz strong-monotone property from a diffeomorphism perspective, which guarantees continuous-discrete equivalence for the neural operator and provides strong theoretical support for discretization-invariant operator learning.\n\n3. The RNO model is extensively evaluated against a wide range of baselines on multiple benchmark datasets. The experimental results demonstrate superior accuracy, showcasing the effectiveness of the proposed method.\n\n# Weakness\n1. RNO appears to struggle with handling unstructured grids. Although the authors mention the use of masking as a workaround, this may not fundamentally resolve the problem. The experiments are primarily conducted on datasets used in FNO [1] and Geo-FNO [2], but do not include the Elasticity dataset, which represents PDE fields with scattered points and poses a more challenging setting for operator models to robustly handle non-uniform meshes.\n\n2. RNO lacks evaluation on real-world and 3D datasets. For example, FactFormer [3] involves 3D time-dependent PDE problems, and Transolver++ [4] targets large-scale industrial simulation problems with millions of mesh points. Evaluation in such settings would better demonstrate the scalability and practicality of the proposed method.\n\n[1] Zongyi Li et al. Fourier neural operator for parametric partial differential equations. ICLR 2021.\n\n[2] Zongyi Li et al. Fourier neural operator with learned deformations for pdes on general geometries. JMLR 2023.\n\n[3] Zijie Li et al. Scalable transformer for pde surrogate modeling. NeurIPS 2023.\n\n[4] Huakun Luo et al. Transolver++: an accurate neural solver for pdes on million-scale geometries. ICML 2025.", "was_revised": true, "final_justification": "I believe this is a paper with high theoretical value, focusing on addressing the issue that FNO and WNO tend to emphasize low-frequency information. By paying attention to high-frequency features, it achieves strong experimental results across a range of benchmarks. I recommend accepting this paper.", "rebuttal": "> **W1: Handling Unstructured Grids**\n\nWe 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.\n\nWe 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.\n\n*batch size 20, single RTX-4090, angles 32*\n\n| **Model**    | **Radon Method** | Transolver      | FNO     | Geo-FNO     |\n| ------------- | -------------- | -------------- | ------- | ----------- |\n| **Relative L₂** | **0.0051**🥇 | 0.0064🥈      |    🚫   | 0.0229 🥉  |\n\n> **W2/Limitations: About real-world and 3D datasets**\n\nWe 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.\n\nWe 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.\n\nAs 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.\n\n| Method    | Error |\n| --------- | ----- |\n| WNO       |0.0155  🥈|\n| RNO (Ours) |0.0121 🥇 |\n\n> **Q1: About sinogram domain convolution**\n\nWe 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.\n\nThe 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.\n\n> **Q2: About ablation experiments**\n\nWe 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.\n\n*Darcy 85×85 grid, batch size 2, single RTX-4090*\n| Configuration      | Relative L₂  |Memory (GB)|Time (s)\n| ------------------ | ------------ |-----------|-------|\n| RNO w/o P.A.       | 0.0093       |2.610      |19.65  |\n\nAs 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.\n\n*Darcy 43×43 grid, batch size 2, single RTX-4090*\n\nTips: 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.\n| Stage    | **RNO (ours)** | **FNO(without GPU optimization)**|\n| --------------------------- | ------- | -------------- |\n| **Training time per (s/epoch)** |32.87   | 196.79          |\n| **Inference (s)**    |3.02    |  14.17         |\n\n[1]Fourier neural operator with learned deformations for pdes on general geometries, JMLR 2023\n\n[2]Geometry-Informed Neural Operator for Large-Scale 3D PDEs, Neurips 2023\n\n[3]Scalable transformer for pde surrogate modeling, NeurIPS 2023\n\n[4]Transolver++: an accurate neural solver for pdes on million-scale geometries, ICML 2025\n\n[5]Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems, COMPUT METHOD APPL M"}]

Author rebuttal:

Meta-review: This paper introduces RNO (Radon Neural Operator), that uses an interesting parametrized Radon transform to replace DFT. Please update the comparison with the state-of-the-art (such as Transolver) and other suggested changes in the discussion.

Personally, regarding the statement "RT has less popularity enjoyed versus FT" in the paper, I am quite disappointed that the authors did not construct any new dataset related to inverse problems in imaging/tomography that actually uses Radon Transform. RT has its niche usage in this field and is much more popular than FT when the features are provided on hyperplanes.

Suggested revisions that reviewers did not mention:
- Typo in Figure 1: "Gloabal".
- The term "Promotion" sounds non-idiomatic, it should be "Improvement".
- The letter $W$ is used to denote the width, while $W$ also denotes the linear layer used in the NO. Please change this.
- Please state clearly where in the formula the physics-attention block is used (instead of just using a figure).
- Radon transform is an isometry is long known, no need to prove it. To fit the context, it would be interesting to prove the nonlinear version in which $R$ is composed with the MLP angular weights.
Decision: Accept (poster)