Tucker-FNO: Tensor Tucker-Fourier Neural Operator and its Universal Approximation Theory
Authors: Guancheng Zhou
Zelin Zeng
Yisi Luo
Qi Xie
Deyu Meng
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Operator
Implicit Neural Representation
Functional Tensor Decomposition
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose Tucker-FNO, an efficient neural operator utilizing decomposition, and further prove its universal approximation theorem.

Abstract: Fourier neural operator (FNO) has demonstrated substantial potential in learning mappings between function spaces, such as numerical partial differential equations (PDEs). However, FNO may suffer from inefficiencies when applied to large-scale, high-dimensional function spaces due to the computational overhead associated with high-dimensional Fourier and convolution operators. In this work, we introduce the Tucker-FNO, an efficient neural operator that decomposes the high-dimensional FNO into a series of 1-dimensional FNOs through Tucker decomposition, thereby significantly reducing computational complexity while maintaining expressiveness. Especially, by using the theoretical tools of functional decomposition in Sobolev space, we rigorously establish the universal approximation theorem of Tucker-FNO. Experiments on high-dimensional numerical PDEs such as Navier-Stokes, Plasticity, and Burger's equations show that Tucker-FNO achieves substantial improvement in execution time and performance over FNO. Moreover, by virtue of the compact Tucker decomposition, Tucker-FNO generalizes seamlessly to high-dimensional visual signals by learning mappings from the positional encoding space to the signal's implicit neural representations (INRs). Under this operator INR framework, Tucker-FNO gains consistent improvements on continuous signal restoration over traditional INR methods in terms of efficiency and accuracy.

Published as a conference paper at ICLR 2026
TUCKER-FNO: TENSOR TUCKER-FOURIER NEURAL OPERATOR AND ITS UNIVERSAL APPROXIMATION THEORY
Guancheng Zhou!2: Zelin Zeng' Yisi Luo'! Qi Xie', Deyu Meng!.3t
"i'an an Jiaotong University. *Shanghai Innovation Institute. Macau University of Science and Technology
ABSTRACT
Fourier neural operator (FNO) has demonstrated substantial potential in learn- ing mappings between function spaces, such numerical partial differential equations (PDEs). However, FNO may suffer from inefficiencies when applied large-scale, high-dimensional function spaces due the computational over- head associated with high-dimensional Fourier and convolution operators. In this work, we introduce the Tuc,er-FNO, an efficient neural operator that decom- poses the high-dimensional FNO into series of 1-dimensional FNOs through Tucker decomposition, thereby significantly reducing computational complexity while maintaining expressiveness. Especially, by using the theoretical tools of functional decomposition in Sobolev space, we rigorously establish the univer- sal approximation theorem of Tucker-FNO. Experiments numerical PDEs such as Navier-Stokes, Plasticity, and Burger's equations show that Tucker-FNO achieves substantial improvement in execution time and per- formance over FNO. Moreover, by virtue of the compact Tucker decomposition, Tucker-FNO generalizes seamlessly to high-dimensional visual signals by learn- ing mappings from the positional encoding space to the signal's implicit neu- ral representations (INRs). Under this operator INR framework, Tucker-FNO gains consistent improvements on continuous signal restoration over traditional INR methods in terms of efficiency and accuracy. The code is available at httpss//giiibbcoommmmmmmee/////e
high- lige--imensional
INTRODUCTION
Neural operator (NO) methods have demonstrated great potential in diverse fields of science al. 2020) and engineering (Pal al. 2024p. Compared with traditional neural network methods,
Li
which learn the mapping from signal spaces, NO methods (Li et al., 2021 2023a Li & Ye, (025)
design neural networks learn mappings between function spaces. By discretely sampling a finite number of observations, the function space can summarize information in tensor form, enabling the
implementation of NO methods in a discretized version. Equipped with the universal approximation theorem (Kovachki et al.) 2202I). NO methods can be applied to variety of tasks, including fast
solution of partial differential equations (PDEs) (Li et al. 2021) and signal restoration (Pal et al.
2024
Liu & lang 2025).
As representative examples, PDEs describe the phenomena in physics, engineering, and other fields from mathematical perspective. Traditional numerical solvers (e.g., finite element methods and finite difference methods) face the challenge of balancing solving speed and accuracy. In contrast, to obtain the PDE solution for greatly accelerates the solving speed. The universal approximation theorem of neural operators (Li al. 202I) further guarantees the performance of NO methods learning PDE solutions.
condition, NO only needs run the neural networks once, which
Another example is the implicit neural representation (INR) using NOs. Specifically, (Pal et al.
2024] has shown the ability of NO in signal restoration by leveraging NO-based INR, uncovering
"Equal contribution.
Corresponding authors.

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Published as a conference paper at ICLR 2026
the potential of NO for general signal processing. The INR methods parameterize data through learning the mappings from the coordinates to the value of the signal, which can be considered as coordinates-valus transform. NO-based INR methods learn the mapping between two function spaces, which are related to the coordinate space and signal space. By learning data representations
in such a function space transformation, the NO-based INR (Pal et al. (024) can be effectively trained and evaluated, and can uniquely control the spatial interpolation behavior explicitly.
Traditional NOs based on data-domain integral operators may lack the efficiency comparable that
of traditional neural networks. Fourier neural operators (FNOs) (Li et al. 2021) address this by learning mappings in Fourier space, showing promise for efficient function mappings. However, the Fourier transform is still computationally expensive in high dimensions, with complexity reaching Od log n²²) for 3-dimensional PDEs on an n x n X n grid and dy-ddiensiooall latent space. This cost makes FNO relatively inefficient for high-dimensional PDEs. Furthermore, both conventional NOs and FNOs suffer from the curse of dimensionality for representing high-dimensional visual data, which may limit their efficiency in related applications using INR (Pal et al. 2024).
Low-rank decomposition can effectively reduce model dimensions and enhance computational and parameter efficiency (Kolda & Bader 2009a; Luo et al. 2024). Motivated by the compactness of low-rank decomposition, we propose Tucker-FNO, a novel NO that leverages Tucker decomposition to enhance operator learning. Especially, we extend the recent functional tensor decomposition (Luo et al. 2024) to the operator learning level for the first time, and decompose a high-dimensional FNO into series of 1-dimensional FNOs in the Tucker format. This splits the high-dimensional Fourier transform into multiple 1 I-dimensional transforms, reducing computational complexity from O log n²*) (3d,,n log for 3-dimensional PDEs. Furthermore, using the Stone-Weit Weierstrasss and universal approximation theorems, we formally derive a tensor product decomposition of high- dimensional functions and prove the universal approximation capability of Tucker-FNO the first such result for tensor- mensor-decompseed NO. The contributions of this work are as follows:
We propose a novel Tucker-FNO method. Different from traditional FNO methods, our methods utilize Tucker tensor decomposition to decompose the high-dimensionall FNO into multiple 1-dimensional FNOs, which accelerates the FNO through operator decomposition.
By utilizing theoretical tools of functional decomposition, establish the universal ap- proximation theorem for the decompasition--aat FNO, which provides a theoretical guar- antec for the representation ability of Tucker-FNO.
Extensive experiments demonstrate the superior performance and efficiency of our methods for numerical PDE solutions. We also extend the Tucker-FNO to high- -igh-dimensional signal restoration tasks. For signal restoration tasks, our method shows performance that exceeds that of traditional INRs and similar NO-based INR.
2 RELATED WORK
2.1 NEURAL OPERATORS
Operator learning constructs mapping from parameter function spaces to solution spaces, and is rep- resented discretely on grids via finite observations in practice (Lu et al., 2021 Raissi et al. 2019]
Mishra & Molinaro| 2023 Kovachki al.) 2023). While conventional neural operators mesh- dependent and require retraining for different resolutions, Li et al. (2020) introduced a resolution- invariant graph neural operator (GNO), later improved via fast Fourier transforms (FFT) construct the Fourier neural operator (Li et al. 202T). Subsequent variants like Geo-FNO (Li et al. 2023a) handle arbitrary geometries, F-FNO Tran et al. (023) uses separable spectral layers reduce pa- rameters and enable deeper architectures, and T-FNO (Kossaifi et al. (024)) decompose the weights in FNO to reduce the number of parameters. While these methods improve model performance and reduce parameter number, they incur additional computational overhead. Recently, D-FNO (L & Yel 2025) observes that FFT calculations dominate the overall computational complexity in FNO and introduces decomposition to mitigate this cost in FNO. However, such decomposition may compromise the representational capacity of FNO (e.g.. its universal approximation property), potentially leading to performance degradation. Our proposed Tucker-FNG applies tensor Tucker decomposition to decompose the high-dimensional FNO, replacing high-dimensional FFT with mul- tiple I-dimensional FFTss while preserving expressiveness (in most cases, enhancing performance).
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Especially, theoretically demonstrate that Tucker-FNO still possesses the universal approxima- tion theorem.
2.2 IMPLICIT NEURAL REPRESENTATION
Implicit neural representations have garnered great success in continuously representing signals across various tasks and modalities, such as signal restoration (Chen & WWang 2022 Shi et al.
2024
Liu et al., 2024 Saragadam et al. 2022
2023: Luo et al.] 2024). generation (Skorokhodov
al. 2021, Chen & Zhangg 2019), and object detection (Zheng et al. 2024). NeRF (Mildenhall
et al.
2020 and subsequent works (Feng et al. 2024 Takikawa et al. [2021] Yariv et al. [021]
leveraged INRs to predict RGB values and density for any given coordinates and view directions
within a 3-dimensional scene. To mitigate the computational overhead associated with traditional
INR methods, incorporating tensor decomposition based approaches (Luo al.) 2024) can yield
superior and more robust results. Recent research (Pal et al. (024)) studies the potential of NO in signal restoration tasks through combining NO and INR. Specifically, OINR (Pal et al. (022)) learns mapping from the positional encoding space to the signal's 's INR space, which improves the repre- sentation ability and can effectively handle downstream signal processing tasks such as denoising.
