KANO: Kolmogorov-Arnold Neural Operator
Authors: Jin Lee
Ziming Liu
Xinling Yu
Yixuan Wang
Haewon Jeong
Murphy Yuezhen Niu
Zheng Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Operator
Operator Network
KAN
SciML
AI4Science
Interpretable AI
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: KANO: an operator network expressive and efficient over a generic position-dependent dynamics with intrinsic symbolic interpretability.

Abstract: We introduce Kolmogorov–Arnold Neural Operator (KANO), a dual‑domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over a generic position-dependent dynamics for any physical input, whereas FNO stays practical only to spectrally sparse operators and strictly imposes fast-decaying input Fourier tail. We verify our claims empirically on position-dependent differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground‑truth Hamiltonians in closed-form symbolic representations accurate to the fourth decimal place in coefficients and attains $\approx6\times10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $\approx1.5\times10^{-2}$,  by orders of magnitude.

Published as a conference paper at ICLR 2026
KANO: KOLMOGOROV-ARNOL NEURAL OPERATOR
Jin Lee
Ziming Liu
Xinling Yu²
Yixuan Wang
Haewon Jeong'
Murphy Yuezhen Niu'
Zheng Zhang'
1 University of California, Santa Barbara
Massachusetts Institute of Technology California Institute of Technology
ABSTRACT
We introduce Kolmogorov-Arolld Neural Operator (KANO), a dual-domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over a generic poitionn-dependet dynamies (variable coefficient PDEs) for any physical input, whereas FNO stays practical only to spectrally sparse oper- ators and strictly imposes fast-decaying input Fourier tail. We verify our claims empirically position-depenndnt differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground-trun truth Hamiltonians in closed-fore symbolic represen- tations accurate to the fourth decimal place in coefficients and attains z x 10 state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, Rs 1.5 x 10-2, by orders magnitude.
INTRODUCTION
Contemporary science and engineering increasingly operate in regimes where the effective dimen- sionality and complexity of phenomena and data overwhelm human-designed calibrations and approximations. This motivates data-centric modeling of governing dynamics from observations (Kar niadakis et al. 2021 Wang et al. 2023 Carleo et al. 2019). For learned model to be constituted as a scientific law, it should first generalize universally over a well-defined domain, and also should be interpretable so that the learned representations can be extracted and reused for verification, testing, and downstream simulation. Mathematically, physical dynamics are generalized as operators as they are often formalized through PDEs (Courant et al. 1963 Evans 2022). large and practically important subclass consists of variable coefficient PDEs, in which at least one term has coefficient that varies by its variables (Gilbarg et al.) 1977): we define physical dynamics governed by such PDEs as desieden-deeenneent dynamics, when one of the variables that varies the coefficient is position. Examples include fluid flow in media with spatially varying viscosity or conductivity (Kundu et al. 2024), and the Schrödinger equation with potential that is function of position operators (Sakurai Napolitano| 2020). Scientific AI such as operator networks (Kovachki et al. 2023 Lu et al. (2021) that efficiently approximate a generic position-depender dynamics with tractable interpretability therefore valuable, which we recognize the absence and aim to fill the gap in this work.
An operator network approximates an arbitrary mapping between spaces by first encoding functions into finite latent vectors and then learning the iatent-to-laten map that represents the target operator (Lanthaler et al.. 2022). DeepONet of Lu al. (202T I 2019) implements the most general dense operator network where two neural networks learn both encoding and latent mapping directly from data, based on the theoretical foundation laid by Chen & Chen (1995). Fourier Neural Operator (FNO) of et al. (2020), on the other hand, hard-codes the encoding as pseudo-spectral projection with its spectrally diagonal kernels. FNO is provably and empirically superior when its hard-coded sparsity is optimal (Li et al. 2020 Kovachki et al.
function
"hojingucsb.edi "hhengzhang@ece uesb. edu

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Published as a conference paper at ICLR 2026
2021p, but this spectral sparsity becomes maladaptive for position-dependen dense dynamics (Koshizuka et al. 2024 Qin et al.) 2024). ). In such cases, the model size required for target accuracy grow super-exponentiall (Kovachki et al. 202I). and although the universal approximation guarantee still holds, realistic size FNO may only converge on an in-sample mapping that fails outside the training distribution. Numerous variants of FNO attempted break this spectral bottleneck. Some have broadened spectral coverage by exploiting factorized (Tran et al.) 22021) or multi-scale YYou al. 2024) spectral kernels, and others have injected local spatial kernels alongside the original spectral ones (Wen et al. 20222] Liu-Schiaffini et al. 2024 Liu et al. (025). Yet all prior
otherwise spectrally
works still privilege the spectral basis and cannot achieve optimal sparsity the spatial basis.
In parallel, interpretability has recently pivoted around Kolmogorov- Arnold Network (KAN) (Luu al. 2024b a). whose edges are trainable univariate functions and thus amenable to human inspection.
Several works demonstrate data-driven scientific modeling with K KAN: Chiparova & Popov (2025);
Gashi et al. (2025) use KAN for system identification, Koenig et al. (2024) replace the ML.s
in Neural ODEs (Chen et al. 2018) with KANs, each reporting symbolic recovery of benchmark equations and parameters. KANS have also been explored within operator networks: Abueidda
et al. (025) employed KANs instead of MLPs in DeepONet and Xiao et al. (2024); Wang al.
(2025) augmented FNO with KANs. Despite performance gains however, prior KAN-based operator networks have not reported symbolic recovery of the learned operator, leaving the avenue of an interpretable operator network largely unexplored.
To address these research gaps, we introduce the Kolmogorov-Aroold Neural Operator (KANO), an interpretable operator network jointly parameterized in both spatial and spectral bases with KAN sub-networks embedded in pseudo-differential operator framework (Hörmander 2007 Kohn & Nirenberg 1965). The key insight is to represent each component of the operator in the basis where is sparse: differential terms spectrally, localized terms spatially, to achieve the most compact and tractable representation. Our work offers three main contributions to the scientific Al community.
First, we demonstrate the pure-spectral bottleneck of FNO with an illustrative example and the-
oretically analyze why FNO cannot converge closely as desired to generic position- -eeenen dynamics (variable coefficient PDEs) with a practical parameter complexity.
Second, we propose novel framework of KANO that is expressive over generic position- dependent dynamics with intrinsic symbolic interpretability. We provide theoretical analysis on KANO's dual-domain (spatial and spectral) expressivity along with the empirical evidences of KANO robustly generalizing on unseen input subspace when FNO fails to.
Finally we validate the performance of KANO on some synthetic operators and a quantum simula- tion benchmark. KANO successfully recovered the closed-form formula accurately to the fourth decimal place in coefficients. Compared to the FNO baseline, KANO used only 0.03% of the model parameters, but achieved an order lower relative loss & in our synthetic operator benchmarks, and a four-order lower state infidelity in the quantum Hamiltonian learning benchmark.
To the best of our knowledge, our work is the first to demonstrate and quantify the symbolic recovery via KAN in operator learning. We shift the paradigm from mere universal approximation in operator learning toward the universal generalization of an operator network. Different from DeepOKAN (Abueidda et al.] (2025) which replaces MLPs with KANs in DeepONet, our work achieves generalization over disjoint out-of-distribution subspace via novel architecture design.
2 BACKGROUND
2.1 OPERATOR LEARNING AND FOURIER NEURAL OPERATOR
Operator learning approximates mapping between from function pairs ((a, E A, u, = (a.) ) E { 2 Kovachki et al. 2023 2024b). An operator network Go first encodes input a; via encoder E.. : : cm into a latent vector, then learns the latent map To C mm which the output is reconstructed approximate the label u; via reconstructor Ryn Cm U: i.e. Go- Rm Tg Em (Lanthaler et al. 2022). For fixed (En,: Rm,)), ), we can
function spaces, 9 A - I
A and L/ are Banach function spaces (e.g., Sobolev spaces) defined on hounded domain D C Rdd. In practice, each function is sampled discretized grid in and stored vector.
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define the projection II of operator g as
gI( Rn T Em where T
arg arg min C +mmm
Rm - T To Em]l
(1)
Dee Lu al. 2021 2019) learns m, Rmn and To all with two sub-networks. FNO (Li et al. 2020), on the other hand, hard-codes mm to be the truncated Fourier transform and R.m. to be its band-limited inverse.