3 PROPOSED METHODS
3.1 NOTATION AND PRELIMINARY
Scalars, vectors, matrices, tensors, sets and operators are denoted by fonts z, x, X, X, A, and g, respectively. The i-th element of vector x is denoted by xl(a). and it is similar for matrices and tensors, i... X (11 and X (2,, The unfolding operator of tensor XER" XX2 along the d-th mode (d 1, 2, 2........ .N) is defined as unfolda(-) R=1xxxXN which returns the unfolding matrix along the mode d, and the unfolding matrix is denoted by (d) : unfoldu(X). folda(-) denotes the inverse operator of unfolda(-). The mode-d (d = 1, ,N) tensor-matrix product defined as X Xd A := fold, (AX(())) (d)), which re- turns tensor. T R is the periodic torus, identified with [0, 2nr. The Tucker rank of tensor X' E R" XxN is a vector defined as ranky(X) == prank( [ (X))... ... rran (N)] (Kolda
R
& Bader 2009b).
Lemma 1 (Tensor Tucker decomposition (Kolda Bader 2009b)). Let X' E RT XX2X XAN be tensor. the Tucker rank of X is "i....,TN]] then there exist N factor matrices U1 UN E TR KrN and a core tensor E prixxax--XN such that
RRix xM1 U2 E R22xre
X' = x1 U X2 U2 X3 ... XN UN,
(1)
which is called the tensor Tucker decomposition and can be written in an equivalent form as
X X(( = X1 U1(Gn X2 ... XN U UN|(SN))
3.2 THE SETTING OF FNO
The operator is a mapping from two infinite-dimensional function spaces, i.e., S ::A. + U where A C HH HT(T,, and U C (TT, Rd)) are two function spaces. The representation of these function spaces specifically depends on the particular problem. In general, the solution problem is a function u, which satisfies constraint of the PDE form with condition a, and there exists an operator to learn that maps from condition u to the target function u. Specifically, we consider PDE problem
PDE
(x) O, uxx)=
X DE
x x OD,
where u E HTd, Rdd.) ) is the target solution function, La is a differential operator which depends on the condition function a E (T; Rd),, ). D C T denotes the physical domain, and ØD C D denotes the boundary condition domain. Following the previous work (Kovachki al. 2021
assume that d, = d, 1. Otherwise, we can re-formulate the target function a T de and T Rdd r(x,r) u(x)](r), which satisfies da d, 1. The goal of operator
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Tecler-FINO
Layer
Layer!
X
Layer
Layer!
Layer
Iopurt
Layer
Layer
Layer
Ouquat functionniion
Pre-Lifiig
Projectoo
Port-Prajection
X,
MAM
EXP
AXYXT
nxr
Fighnweight 1-D END loyer
Figure I: The architecture of the proposed Tucker-FNO. Given sample from the condition function, the pre-lifting module extracts the imput into series of factor inputs. Then, the inputs are processed by the factor I-dimensional FNOs, and the outputs are aggregated by Tucker decomposition. Note that the last post-projection does not influence the universal approximation.
learning is to learn an operator & equation B
that can efficiently approximate the PDE solution in
NO methods (Kovachki et al. 2023 Li al. 2021 construct map iteratively from to
9 (a) (22 L o L' (a).
where denotes the function composition, L is the number of iterations, 9 is the lifting operator that maps the input to the dy-dimensional latent representation, _e(1 is I-th non-linear operator layer, and 2 is the projection operator that maps the d, d,-dimensional latent representation to the du di-dimensional output. For FNO, the non- non-linear operator layer L( are defined as
(1)(( o ((x)) (x) F- E (b)(kk (K)))))
(5)
where & and 50 denote the Fourier transform and the inverse Fourier transform (see Lemma in Appendix B ( E Rd, (()) define a point-wise affine mapping, E 2d denotes the non-local linear mapping via
frequency, and R zd xd. defines coefficients Fourier transform. For an efficient implementation, FNO keeps only the top-k Fourier frequency for each dimension. Specifically, we only consider the frequency in (k 2d k((s) S (i)x 1,2, .d).
To learn operators numerically, NO-based methods assume access only to point- fhint-wise evaluations to work with and u numerically, and utilize grid discrete sample I and on functions a and u to represent the function information, i.e.,
I : {a(x): x S), O = ((i
(6)
j.ddd) A. 2n/nk, js E f0,1, ,n,,, 1,2,
The sample set S is ITd (n, + -point discretization of the domain T Therefore, we have obser- vation I and O for finite collection of input-output pairs. Therefore, the operator learning task can be formulated discretely as learning an operator / satisfying O = 9 (I). The Fourier transform is realized through FFT, which is discrete Fourier transform. The discrete Fourier transform is com- puted efficiently via FFT. For a 3-dimensional PDE estimation problem of sample size n X n X n, the complexity of the Fourier transform is O(d,n" log(n))), which relatively large w.r.t. n.
3.3 THE SETTING OF TUCKER-FNO
Inspired by the tensor Tucker decomposition in Lemma mated by a functional tensor Tucker decomposition.
continuous function can be approxi-
Theorem (Universal approximation theorem of functional tensor Tucker decomposition). Let a
Ω R be continuous function defined compact subset Ω [$1, (4) x...X (8d, Lu] C Rdd For
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any O, there exists core tensor C R and continuous functions 9s : [s,,t]] - R", such that
xen fx() C X1 (x((()) x xxa <,
sup
where > 0 is the Tucker rank.
Sketch of proof. The detailed proof is proposed in Appendix C From the Stone- Weierstrass theo- rem, the function can be approximated by sufficiently high-order polynomials, and the high-order polynomials can construct the Tucker functional tensor decomposition.
The above theorem proves that an arbitrary continuous function can be approximated by the func- tional tensor Tucker decomposition arbitrary precision. Therefore, we can estimate the target function u by
u(x) = x 91 (xl()) X2. X (a())
(8)
and the operator learning task transforms from estimating operators u estimating operators on gi- Motivated by this insight, we propose the Tucker-Fourier Neural Operator (Tucker-FNO), which decomposes the nigh-dimensional FNO into series of 1-dimensional FNOs via tensor Tucker decomposition. Specifically, we define the Tucker-FNO N : + u as:
(a) = x1 (x) (1)) x X (a(a)(xx((()
where C E Rn X (ri,... ,ra) N is the rank of Tucker-FNO, and M3 is the I-dimensional
FNO described in equation S
Since we decompose the d-dimensional FNO into d 1-dimensional FNOs, correspondingly construct a pre-lifting module, which transforms the d-dimensional representative input tensor
T( (2) == 2m. 2 E R X-na into a series of 1-dimensional representa- tive factor matrices I, Motivated by D-FNO Li & Ye 2025). the I-dimensional representative factor matrix I, can be extracted linearly through pre-lifting operator 3. expressed
nu
(II)=(II...-- Ia), where unfold
(10)
where W. E R [llaymyyxd xd. is the pre-lifting weight, dg is the latent dimension, and I, Raxxd. is the i-th factor matrix. Therefore, each representative factor matrix I, contains partial information of the function 2 after the Tucker-FNO, where 2 utilizes the structure of multi-layer perceptron (MLP) for the discrete setting. In general, we set its input dimension as d,. Under discrete numerical computing, the Tucker-FNO equation can be expressed as
To enhance the representability of Tucker-FNO,
empoly a post-projectio
O= 2(C X1 (((11) X2 X XaLa)
(11)
where each Ny is a 1-dimensional FNO. this way, we effectively decompose the d-dimensional FNO into series of 1-dimensional FNOs by Tucker decomposition.
3.4 THE UNIVERSAL APPROXIMATION THEOREM FOR TUCKER-FNO
The classical FNO (Kovachki et al. 2021 enjoys the universal approximation theorem for any con- tinuous operator learning, as stated below.
2021 Let W Lemma 2 (Universal approximation theorem of FNO (Kovachki et al.
H (T,, Idda) + H² (Td, Igd..) be continuous operator. Let F C HT Rdd be compact sub- set. Then for any e O, there exists FNO N Ha (Td, (Td, ) Hd (Td, Rdd of the form equa- tion which is continuous as an operator H H² such that sup/er (1 Mf S.
Similarly, we aim to demonstrate that Tucker-FNO also possesses universal approximation capabili- ties. That is, we show that Tucker-FNO can approximate a broad class of nonlinear operators con- tinuous function spaces. The overview of our proof is presented in Appendix A Especially, we first establish the universal approximation property for the operator Tucker decomposition, Specifically, according to Theorem [ multivariate function can be decomposed into series of 1-dimensional factor functions using functional tensor Tucker decomposition. This decomposition can be further extended to the operator level, stated below.
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Theorem 2 (Universal approximation theorem of operator Tucker decomposition). Let H S HT(TR)) R) be a compact set of contimuous functions with s > d/2. Assume that all E H are analytic T Then, for every h H and any >> O, there exists ranks (ri, .ra) N continuous operators 44 HH H (TT; and core tensor C &RRxx xRrxx such that
sup [[h(-) C x 24h(11) X2 Xa hCH
Sc.
(12)
Sketch of proof. The detailed proof is placed in Appendix E By Theorem | and the finite- dimensional function space constructed by the Fourier basis, the function in the finite-dimensional function space can be approximated by the Tucker decomposition form (Lemma [0 in Appendix D). Therefore, the nifnite-dimensional function space can be further approximated by the finite- dimensional function space. Through the above approximation and Lemma I0 the proof is done.
In Theorem the factor operator V4 can be further approximated by a I-dimensional FNO using Lemma 2 and thus the universal approximation theorem of Tucker-FNo equation iis established.
Theorem (Universal approximation theorem of Tucker-FNO). Let H (TT
H HTTTT (T²,R)
be a continuous operator. Suppose F C H (TT, IR) be a compact set and H == ((F). Assume that 8, d/2 and that all h H are analytic on Tr. Then for all > O, there exists rank (r1, ... .ra) E N4, Tucker-FNO N F - H of the form equation and a core tensor C E R Xrr,, such that
sup
( FEF
(13)
The detailed proof of Theorem 3 is proposed Appendix Thus, we demonstrate that Tucker- FNOs are universal approximators, capable of approximating any continuous operator to desired accuracy. is important to note that, owing to the universal approximation properties of MLP (Rosenblatt 1958). , the inclusion of the post-projection 2 in equation universal approximation theorem of Tucker-FNO.
does not influence the
3.5 TUCKER-FNO SIGNAL RESTORATION
Following OINR (Pll et al. 2024). which demonstrates the potential of NO in signal processing through NO-based INR, we further introduce Tucker-FNO to nigh-dimmnssonal signal restoration tasks. Specifically, the natural signal is continuous and can be considered as a continuous function in signal space H (T,, TR). Traditional INRs learn a map from the coordinate positional encoding to the signal space, and operator- based INRs learn operator ạ from the input function space to signal function space H. where F, E ( I) and 52 is the domain of definition (c.g.. 2-dimensional plane for images, and a 3-dimensional cube for volumes). Following OINR, we define our input function in terms sinusoidal positional encodings (Mildenhall et al. 2020) which involve sinusoids across many frequencies. Specifically, for a 2-dimensional plane for images, define the input space F as
F = (fs(z. S>>8200 0}.
(14)
cos((2 .. E (0, 1.... SS111 1}, where where fs(x, y) = (si R.), cos( (2' (2 sin (2' S denotes the encoding length. Therefore, the operator-based INR can be defined as:
h(x) = (x),,
(15)
where f E F is the input function, E H is the signal function and & is the operator mapping from F to H. For Tucker-FNO, the d-dimensional sinusoidal positional encodings can be reformulated as
fs fx(x x (2), xl(2)) = [f ( (z) (2)) .ag....ffggx(/le fx(()))))
(16)
where fz(x) [sin (2 R.), cos(2' ..... 1.1 (0. 1.... S 1}. Then, the sinusoidal positional encoding can be decomposed across each dimension, which similar to Tucker decomposition. Therefore, thanks to the decomposition property, Tucker-FNO can remove the pre-lifting operator and be simplified to
fffs] = - (C x1 (ff ... U.ff)).
(17)
Similar to Theorem 3 the Tucker-FNO here enjoys universal approximation ability to learn any high-dimensional signal structures (e.g., image signals).

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Table 1: The comparisons of computational complexity and parameter number for FFT-related cal- culations. dy is the latent dimension, & is the frequency truncation threshold, is the input size, is the maximum of decomposition rank.
Components
FNO
T-FNO
Tucker-PNO
FFT
O(d, log(n))
Old, dn" log(n)) O(fd. Old, (ng())
Oduddo log(n)) O(df,k)
Multiplying Weights
O(dex=))
Inverse FFT
Old, log(n))
O(d,dn log(m))
Params
O(d.t*)
O(Fat2 + f(du + k))
O(d,ak)
3.6 COMPUTATIONAL COMPLEXITY ANALYSIS
By decomposing the d-dimensional FNO into d' 1-dimensional FNOs, Tucker-FNO can efficiently process the high-dimensional data. Although recent tensorized FNOs (Kossaifi et al. 2024 Tran et al. 2023) also incorporate the tensor decomposition to improve FNO by decomposing the model weight to reduce the model parameter, it leads to higher computational complexity. As noted in a recent study (Li & Ye 2025), FFT calculations dominate the overall computational complexity of FNO, indicating that the key to accelerating FNO lies in reducing the complexity of FFT-related calculations. In FNO, the FFT-related calculations includes FFT, multiplying weights, and inverse FFT. In the following analysis, we evaluate the computational complexity of FNO based on these components, using d-dimensional grid of putational complexity. The complexity comparisons among FNO, T-FNO (Kossaffi et al. (2024). recent tensorized FNO), and Tucker-FNO are as follows:
n as the example to calculate the com-
FNO. The computational complexity for FFT and inverse FFT is Oldon" log(re)) = ddd log(n)), where d, is the latent dimension of FNO, and the computational com- plexity for multiplying weights is ((d where A is the frequency truncation threshold.
T-FNO. T-FNO decomposes the model weight R Rex xd. where k is the fre- quency truncation threshold and de is the latent dimension. Specifically, the weight R can be decomposed
R- = x1 U1 X2 .... Xd+2 Ud+2, where C E IR"1 x x..xxxxxpp xPa+2 is the core tensor, (f1, ; rd+2) is the rank, and U; E RRxxss (for i I d) or U, E Rddexrs (for >> d) are the factor matrices. This weight decomposition does not influence the com- putational complexity, i.e., the computational complexity for FFT and inverse FFT is still (d, doggd)Log)())) For multiplying weights, the computational complexity for multiplying weights increase to Of + d2,k)) O(fdhh4), where f max(r1,".. (d+22 is the maximum of rank, and ((f is the computational complexity for the decomposition of R. Due to the computation for the the decomposition of R, T-FNO incurs increased computational complexity.
Tucker-FNO. By decomposing the d-dimensional FNO into series of 1-dimensional FNO, Tucker-FNO processes 1-dimensional FFT and inverse FFT for d times, i.e., the computational complexity for FFT and inverse FFT is reduced to On, log(n)). Simi- larly, the computational complexity for multiplying weights is reduced to OdL.kk), where k is the frequency truncation threshold.
For clarity, the computational complexities of FNO, T-FNO, and Tucker-FNO are summarized in Ta Notably, Tucker-FNO significantly reduces the computational burden associated with FFT- related operations. Furthermore, in Tucker-FNO, decomposition is performed after all FFT-related computations, with a complexity of O(fd,n"). where f is the maximum of rank. While this com- plexity may appear to grow rapidly, it only becomes dominant when the rank is extremely large, which is validated in Appendix [ Beyond computational efficiency, Tucker-FNO also reduces the number of model parameters, as shown by the weight scales reported in Table []
Remark. In summary, although both T-FNO and Tucker-FNO employ tensor decomposition, their implementations and motivations differ. T-FNO applies decomposition to the network parameters to reduce the parameter number, introducing an paraneterization scheme for FNO. In contrast, Tucker- FNO applies decomposition directly the operator, aiming to reduce computational complexity. Thus, T-FNO focuses on parameter efficiency, whereas Bucker-FNO proposes a novel FNO-based paradigm centered on computational efficiency.
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ResNet
FNO
Com-FNO
D-FNO
Tucker-FNO
Ground-
Figure 2: Error maps comparisons of different NO methods for Navier-Stokes (the first row) and Burger's s equation (the second row). EXPERIMENTS
4.1 EXPERIMENT SETTINGS
Task. To validate the effectiveness and efficiency of Tucker-FNO, we evaluate the performance on the NO-based PDE approximation task and INR-based signal restoration task. For PDE approxima- tion tasks, we compare the proposed Tucker-FNO with multiple neural network architectures and NO-based approximation methods on the Navier-Stokes equation, the Plasticity equation, and the Burger equation. The details of PDE datasets are presented in Appendix G
For signal restoration tasks, we utilize NO-based INR architectures et 2024) and compare proposed Tucker- FNO-based INR in equation equation 17 with multiple INR signal restoration methods and other NO-based INRs on the image signal denoising, inpainting, and 3-dimensional volume representa- tion tasks for different signals, e.g., multi-spectral images (MSIs), nalei-dimensional images, and 3-dimensional volume. For image denoising. the testing data are MSIs from the benchmark CAVE datase Yasuma al., 2010), including Cups, Balloons, Cloth, Toys, Clay, and Fruits. We consider three noisy cases containing Gaussian noise with different noise deviations of 0. 0.15, and 0.2. For image inpainting, testing data consists of two videos (Foreman and Carphone We consider random missing with observed sampling rates (SRs) 0.1. 0.15, and 0.2 for image inpainting, For the 3-dimensional volume representation task, the occupancy volume is sampled 512 512 512 voxel grid following OINR (Pal et al.) 2024)), where each voxel within the volume is assigned a value of inside an object and outside an object.
Benchmark. For PDE approximation tasks, we compare our methods with ResNet18 (He et al. 2016), FNO Li et al. 2021), Com-FNO (Li al. 2024). T-FNO (Kossaifi et al. 2024), D-FNO (Li & Ye 2025). For signal restoration tasks, we compare methods with sinusoidal representation networks (SIREN) (Sitzmann et al., 2020), multiplicative filter networks (MFN) (Fathony et al, 2020 low-rank tensor function representation (LRTFR) (Luo et al. 2024). OINR Pal et al. 2024)
and FNO, which replaces NO with FNO in OINR.
Metric. For the PDE approximation task, following F-FNO Tran et al. 2023 the results quan- titatively evaluated by normalized root mean square error (NMSE) and vorticity correlation error (VCE, - vorticity correlation). For signal restoration tasks, the results quantitatively evalu- ated by peak-signal-to-noisse ratio (PSNR), structural similarity (SSIM), and NMSE
Table 2: Performance (NMSE U VCE L) on Navier-Stokes, Burger, and Plasticity equations among different methods. The best and second best values are highlighted.
Stvier-Stokes
Config.
Plasticity
Burger
20
30
Metric
NMSE
VCE
NMSE
УCE
NMSE
VCB
NMSE
VCE
ResNet
0.2582
0.0351
0,1015
0.0587
0.0941
0.0651
FNO
0.0034
1031Ile55
0.0072
3.9776e-5
0,0080
1.9514e-4
0.0087
4.4465e-5
Com-FNO
0.0036
1.1265e-5
0.0071
3.8146e-5
0.0084
2.0897c-4
0.0063
2.4543e-5
T-FNO
0.0035
8.8214c-6
0.0065
3.0354c-5
0.0079
2.0076c-4
0.0075
2.9672e-5
D-FNO
0.0063
3.0994e-5
0.0132
9.14266-5
0.0097
2.2150e-1
0.0071
2.8610e-5
2.9742e-5
2.0050e-5
4.2 TUCKER-FNT FNO IN PDE APPROXIMATION TASK
Setup. The details of the settings of the PDEs are presented in Appendix G training for 500 epochs using batch size of 20 using the Adam optimizer Kinggaa
Here, we perform
2014) with
https: htt::://wwwwcoo cs. co uumbi http://waw..a../aaaaaaaaaee///// tisetttral)
http: /ttp:///tace asu wau.edu/yuy
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Table 4: The average quantitative results (PSNR 1 (dB), SSIM t. MSE 1) by different methods for multi-spectral image inpainting. The best and second-best values are highlighted.