Fourier Neural Operator (FNO). Let the domain D C R be periodic and write the Fourier transform FF of function a(x) as â(E):
/ a(x) -iux dx,
(F) = a(E) =
w = 2 E Zd
(2)
For fixed set retained modes & E 1 Em. 2d, truncated Fourier transform Fm A + cm and its band-limited inverse F 0 -lll can be defined as:
Fm aa) (a(E)) ..... a{mm))
(Fm((a))
Σ a(E)) 2
j=1
with slight abuse of notation. A single Fourier layer LIno of FNO is written as:
(a)) o (F.' Ro, (E) Fm((a(E)) (x) + Wo, (o,))(x)
with learnable spectral block-diagonal multiplier R(E), parametrized linear transformation W, and point-wise nonlinear activation a. FNO is comprised of iterative Leno between lift-up (P) and projection (Q) networks:
(a) 2 FNO
C P(a).
In the perspective of the operator network formulation I Lanthaler et al. 2022). FNO hard-codes its encoder e. as Fi and reconstructor R.m F. then learns the latent map by its iterative layers of parametrized linear kernels interleaved by non-linear activations (Kovachki et al. 2021).
2.2
KOLMOCOROV-ARNOLI NETWORK
KAN (Luu al. 2024b replaces fixed node activations of traditional MLP with simple operations and train the learnable univariate 1D functions & on edges. With layer width ni - 74411
and input field x(!) (++1), a KAN layer yields a function matrix (()
(1) at 7th layer as
((+1) (03
s (1)
ppl....
(1+1)
u. (a
(6)
2m1
so each output channel a sum of edgewise transforms of the inputs (Liu et al. 2024b In the original KAN each edge function is spline expansion
(1 q.p b(1) +
(6) g.p. Bc(l),
with learnable coefficients for a fixed base ID function b(-) and ID B-spline basis (B.). Because every 0g.p is a 1D curve, KANs are directly inspectable and amenable to visualization followed by symbolic regression. On expressivity, Wang et al.) (2024) theoretically prove that KANs match MLPs up to constant depth and width factors: empirically, with appropriate optimization recipes, KANs and MI.Ps exhibit comparable scaling on PDE and operator benchmarks (Shuklaa et al., 2024). Hence, swapping latent MIP for KAN preserves expressivity while enabling symbolic readout.
3 THEORETICAL ANALYSIS ON FNO'S PURE-SPECTRAL BOTTLENECK
This section first illustrates the pure-spectral bottleneck of FNO. Then we provide a theoretical analysis and prove that FNO suffers the curse dimensionality for position-dependent dynamics. FNO is proven to have the universal approximation guarantee over any arbitrary non-linear Lipschitz operator (ovvahhii et al.) 2021 Lanthaler et al. 2025). This section does not disprove the universal approximation ability of FNO: it illustrates the limitation the generalization ability of FNO stemming from its spre-spectral bottleneck on spectrally dense operators.
Akin to matrix-vectto multiplication but follows the third equation Eq. 6 instead of row-vector inner product.
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3.1 THE
BOTTLENECK OF FNO
We consider the ID quantum harmonic oscillator Hamiltonian example:
Ha(x) = a()))(a) + x² a(x).
(8)
Multiplication and differentiation have a dual relationship under the Fourier transform:
F((-02=a)](E)
a(E),
FI(2² ())
dgçā(E).
(9)
In spectral domain, the spatial differential Ozw is spectral multiplier 5², whereas the spatial multiplier x² becomes spectral differential dge- Consider a truncated polynomial basis {1, 2. ,.,...... x= x=-1} .....
and a truncated Fourier basis (0) eike = O...., 1, on periodic domain. In the spatial (polynomial) basis, the map a(x) x²2 a(x) acts two-step up-shift sparse matrix
0 ... 0
0
S
'.. 1
(10)
0
0
while the spectral (Fourier) basis it a dense Toeplitz matrix (Morrison 1995
co
C-1
c-2
C-n+1
C1
Co
C-1
C-n+2
1
2x |
Tx(x
=
C2
Co
C-n+3
Gm
imm d0.
(11)
2T
Cn-1 Cn-2 Cn-3
Thus each term in H. is sparse in one basis and dense in the other (Morrison 1995).