Sampling rate
0.15
0.2
Data
Metric
PSNR
SSIM
NMSE
PSNR
SSIM
NMSE
PSNR
SSIM
NMSE
Observed
5.20
0.023
0.949
5.45
0.035
0.922
5.71
0.047
0.895
SIREN
0.109
0.105
Foreman
MPN
22.47
0,778
0.132
24.16
0.801
0.108
26.20
0.828
0.083
Caryhone
LRTTR
26.96
0.813
0.071
28.75
0.870
0.064
30.22
0.903
0.056
(144x|766x100 76x(000)
OINR
23.32
0.738
0.108
24.04
0.791
0.095
25.17
0.840
0.083
OINR-FNO
24.53
0.803
0,080
25.02
0.837
0.076
26.27
0.857
0.072
Tucker-FNO
0.074
0.047
PSNR 3.61
PSNR 23.26
PSNR 21.58
PSNR 26.24
PSNR 21.36
PSNR 24.06
PSNR 27.54
PSNR Inf
PSNR 7,05 Observed
PSNR 25.27 SIREN
PSNR 25.86 MFN
PSNR 28.11 LRTFR
PSNR 25.8 OINR
PSNR 25.38 OINR-FNO
PSNR 30.17 Tucker-PNO
PSNR Inf Original
Figure 3: The results of multi-dimennsioal image inpainting by different methods on video Foreman (SR=0.1) and Carphone (SR=0.15).
learning rate of 0.001. The latent dimension de is set 32, and the Tucker rank of Tucker-FNO is set to the same as the latent dimension. The Tucker rank rate for T-FNO is 0.4, which indicates that the Tucker rank of each dimension is 0.4 of the size of the dimension. The Fourier frequency truncation threshold is kmax (12, 12, ...) for each dimension. The depth of NO-based methods is 4. And the post-proeection 2 is 2-layer MLP.
Table 3: Comparisons (Params J. Time/iter + (seconds)) of parameter number and running time in 256 256 data.
Result summary. The results of PDE approximation task are Tucker-FNO outperforms the
shown in Table and Fig. baselines FNO, T-FNO and D-FNO, achieving the best perfor- mance. From the Fig. 2 the comparisons of the error maps on Navier-Stokes and Burger's equations and the results on the Plasticity equation demonstrate that Tucker-FNC FNO performs better on regions with significant numerical variation. More qualitative results are demonstrated in Appendix H Additionally, the pro- posed Tucker-FNO demonstrates outperforming advantages in ef- ficiency. From Table 3 we can observe that the proposed Tucker-FNO is more efficient than FNO, T-FNO and D-FNO in terms of the parameter number of model parameters and the execution time. In 256 x 256 data, Tucker-FNO reduces the parameter by 50% and the execution time by 48.6% compared FNO. T-FNO reduces the parameters but increases computational complexity. While D-FÑO reduces the computation complexity, it compromises performance as shown in Table 2 In contrast, Tucker-FNO excels in both performance and efficiency.
Method
Params
Timefiter
0.0685
TENO
232.6 K
0.0746
0.0382
TuckerPPNO
293.5 K
0.0352
4.3 TUCKER-FNO IN SIGNAL RESTORATION
Setup.
The rank of Tucker-FNO set (r1, r2, r3) = (n1 n2, [23/2]].
Table 5: Comparisons (Params J. Time/iter + (seconds)) of parameter number and running time in the signal restoration task with 1024 x 1024 x 31 data.
where nas (d = 1, 2, denote the sizes of the observed data. We parameterize each I-dimensional factor mapping as DNN, which consists of one lifting layer with sine activation, two FNO layers with a latent dimension of d, = 32 and a final lin- projection across all experiments. The factor network takes sinusoidal positional encodings described equation [6 with S 10. The post-projection is 2-layer MLP. We the Adam
Mathol
Timefter
OINR-FNO
1.06 0.2322
FNO
553
0.0963

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Table 6: Ablation experiments of latent dimensions and frequency truncation threshold for PDE solving problems in Plasticity equations. The Tucker rank is dy
d,
NMSE
VCE
NMSE
VCE
NMSE
VCE
16
0.0098
2.07e-4
0.0092
2.06e-4 2.00e
2,06e-5 0,0092
64
0.0068
2.45c-4
0,0064
2.04c-4
0.0062
2.00e-4
Table 7: Ablation experiments of Tucker-FNO rank and the latent dimensions for PDE solving
problems in Plasticity equations.
Rank
16
32
64
NMSE
VCE
NMSE
VCE
NMSE
VCE
2.06e
0.0087
1.97e5
32
0.0079
2.03e-4
0.0074
2.00c-4
0.0071
1.78e-4
0.0064
1.87c-4
0.0064
2.00c-4
0.0063
2,04c-4
optimizer with learning rate of 0.001 and weight decay. To determine the optimal frequency truncation threshold kmax = (k. k k) which controls the number of retained Fourier frequencies in
each spatial dimension, we consider range of candidate values for k {6, 12, 18, 24, 30, 363 to balance model expressivity and computational efficiency.
Result summary. The quantitative results of multi-dimensional image inpainting summarized in Table Our proposed Tucker-FNO achieves better performances and surpasses the baselines on video restoration with considerable margin. As illustrated in Fig. 3 Tucker-FNO holds superior edge preservation and smoothness recovery for different videos. The results of image denoising and 3-dimensional volume representation are demonstrated in Appendix H Similar the PDE approximation task, Tucker-FNO also demonstrates advantages in efficiency. From Table [3] the proposed Tucker-FNO achieves significant execution time reduction and parameter reduction in signal restoration as compared with OINR (Pal et al. 2024). The results of Tucker-FNOO for these high-dimensions visual data validate its superior efficiency and applicability.
4.4 ABLATION EXPERIMENTS
Table 6 demonstrates the influence of latent dimension dy and the frequency truncation threshold k, where the rank is the same as dg- Table 7 further examines the influence of the rank and d, while keeping & fixed. From these results, Tucker-FNO's de, rank, and exhibit positive correlation with the Tucker-FNO's performance, and the rank and A demonstrate relatively greater robustness.
PSNK
Figure 4: The PSNR results of Tucker-FNO w.r.t. different d, and k of the network for Fore- man (SR 0.15) inpainting.
In the Carphone video reconstruction task, our method performs robustly across various & and latent dimension d, in Fig. 4 Per- formance initially increases then decreases with these hyperpa-
rameters, peaking at d, 32 and & 24. This pattern arises because excessive parameters lead over-fitting, while moderate width implicitly captures signal low-rankness, facilitating faster and better convergence.
CONCLUSION
We proposed an efficient neural operator based on Tucker decomposition, termed Tucker-FNO, which decomposes the high- nigh-dimensional FNO into multiple I-dimensional FNOs, thereby largely accelerating the classical FNO while maintaining expressiveness. Utilizing theoretical tools of func- tional decomposition, we establish the universal approximation theorem for Tucker-FNO, which provides a theoretical guarantee for the representation ability Tucker-NNO. Extensive experiments demonstrate that Tucker-FNO achieves substantially improved performance in terms of both effec- tiveness and efficiency for the PDE approximation task and the signal restoration task. Moreover, Tucker-FNO has the potential in future research to handle problems. The detailed optimization dynamics and implicit regularization of Tucker-FNi also war- rant further exploration.
PDE approximation
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ACKNOWLEDGMENTS
This work was supported in part by the National Key R&D Program of China (2025YFA101640)) the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (JYB2025XDXM101) the Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (12426105), and the National Natural Science Foundation of China (124B2029 and 62476214).
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A THE SKETCH OF PROOF FOR UNIVERSAL APPROXIMATION THEOREM OF TUCKER-FNO
Preliminary Lemma (Appendix B)
The Existence of Tensor Tucker Decomposition (Lemma
UAT of Functional Tensor Tucker Decomposition (Theorem 1, Appendix C)
UAT of Operator Tucker Decomposition In Finite- Dimensional Space
(Lemma 10, Appendix D)
UAT of Operator Tucker Decomposition
UAT of Tucker-FNO (Theorem 3, Appendix F)
(Theorem Appendix
UAT of FNO
(Lemma 2, Kovachki et al. 2021)
Figure 5: The sketch of proof for the universal approximation theorem of Tucker-FNO. For simplic- ity, the universal approximation theorem abbreviated as UAT.
Our proof follows general approach: starting from the tensor level (Lemma we extend to the function level (Theorem I. then operators in mnite-dimensions function functiioon spaces (Lemmal and finally demonstrate universal approximation for operators in the general function space (Theorem 2)
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Specifically, in Appendix B we demonstrate some preliminary auxiliary lemmas for the following proof. Then, in Appendix C through Stone- Weierstrass theorem, we propose the universal ap- proximation theorem of functional tensor Tucker decomposition. Therefore, in Appendix D further propose the universal approximation theorem of operator Tucker decomposition in finite- dimensional function space. In Appendix [ through the universal approximation theorem of FNO in Koovahhii et al. 2021). we finally propose the universal approximation theorem of Tucker-FNO.
B PRELIMINARY LEMMA
Lemma 3 (Fourier transform and inverse Fourier transform). For any such function IT there exists Fourier Transformer
-27jj L v(x)e ik -k.ddx
{V)( = (2T)
(18)
For k E Zd, the k-th Fourier coefficient of is denoted by uk 9 (k).
The inverse Fourier Transformer defined as:
(0)(x)
Σ 0x
(19)
kEzd
Definition 1 (The H norm). Let h E H C HT(T,) where T = =10,27 U its Fourier series expansion is given by:
h(x)= = Σ cke
Ck =
(2x)d [ h(x)e -(x))-ikxxxxx
1
(20)
kE
where k- = -kk(a)) k(a)) E zd
(0}. The H*-norm of h is
||0|/2
Σ (1+
(21)
kE22
To prove that the combination the Fourier basis can approximate any function HT ((TT,,,) R). we propose Lemma and then propose the approximation lemma of the Fourier basis in Lemma g Lemma 4, For any 2 and Rdd \\ {0), we have
(1 + k(3)" s (
Σ
k 218
(22)
1815
Proof.
According to the binomial theorem, we can expand (1 + |k/2²* follows:
k 2m
(23)
The binomial coefficient (*) achieves its maximum value when m = [s/2] or m
Lt/2], thus we
have
(1 + [k|3}*
[s/21/ km
(24)
m==
Since
d-1 2 for all m 2 0. we have
[s/21) )
m + d -
[s/22) k 2m
k 2m
d
m=0
m==
[s/2])
s
Σ
1)/k|2m 2m
(25)
m=0 0
k 2|,1
[s/2]
1315.
Combining equation 24 and equation 25 the proof is completed.
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Lemma 5. For any 2 1, 0 and k R \ {0}, have
k
2P I (d + k)
(26)
S
Proof. The direct computation is as follows
Σ 3
k
2
1)(k| 2m
m=U
m + d 1)
d - 1 1) |k| 2m
|
m + 1)
|k|
(27)
d d-1
m
k
+ 4)
d
K|/a* s 1)
Therefore, the proof is completed.
Lemma 6. Let h E H C RR, where T = 2T. Its Fourier series expansion is given by:
Oke dxx cke Ok (2 [ M(x)e ik dx,
1(x)= = E
(28)
KEZd
where k (kl())...... k()) and |k| /k2la)
(a)(a)- We aim prove that its Fourier
coefficients Ok satisfy:
lok S Cmilk| Vm N, k zd {0},
(29)
where Cm constant depending only on the m-th derivatives of h.
Proof. Choose j such that [klo)] = maxisiça kko. then [k|c)| 2 [kl/vd. Perform mith inte-
gration by parts along the j-th direction:
(2 L
c
(i)
e -lex dx,
(30)
where x; denote (( here for simplicity. Estimate and take absolute values:
2Th h>x
1107h fll 7
lckl
dx
(31)
klonl
(2rr)d
Ox m
k k(()] m
Using k (3) 2 k///d, we can conclude lox| I Cm/ [k| with Cm fore, the proof is completed.
There-
Lemma 7. For any 2 1, and Rd \ (0). if m 2 d/2, have
xx] |x| 2( < Cm,..an 2mm=)-d/2)
(32)
where Cm.e.d
2ddd
II
Proof. Since |xx| = max;=1..... as follows
| thus in d-dimensional space, we can rewrite ||x|| >
|x||oc > n =
U |x(|| > n, |x|مr| < n for j # i).
(33)
i=1
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By symmetry, the contribution from each region where xx())| >
is identical, specifically
M [x] 2(m 1omm=sdx dx
d
x
2(m-a)
(34)
|xlex|| >m
It is obvious that |x|
>
x (1)) thus the integral can be bounded and directly calculated as follows:
/
d
|x] 2(m dxa dx
1x1 2n
n.
<d
[x]
2(m
8)
dx dxydx1
|x1 Sn.
m
-d(2n) d-1
|x:/>>m [x]
-2(m-
dx dxi
(35)
-2d
X1
-2 dxi
n2 (2m-)))))
-2d
2(m - 5) -
2d d
n1 2(m sd/22) 2(m 8) 31-1
where x; denotes ((( for simplicity. The proof is completed by combining equation 34 and equa- tion 35
Lemma 8 (The approximation lemma of the Fourier basis), Let h H S H* R), where T [0, 2m)d d. Its Fourier series expansion is given by:
Σ akx Ok (2x)2 I. h(x)E
h(x)=
fike-ixxxa -kx dx.
(36)
kEZa
Let the truncated Fourier series be
(x)) = Σ Cke xex
(37)
[|k]l Sn
and define the remainder as Ra, = l((xx( - (x) Thus we can conclude that
lim
| (h))ll = O.
(38)
+00
Proof. We have
R2 (h) = Σ (1+ [k?))///k
||||/s0
8
Σ
Σ
2|
1axl
[s/21/
||k]]l 2m
1915
(39)
[s/21)
(d + 1)a
|k 20 lck"" 22
|| loo >n
C2 (d 1)*
k 2(m-s)
[s/21
||k| Dn where the first inequality follows from Lemma the second inequality follows from Lemma 5 and the last inequality follows from Lemma 6
The series estimate:
ΣbEza [k] 2((m converges if m s + d/2. According Lemma
we further
Σ
k o0
[x] 2(m- < Cm.s.am -2(m-s-d/2)
(40)
|x|l
[k]l >n
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Thus have
( I am (mm-o-/aa) d/2) where Mm, v (1/22) VCm,s, Cm (d 1) For any e O, exists = s + d/2 + [log2 (,,,s,a/ c]], such that
(41)
|R S.
(42)
Therefore, the proof is completed.
L
Lemma 9 (The relation between Loo norm and the He norm), Let 8 > O, and compuct set F H IR) where Ω I& is compact subset. Here, As E F is analytic function Z. There exists A, O, such that
IIfllH < A, sup [f(x)..
(43)
xcS
Proof. From f being analytic in compact set Z2 and the Cauchy incquality, there exist constant C, > that are independent of f. such that
DD f S CM
(44)
Because P compact set, define A = Sn 1dx 0. Then, we have
llflle, S VAlfflo S VACMlolIa|!
(45)
For [|f we have
|| = Σ SI A Σ (CM la!!)?
(46)
lalss
Ja|Sa
Because F a compact set and f is analytic, there exists constant A, O, such that
Σ
(CM))aa))) s fl
(47)
la|s
Therefore, have
||f S VAA.Iff
(48)
The proof is completed.
C PROOF OF UNIVERSAL APPROXIMATION THEOREM OF FUNCTIONAL. TUCKER DECOMPOSITION (THEOREM I
Proof of Theorem Due S2 is a compact set, the Heine-Cantor theorem is valid. So, f being continuous on Ω implies uniform continuity, i.e., for any >> 0, there exists d >> O, such that
x'1l 8 llf(x) f(x')l <.
(49)
Therefore, from the WeieeWWeessssrsss Theorem, for any 0, there exists polynomial P(x)
Ei Era-1 an sa III
x (g), such that
sup - P(x)] xEi
(50)
Define index sets I, (0,1, 1... .r [x x 1: 11"" Therefore, P can be rewritten as
1}, core tensor C T x. and basis functions (x<x)
P(x) (x1) x.... x (xa)..
(51)
Then, from equation 50 we have
sup f(x) x1 gx((()) x xaga(x((a)) <.
(52)
xen
The proof is completed.
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D PROOF OF UNIVERSAL APPROXIMATION THEOREM OF OPERATOR TUCKER DECOMPOSITION IN FINITE-DIMENT DIMENSIONAL FUNCTION SPACE (FOR THEOREM 2)
Definition 2. Let d be the number of variables and n the maximum Fourier degree. The basis functions are @r (x) = ek for all multi-indices k (k) (1)+ kl (2), ... k(.) E 2 with || |oc n, then we can define
A. = span fok (x)]k| Sn). For any coordinate vector == (al(k) ) >n E R the mapping H. RN An (where N (x]l(l [n} -2n 1) ) is defined as
(53)
(x)) = Σ (l())
(54)
where the basis is ordered by ascending total degree, and lexicographically within the same degree. Each component al (k) corresponds uniquely to the basis (xx)).
Lemma 10 (Universal approximation theorem of Tucker operator decomposition finite-dimen- sional function space). Let H S H*(TM, R) be compuct subset with
>> d aand define
n). Assume that all h H are analytic T Then for all h H, e > O, there exists ranks (ri,.:: rr)) E Nd, continuous operators W, An O - (T(T)RD))
An (pan(
and a core lensor C X Xra, such that
[lh(-) C x1 (4((1) X2 Xa Vahl-a)lHH- <e. heh
sup
(55)
Proof. Define the basis function of An is {0i,... ,N, i.e., A, span( NN). From
Theorem for >> O, there exist tucker rank (r1,-.. .ra). the core tensor ((ú) E Brixxxrss, such that
16,(x) -(() X1 91 (x)()) x2... xa I6 (x)a)) S N
(56)
For h E An, we have EN aj di. where (x(xx = e- and a; (b) :=< h, di > continuous for h. Therefore, define
(1)
C: diag((() .... ,((})
(57)
(xx) = 1V at o/ (xla),, a 96 (x((
(h)
have
[h(x) Cx1 21 (1)(((()) X2. xa (2(x((())
N
N
d
(h)
(x(x)
Σ
a
c() x1 96 (x(()) x2. xa gd (x (a))
(b)
N
a
(N)
(b) - -ct,) x1 g (3)
(x((1)) x2. ... Xd 96 (x)(a)))
(58)
N
(h) (xx(x) - -clo) x1
(3) (x(1)) x2 ... x-xx Sa (2) (x (x))))]
1=1
(h)
N
For H is compact set and a (1)) =< h Φb >, there exist constant A > 0, such that a0 A. Therefore, from equation 58 we have
[h(x) C x (h)((((()) X2. xa [x(x((a))1 S Ae,
(59)
i.e., [[h(.) C x1 (.) X2 X, (aa(:a)le loc Ac.
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From Lemma 9 there exists Ag, such that I| A.llflloo Then, we have
sup ||(-)) - C x1 (/h(11) .. Xa U
hcH
A sup [1 - C x1 (1h(11) x.. ... x Ugh|lo
(60)
heH
Ae,
Therefore, the proof is completed.
E PROOF OF UNIVERSAL APPROXIMATION THEOREM OF FUNCTIONAL. TUCKER DECOMPOSITION (THEOREM 2)
Proof of Theorem
2
From Lemma.
for all > there exists n such that
| (h(h)] S 2 E
(61)
where Pn is define in equation 37
From Lemma.
10 there exist continuous operators V4 An H²(TT; and core tensor such that
E Xrrd,
sup (x) -c 26 (x)(())())) X X Uahn (x (x(
(62)
h.E
2
We define U4 := % For (x))))x) = [X)])) c and c = (2n) S h(x)e x dx,
1
we have Ck is continuous for and P. is also continuous for h, i.e., Pn is also continuous operator. Therefore, From the continuity of 24 and Pn. U. Wi o Pn is also continuous operator.
For all H. we have
[[((-) - C X1 24h(11) ... ... Xa
[[((-) - Cx (Prh(()) X2 xa Va (a((nh(-a))ll
(63)
S |/ P- + (b) x, (Pah(-i) x2... Xa (a(Pa,
S
i.c.
sup ||A(-) C 1 (h((-1 X2 Xa Wahl-a)) S.. ACH
(64)
Therefore, we have continuous operators 24 satisfying the lemma.
F PROOF OF UNIVERSAL APPROXIMATION THEOREM OF TUCKER-FNO
(THEOREM 3
Proof of Theorem 3 For 9 is continuous and F is compact subset, H is also compact subset on H. (Td:I) Therefore, from Theorem 2 and the constraint of FS-rank, for all >> 0, there exist 2 H H² (T;RR and the core tensor I Xr. xra satisfying
[[(()) 1 h(1() X2.... X Lah( Sei H.
(65)
Therefore, we define 2, = 2/ a G, such that
[-) - C x1 (1f(1) X2... X Haf (-af(=))ll
f() 1 th f(1) x Xa 2 f(-a)l
(66)
Because Wr are continuous operator define in the compact set F, there exist C, such that
sup fEF
(f)ll $ C.
(67)
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From Lemma 2
there exist continuous FNOs N/; - H (T) such that
[| M(())lH> S V22 > 0.
(68)
From equation 67 we have
sup SEF
||A(f))l c
(69)
Therefore, we have
|C x I X2 ... Xa Vo5 - C X1 MI X2 ... Xd Nafll
< CC 1 V x X. Naf C1 7 X2 ... Xd-1 Va-1f of X Nafl
+|C X1 ff X2 ... X Naf - C X1 W X2 .. 10-11 No-1 x NafllH
+|C 1 Vf X2 42 X2 ... X Naf -C X1 Mf X2 xa NafllH
(70)
|C 1 1f x-. X (Us -a))
+IC X1 ff X2 ... Xd-1 ( - 1 X Naf
+|C 1 (4/ - Mi)f X2 U2 X2 x NafllH
By defining
Til = C U. X2 Xi-1 2-11 Xi41 Nowl Xi42 Xa Naf.
(71)
have
||C 11 24 X2 ... X Uaf Cx Kif ... Xd . Xa < aafaf x (Us -aa) + 11- x - (a--) 1...flll.....11................. +I. x (% M))
(72)
From Sobolev Embedding Theorem and Kato-Ponce Inequality, for f2 H*(T,R"), there exists Sf. > 0, such that,
IIf x fell= S CCnsaa ffllll
(73)
Therefore, there exists Ć > 0, such that
|1 x (24 - Kf)ll S &1I,211l -fller
(74)
For each Tr we have
C X1 2/ f X2 ... X:-1 U-1I X441 Nor I Xi42 Xa Nalllm
201 Wfll 1l ... |1 M441fllH |aaaf
(75)
500'-1 (C+ (2)
sŌ(C+
where Ō > 0 is existed from the analysis from equation 73
Therefore, we have
110 x1 2h x....
... X Waf x1 Kf ..... Na
|| 11 H 24 NrflHH
(76)
< (c) ) d(d
+ 0. Therefore, for all 0, there exist Obviously, when e2 - O, have (d + (2+)))dd)) d(d-1))
continuous FNOs M². H² (Id, RT:), such that
[C x1 24 f X2 ... X Uaf C x1 Nif X2 ... x xa S 2
(77)
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Then, from equation and equation
have
| - X1 (1)(1)) X2 ... x (af)
< [() - x1 f(11) X2 X (af(-a)l
+ +|CC x1 (f((1) X2 .. Xa Waf(:a) x1 (1f((() x xa Naf(-a)l
(78)
- 2 =E
Therefore, we have Tucker-FNO
M()(x) = =- (I x xa Xaf
(79)
such that,
- - -(0) sc.
(80)
The proof is completed.
G THE SETTINGS OF PDE
For PDE approximation task, we evaluate Tucker-FNO in three PDEs, e.g., Navier-Stokes, Plasticity and Burger's 's equations.
The Navier-Stee Stokes equation for viscous, incompressible fluid in vorticity form on the unit torus:
aw(x,t t) 01 + u(x,t) Vw(x, t) = pdw(xx)) f(x), x (0, E [0,T].
(81)
V u , t) = O,
X (0, 1), [0,
w(x, 0) = (x))
E (0,
where u is the velocity field, w = V u the vorticity, we is the initial vorticity and μl is the viscosity coefficient. f is the force term, which is fixed f(x) 0.1 + x22) + cos ( (21 + x2))). The initial vorticity wo follows the Gaussian random field with a covariance decay of 2.5 and scale parameter of 7. The viscosity coefficient is set as le 5. Following FNO (Li al.) 2021),
interested in learning the operator mapping from the vorticity up to time 10 to the vorticity up to some later time T' 10, i.e., y (0.1)xx [0.10 10 + (lo01))xx and we fix the sample resolution as 64 x 64. The data are from (Li et al. 2021). and we utilize 1000 samples for training and 200 samples for testing.
The Plasticity equation is a non-linear PDE for the Plastic forging problem, where block of material impacted by frictionless, rigid die at time The governing equation is the same as the previous example given by
u
==
02
o = C: (c-Ep)
(82)
6 AV)((a)
13 f(a) 1(0)= /3 (r(()) T| -a,
where the PDE detail setting is the same as Geo-FNO(Li et al. 2021) To validate the efficiency and performance of Tucker-FNO, we extend the I-dimensional input to 2-dimensions by duplicating the input to align it with the output, which is different from (Li et al.) 2021) Our dataset is also from (Li et al. 2021) and we utilize 900 samples for training and 80 samples for testing.
The Burger's equation is non-linear PDE model the 1-dimensional flow of viscous fluid, which is formed as
@u(x, 1) + u(x, t)
Ou(x,!) = Pu(a,!) () dr
Or² I E 10, E (0. 1),
Øt
(83)
u(x,, 0) = mo(x),
E (0.
where Ug is the initial condition and ب = 5e is the viscosity coefficient. Following FNO L et al. 2021), solution u, i... 3 uhg + u. The dataset from (Li et al. 202I), and we utilize 100 samples for training, and 20 samples for testing.
interested in the operator mapping from the initial condition un to the target
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H
MORE NUMERICAL EXPERIMENTS
ResNet
FNO
Com-FNO
T-PNO
D-FNO
TuckerFNNO
Ground-Truth
Figure 6: Error map comparisons of different NO methods for the Navier-Stokes equation.
ResNet
FNO
Com-FNO
T-NNO
D-FNO
Tucker-FNO
Ground-Truth
Figure 7: Error map comparisons of different NO methods for the Burger's equation.
ResNet
FNO
Com-FNO
T-FNO
D-FNO
Tocker-FNO
Ground- Truth
Figure 8: Qualitative comparisons of different NO methods for the Plasticity equation.
More results of PDE approximation task. Here, we demonstrate more qualitative results of the Navier-Stokes, Plasticity, and Burger's equation in Fig These qualitative results further demon- strate the superiority of our Tucker-FNO in regions containing high-frequency signals.
More results of large-scale PDE approximation task. To investigate the influence decomposition on FNO for large-scale PDE problems with complex geometry, we replace the FNO in GINO (Li et al. 2023b) with our Tucker-FNO, and evaluate it in Car-CFD task (Li et al. 2023b), which is Navier-Stoke PDE problem with velocity 20m The dataset takes 611 I weight-tight shapes out the 889 instances, and divides the 61 instances into 500 for training and 111 for validation. For Car-CFD task, the goal is estimate the full pressure field given
3 3-dimensional
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FNO
D-FNO
TuckerFNNO
Ground- Truth
Figure 9: Error map comparisons of different NO methods for the Car-CFD. The first demon- strates the results with kmax = (12, 12, 12), and the second row demonstrates the results with kmax (16, 16, 16).
Table 8: The average quantitative results (PSNR 1 (dB), SSIM 1, NMSE 1) by different methods for multi-spectral image denoising. The best and second-bes best values are highlighted.
Noise level
0.1
0.15
0.2
Data
Metric
PSNR SSIM NMSE PSNR SSIM
NMSE PSNR
SSIM
NMSE
Observed
20.00 0.313
0.922
16.48
0.202
0.940
13.98
0.140
0.953
MSIs
SIREN
26.23 0.821
0.173
25.92
0.795
0.180
25.82
0.730
0.185
Ballons
MFN
28.43
0.833
0.151
27.23
0.794
0.163
26.14
0.779
0.170
Fruits
LRTFR
30.60 0.896
0.139
29.51
0.866
0.144
28.63
0.839
0.150
Toys
OINR
30.48 0.880
0.140
29.14
0.861
0.150
28.27
0.821
0.155
(256x256x31)
OINR-FNO
31.19
0.909
0.122
29.40
0.880
0.134
28.94
0.838
0.142
Tucker- FNO
33.41
0.945
0.092
31.63
0.911
0.113
30.30
0.875
0.130
Observed
20.01
0.231
0.932
16.48
0.152
0.950
13.98
0.110
0.967
MSIs
SIREN
28.55
0.862
0.155
27.83
0.834
0.165
27.04
0.795
0.173
Cups
MFN
30.41
0.877
0.138
27.63
0.824
0.165
25.91
0.776
0.184
Clay
LRTFR
32.30
0.939
0.108
31.27
0.911
0.117
30.24
0.864
0.130
Cloth
OINR
33.18
0.930
0.097
31.65
0.909
0.112
30.25
0.869
0.130
(256x256x31)
OINR-FNO
33.80
0.944
0.088
32.28
0.917
0.101
30.73
0.875
0.122
Tucker-FNO
35.89
0.957
0.061
33.79
0.927
0.089
31.52
0.900
0.112
the shape of the vehicle as input, and we input the meshgrid in the surface, which stores 3586 mesh points for each sample. For the GINO framework, it maps the non-grid -rid data into a grid latent space using encoder GNO, and then FNO performs learning. The latent grid space is then mapped back to the original non-grid space through decoder GNO. Here, we perform training using Adam 0.001. Following the configuration of GINO, the latent dimension d, is 32, and the rank of Tucker-FNO is set to the latent dimension. The Fourier frequency truncation threshold is kmax (12, 12, 12) kmax (16, 16, 16). The depth NO- based methods And the post-projection 2 2-layer MIP. The preliminary results under 10 training epochs shown in Table 9 and the qualitative results demonstrated in the Figure 9 These results indicate that Tucker-FNV remains effective and efficient for the large-scale 3-D task.
optimizer (Kingmal 2014) with learning
More results of image denoising. The results of MSI noise removal are shown in Table S and Fig. 10 It can be observed that Tucker-FNe consistently outperforms all baselines across different noise cases and datasets. According to Fig. I0 Tucker-FNO could obtain better visual results than other competing methods. Specifically, Tucker-FNe FNO could attenuate noise well and also preserve the details of the MSI better.
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I
PSNR 16.48
PSNR 22.21
PSNR 26.44
PSNR 27.31
PSNR 28.09
PSNR 29 44
PSNR 30.94
PSNR Inf
PSNR 13.98
PSNR 21.97
PSNR 24.32
PSNR 24.17
PSNR 24.80
PSNR 24.91
PSNR 26.62
PSNR Inf
PSNR 20.01
PSNR 31.34
PSNR 31.35
PSNR 34.77
PSNR 36.09
PSNR 36.15
PSNR 37.66
PSNR Inf
PSNR 16.48 Observed
PSNR 31,28 SIREN
PSNR 27.5 MFN
PSNR 33.09 LRTTR
PSNR 32.82 OINR
PSNR 32.96 OINR FNO
PSNR 34,72
PSNR Inf Original
Tucker FNO
Figure 10: The qualitative results of MSI denoising by different methods on Toys (0.15). Cloth (0.2), Fruits (0.1) and Cups (0.15).
NMSE 069 SIREN
NMSE 0. 147 MFN
NMSE 1,0 LKTFR
NMSE 0.1 OINR
NMSE 0.133 OINR-FNO
NMSE O. 106 Tucker-FNO
NMSE 0 Original
Figure 11: The qualitative results of 3-dimensional volume representation by different methods.
More results 3D 3D volume representation. INRs are commonly used as a continuous representation of 3D volumes or surfaces. Fig. IT shows that Tucker-FNO performs well in NMSE in this case. Tucker-FNO exhibits superior recovery performance and demonstrates a distinct advantage in high- frequency signals (as indicated by the red box Fig.
More results of video inpainting. To further test the scalability of our method on long and high- resolution video from Zhang et al. (2024). The video dataset has a relatively large spatial resolution of 720 x 960 and contains 600 frames in total. In the experiments, lengths to evaluate the video inpainting performance (sampling rate = 0.1) of our method across different data scales. The results, presented in Table I0, show that Tuc holds advantages in terms of both effectiveness and efficiency than the OINR baseline across varying frame lengths. These findings reveal the scalability of Tucker-FNO relatively large-scale data.
extract frames of varying
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Table 9: Performance (Test Error 4. Training Time + (seconds/epoch), Evaluation Time & (sec- onds/epoch), FNO Running Time (seconds/sample)) on Car-CFD. The test error is the de- normalized L2 error here.
Method
Fourier Mode
Test Error
Training Time Evaluation Time FNO Running Time
GINO GINO-D-ENO
12
0.2047
1832.13
327.18
0.0078
12
0.2187
1801.25
309.24
0.0065
GINO-Tucker-NNO
12
0.2031
1792.43
300.77
0.0062
GINO
0.1382
1858.23
337.28
0.0081
GINO-D-FNO
0.1432
1832.54
329.73
0.0070
GINO PNO
16
0.1297
1826.12
326.65
0.0064
Table 10: Comparisons (PSNR 1 (dB), Time/iter + (seconds)) OINR and Tucker-FNO on a long and high-resolution video.
Frames
Methods
PSNR
Time/iter
OINR
39,6516 46.5796
0.0231
100
Tucker-FNO
0.0173
OINR
0,9024
0.0409
300
Tucker-FNO
42.0626
0.0280
OINR
35.4867
0.0702
600
Tucker-FNO
39.7912
0.0342
I MORE COMPARISONS OF EFFICIENCY
Table 11: Comparisons (Params 4. MACS 4. Time/iter (milliseconds)) of parameter number, com- putation complexity, and execution time in the PDE approximation task.
Size
64 x 64
128 x 128
256 x 256
Method
Params
MACs Time/iter Params MACs
Time/iter
Params
MACs
Time/iter
FNO
593.7 K. 4.01 M
5.62
593.7 K 55.6 M
26.19
593.7 K 217.66 M
68.50
D-FNO 234.1 K 3.95
2.16
258.7 K 23.75
15.24
307.8 K 98.77
38.30
Tucker-FNo -NN 256.6 K. 2.44 M
3.15
268.9 K 16.45 M
14.14
293.5 K 80.23 M
35.23
We conduct more experiments to validate the efficiency of Tucker-FNO on the computation cost here. Table 12 demonstrates the parameters of the model, multiply accumulate operations (MACs), and execution time across different scales of data and various tasks. In Table [I] the results demonstrate that Tucker-FNO can significantly reduce both the number of parameters and the computational complexity of the algorithm in the PDE approximation task. As the input scale increases, the advantage of Tucker-FNO grows accordingly. In Table T2 Tucker-FNO presents a more significant improvement in the signal restoration task than the PDE approximation task.
Here, we further test the running time the lifting, FNO processing, decomposition and post- projection separately in FNO and Tucker-FNO in Tablr 14] where d, 16 and k: 12. We set 2-d PDE problem where the size of input is 128 x 128 with batch size 8. To highlight the efficiency difference, we test the running time on the CPU. The results demonstrate that the bottleneck of the decomposition module occurs when the rank is extremely large (> 1024), which far exceeds the rank settings (~ 32) of our method. Consequently, for Tucker-FNO. FFT calculations still dominate the overall computational complexity in most scenarios, only in extreme scenarios (rank > 1024) might the complexity of decomposition become the dominant factor.
J MORE ABLATION EXPERIMENTS
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Table 12: Comparisons (Params J. MACs 4, Time/iter (seconds)) of parameter number and running time in the signal restoration task.
Size
512 x 512 x
768 x 768 x
1024 1024 x
Method
Params
MACs
Time/iter Params MACs
Time/iter Params
MACs
Time/iter
OINR-FNO 1.06 M 3.338 B 0.0525
1.06 M 7.512 B
0.1292
1.06 M
13.354 B
0.2322 0.0963
Tucker-FNo 486 K 0.538 0.0424
519 K 1.211 B
0.0638
553 K
2.152 B
Table 13: Ablation experiments (NMSE 4, VCE L) Tucker-FNO rank for PDE solving problems in Burger's equations.
k
12
16
d,
NMSE
VCE
NMSE
VCE
NMSE
VCE
16
0.0124
8.3804c-5
0,0071
3.1113e-5
0.0069
2.7835e-5
32
0.0086
4.1782e-5
0.0070
2.0050e-5
0.0063
1.1622e-5
64
0.0082
3.8027e-5
0.0060
1.0669e-5
0.0055
9.4175e-5
Here, we conduct more experiments the width d, and Fourier frequency truncation kmax = (k: k...... Table 13 further demonstrates the ablation results in Burger's equation.
Table 15: Performance (NMSE J.
VCE 4) on Plasticity equation.
Method
NMSE
VCE
Similar to the Plasticity equation in Table 6 performance im- proves with both A and d,- Moreover, Fig. [2 demonstrate the influence of k and d, in the inpainting and denoising tasks. The best performance occurs d, 16 32 and k = 18 ~ 24, which further validates the effectiveness of the low-rank structure in Tucker-FNO.
FNO 0.0080
1.9514e-4
D-FNO 0.0097
2.2150e-4
Tucker-FNO
0.0073
2.0010e-4
CP-FNO
0.0082
2.0021e-4
Here, we compare the different tensor decomposition in the Plasticity equation. By applying the methodological paradigm to Canonical Polyadic (CP) decomposition (Kolda & Bader, (009a), we derive variant of Tucker-FNO, termed CP-FNO. Table 15 demonstrates comparisons between Tucker-FNO and CP-FNO. Although the efficiency of CP decomposition is higher than that of Tucker decomposition, it has relatively inferior accuracy. This can be attributed to the stronger representation ability of the Tucker decomposition.
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Table 14: Comparisons (Time + (seconds), Params 4) of running time and parameter number in the PDE approximation task.
Methods
Time Rank Lifting FNO Processing Decomposition Projection
Params
Type
Total Time
FNO
0,0010
0.4057
0.0935
0.5002
593281
Bocker-FNO
0.0008
0.1761
0.0005
0.0423
0.2197
70593
Tucker-FNO
64
0.0011
0.2753
0.0011
0.0597
0.3372
170529
FNO
256
0.0009
0.2736
0.0088
0.0691
0.3551
1307553
Tucker-FNO
1024
0.0010
0.3114
0.1412
0.0648
0.5184
17652129
28
PSNR
27
36
4,
125
6, 16
A
de
Figure 12: The PSNR results of Tucker-FNO w.r.. different parameters and de for Carphone inpainting with sampling rate 0.15 (left), and Clay denoising with noise deviation of 0.15 (right).
27


Reviews:
[{"review_id": "nzJwdA4aVp", "reviewer": "Reviewer_ofCR", "summary": "This paper proposes Tucker-FNO, a tensor-decomposed variant of the Fourier Neural Operator (FNO) designed to improve efficiency and scalability in high-dimensional operator learning. The main idea is to employ Tucker decomposition to represent a high-dimensional Fourier operator as a composition of a low-dimensional core tensor and multiple independent 1D FNOs. This formulation substantially reduces computational costs, particularly the cubic scaling of high-dimensional FFTs, while maintaining expressive power. In addition to the algorithmic design, the paper establishes a universal approximation theorem (UAT) for Tucker-FNO using tools from functional analysis in Sobolev spaces, providing a theoretical guarantee for its representational capacity. Empirical results on several PDE benchmarks, including Navier–Stokes, Burgers’, and Plasticity equations, as well as continuous signal learning tasks, show that Tucker-FNO achieves improved efficiency and accuracy over standard FNOs.", "questions": "- Can the Tucker ranks be learned or dynamically adapted during training, rather than being fixed a priori?\n- How does Tucker-FNO behave in genuinely high-dimensional PDEs (e.g., 5D or higher)? Are there benchmark results or computational analyses for such settings?\n- For large Tucker ranks, does the core tensor introduce new bottlenecks that offset the decomposition’s computational gains?\n- In the universal approximation theorem, can the approximation error be explicitly related to Tucker rank or decomposition depth?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper’s strength lies in its clear and well-motivated attempt to address one of the key limitations of FNOs—their inefficiency in high-dimensional function spaces. By leveraging Tucker decomposition, the proposed model reduces both memory and computational complexity while preserving the operator’s structural expressiveness. This approach is both elegant and practical, as it connects classical tensor factorization with modern neural operator design.\nThe inclusion of a rigorous universal approximation theorem significantly strengthens the contribution by grounding the empirical observations in a solid theoretical foundation. This theorem not only supports Tucker-FNO’s representational power but also provides a framework for future theoretical extensions.\nExperimentally, Tucker-FNO demonstrates consistent improvement in speed and accuracy compared to baseline FNOs, with notably lower FFT costs in three-dimensional settings. The paper also extends Tucker-FNO to continuous signal learning via implicit neural representations (INRs), showing that the proposed decomposition generalizes well to non-PDE tasks. Overall, the method provides both practical computational benefits and theoretical depth, positioning it as a meaningful advance in scalable operator learning.", "weaknesses": "While the paper makes a valuable contribution, there are several limitations that should be addressed. First, although the method is advertised as being effective for high-dimensional settings, most experiments are conducted on relatively low-dimensional PDEs (2D or 3D). The current results therefore do not convincingly demonstrate scalability to genuinely high-dimensional problems (e.g., 5D+), which is central to the paper’s motivation.\nSecond, rank selection in Tucker decomposition is a critical factor for both performance and efficiency. The paper treats Tucker ranks as fixed hyperparameters but does not provide a mechanism to learn or adapt them during training. Over- or under-factorization can degrade performance or efficiency, and the absence of an adaptive rank selection strategy limits the robustness of the approach.\nMoreover, while the theoretical result is a strong addition, the practical interpretability of the UAT remains limited — it guarantees existence but does not clarify how rank or decomposition depth affect approximation accuracy in practice. Finally, when Tucker ranks are large, the method may still suffer from bottlenecks due to the size of the core tensor, potentially reducing the claimed computational advantage.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FQ4Wr7pm7k", "reviewer": "Reviewer_APdc", "summary": "This paper proposes Tucker-FNO, a neural operator that decomposes high-dimensional Fourier Neural Operators (FNO) into multiple 1-dimensional FNOs using Tucker decomposition. The method targets computational efficiency for partial differential equations (PDEs) and signal restoration tasks while maintaining expressiveness.\n\nTucker-FNO decomposes a d-dimensional FNO into d separate 1-dimensional FNOs arranged in Tucker format. A pre-lifting module extracts factor inputs from the condition function, processing them through individual 1-dimensional FNOs before aggregating outputs via Tucker decomposition. This reduces computational complexity from O(n³ log n³) to O(3 n log n) for 3D PDEs. The authors establish a universal approximation theorem using functional decomposition tools in Sobolev spaces, proving Tucker-FNO can approximate any continuous operator.", "questions": "- I wonder why the authors chose these examples and whether they explored other tensor decompositions. It would also be interested to include such results in the theory section", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Significant efficiency gains: Reduces parameters by and execution time compared to standard FNO on 256×256 data while improving performance​.\n\n- The paper is well-written.\n\n- Rigorous theoretical foundation: First tensor-decomposed neural operator with proven universal approximation capability​.\n\n- Dual applications: Successfully handles both PDE approximation and signal restoration tasks​.\n\n- Good empirical performance: Outperforms FNO, D-FNO, and Com-FNO on Navier-Stokes, Plasticity, and Burger's equations​.\n\n- Better high-frequency signal handling: Shows advantages in regions with significant numerical variation​.", "weaknesses": "- Limited novelty in decomposition: Applies existing Tucker decomposition techniques to FNOs without fundamentally new mathematical insights. Tensorized FNOs, which use Tucker decomposition have already been implemented as part of the neuraloperator library and also published in the literature. While this in itself is not a problem, there is no mention of prior related works such as [1].\n\n- Exprimental evaluation is mainly conducted on toy datasets and more complex examples are omitted. Baselines are also limited and lack other tensorized approaches such as [1] or the factorized FNO. This should also include comparisons to other tensor decomposition than just Tucker (e.g., tensor train, CP decomposition)\n\n- Moreover, it is unclear whether this approach can hold in more complex settings which include geometry. It's quite likely that this approach would fall apart in such a setting, given that it breaks symmetry and even just a single test with a simple geometry would be very informative.\n\n- Restrictive assumptions: Universal approximation theorem requires analyticity on compact sets (Theorem 3), limiting applicability to non-smooth or unbounded domains​.\n\n- Additional hyperparameter sensitivity: Performance depends on Tucker rank selection and frequency truncation threshold, requiring careful tuning​. It's unclear how the rank should be set for a given problem.\n\n- Scalability questions: While efficient for moderate dimensions, scaling behavior for very high-dimensional problems (d > 10) remains unexplored\n\n[1] Kossaifi, J., Kovachki, N., Azizzadenesheli, K., & Anandkumar, A. (2023). Multi-grid tensorized fourier neural operator for high-resolution pdes. arXiv preprint arXiv:2310.00120.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wGpZxqcEIX", "reviewer": "Reviewer_rkTN", "summary": "This paper aims to address the inefficiency of fourier neural operator (FNO) in solving large-scale and high-dimensional PDEs. To reduce computational complexity, this paper proposes Tucker-FNO, a neural operator that decomposes the high-dimensional FNO into a series of 1-dimensional FNOs through Tucker decomposition. It also proves the universal approximation theorem of Tucker-FNO. Experiments are done on PDEs and video reconstruction. The proposed method outperforms previous methods on both tasks.", "questions": "1. Why Tucker rank is directly set to the latent dimension for PDEs? How does the performance change with Tucker rank?\n2. Why not compare with F-FNO since it is already mentioned in the Related Work along with others that are compared with?", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea is simple yet effective.\n2. The theoretical and practical computation decreases significantly.", "weaknesses": "1. Figure 4 looks weird to me. The Carphone video reconstruction gets worse when the Fourier frequency truncation k increases. It is explained in the paper that \"excessive parameters hinder convergence efficiency\". However, it is not the case in the PDE problem in Table 4. Why does this difference exist? What if we simply wait the model to converge?\n2. Related to 1), if the model is not scalable, then decreasing the computation complexity seems not as important as claimed. A figure of scaling would be appreciated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ceGIj4t5Zj", "reviewer": "Reviewer_EBp7", "summary": "This work present the Tucker-FNO, an approach modifies the FNO to support the processing of high-dimensional inputs. Using a Tucker style functional decomposition, high-dimensional inputs may be processed into several low-dimensional inputs and processed in parallel by 1-dimensional FNOs. The authors provide a universal approximation result experiments on PDEs, image in-painting, and image denoising.", "questions": "1. Is it correct that the input does not undergo an explicit Tucker factorization, but rather it is processed in a way that assumes such a factorization exists? And then the inverse of the Tucker factorization is performed at the end, multiplying the factorized inputs by the core tensor? \n1b. If my understanding above is correct, is this process factored in to the complexity calculation? How does this process scale in the case that the Tucker rank is large, even if it is chosen as a hyperparameter? \n2. Is there any theoretical justification for Tucker factorization of the operator/input as opposed to the weights? Or can these be shown to be equivalent?\n3. How does the approach perform on a PDE describing a flow in 3 spatial dimensions? Please provide an experiment with baselines.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Performing tensor factorization on the operator/function space as opposed to the parameter space is an interesting idea.\n- The authors provide a functional Tucker decomposition UAT, which is, as far as I am aware, novel and could be useful in other applications.\n- A broad set of baselines beyond PDEs is investigated, where Tucker-FNO shows relatively good performance.\n- Several ablation studies are also present, which show notable improvements in speed and parameter efficiency in comparison to the FNO and alternative approaches.", "weaknesses": "1. Prior work and novelty. The manuscript does not adequately situate itself amongst existing literature, particularly the Multi-Grid Tensorized FNO [https://arxiv.org/abs/2310.00120]. This work also relies on Tucker-factorization. Although it acts on the weights of the FNO as opposed to the inputs, the works share substantial similarities in their goal to process large inputs and maintain parameter efficiency. Given this similarity, this work must be cited and compared both conceptually and empirically to clarify the novelty.\n2. Limited 3D validation. While this work repeatedly claims applicability to 3D (or general high-D) PDEs, there are only experiments on 2D or 2D+time PDEs. The problems investigated are also relatively simple, and model performance has become saturated in the larger literature. Experiments on more difficult 3D or 3D+time baselines are required to support claims of scaling. Baselines include GINO [2309.00583] and GenCFD [2409.18359].\n3. Implementation clarity and code. I feel many of the details of how the approach actually works in practice are obfuscated by rigorous theoretical derivations. The code provided in the supplementary material, likewise, is incomplete and does not resolve key implementation questions. \n\nMinor typos: line 174 \"construct an mapping\", line 240 \"multi-layer perception\" (I believe perceptron is meant here), line 247 big O notation of O(log(n^3)) should be simplified to O(log(n)).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)