An FNO layer LNo (4) can easily parametrize -Oge by taking R(E) Ps e². However, upproximating the dense off diagonalss in [m[x²] parametrize z² must rely the non-linear activation o()) since RCC) and W are spectrally diagonal and hence incapable of mixing modes. Let z(u) denote the pre-activation for input u, then the Jacobian of Leoo gives the first-order approximation of the learned map and its Fourier transform reveals the spectral off off-diagonals of itself as
F(J(())(E.E'))
(Flo"(z(u.))) ())i))))))))))))))))))) -1)) (w Ro(E)))
(12)
Therefore, all off off-diagonals arise from the spectrum of the u-dependent gate o' (z(u,) )): FNO's nonlinearity can create off-diagonals, but they are tied to the input distribution of u. This the pure- spectral bottleneck of FNO: spectral off- off-diagonals of learned FNO fied to the train subspace, hence FNO can converge only on the in-sapple mapping that fails outside the train distribution See Appendix D for further detailed discussion expanding to the arbitrary higher order contribution and deep layered FNO.
3.2 FNO SUFFERS CURSE OF DIMENSIONALITY ON
DYNAMICS
As explained previously, position operator, a(x) ) H x a(x). ). a highly dense Toeplitz map in the spectral basis (Morrison 1995). Based on the Remark 21 & 22 of Kovachki al. (202T), prove that any position-dependent dynamics induces super-exponential scaling in FNO size by the desired generic position-dependent dynamics with practical model size. hence can only overfit on the in in-sample mapping. We provide de Lemma [] that single position operator already spreads the input spectra too much for FNO stay practical, and Theorem expanding Lemma to an arbitrary composition of position operators.
error bound: FNO cannot converge closely desired
Following from the operator network formulation (T) the error estimate of an operator network Go approximating the ground-truti operator G in an operator norm is bounded as
g Gall
< G 1 Rm, T Emll
|Rm (T To) E, ||
(13)
projection errur:
Intent network error:s.s
by the triangle inequility. Latent network eтrоr Eaet follows the well-established scaling law of conventional neural networks (Hornik et al. 1989. Cybenko 1989). Therefore, whether an operator network efficient in model and sample size to achieve desired accuracy hinges on the scalability
of the projection error Groi (Lanthaler et al. 2022, Kovachki et al.) 2021 2024a).
*This issue of oui-t--distiibuttinn fragility from underspecification is well studied by D' Amour
2022
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Pa E)
FF (Ro(EE) Fx,]a(x)
Opm (p)(())
RE(E)
a
+
(avo(a)
a
a(x)
(a)
W axx)
W
a
(b)
Figure I: (a) LeN architecture. (b) LKAOO architecture.
Reviewing Remark 21 & 22 of Kovachki et al. (2021) Eproj of FNO is governed by the Fourier tail, the sum Fourier coefficients outside the retaind spectrum E truncated by width m (Gottlicb) & Orszag 1977: Trefethen 2000): to achieve the desired 'poo with practical m, both input and output Fourier tails must decay algebraically or faster. However, even in the optimal case of the hand-limited input, if the ground-truth operator is spectrally dense spread out the input spectra and induce algebraic or slower decay in output Fourier tail, yn must scale at least polynomially to suppress Eproj
m (rro where a geometric constant. Meanwhile, as the latent mapping would be also dense, the size of the latent network, Noet. follows the canonical polynomial neural scaling (Yarotsky) 2021 by the desired Ene with its width m" (d is the input domain dimension) as the exponent: Ne ())))))) Consequently, this results as the super- super-exponential scaling in the
2017 De Ryck et al.
latent network size Nuet (Free"" "rr d////a suppress pproj explodes Nuet to achieve the desired net for a generic dense operator.
even with the optimal band-limited input: scaling width m to
Lemma (Position operator elongates Fourier tail). A single position operator, spatial multiplier by x, induces algebraic decay in output Fourier tail when the input band-li
Sketch of proof. Position operator is kernel Ø({) x 1// in spectral basis. Hence, every mode outside the input spectrum picks up a coefficient of size ~ 1/(E,, ending up as |ō(E) ž 1/lE/ in the output spectrum. See Appendix C.2 for restatement and full proof.
Theorem 1 (Curse of dimensionality on position operators ). Any arbitrary composition of position operators requires FNO scale expeenlxxp on its model size by the desired accuracy.
Sketch of proof. Iteratively apply Lemmal] [] then any arbitrary composition of position operators induce algebraic or slower decay in output Fourier tail even for the optimal lind-limited input. This results in super-exponential scaling of latent network size by the desired error as discussed above (Kovachki et al. 2021). See Appendix
C.3 for restatement and full proof.
Remark 1. What is missing from the upper bound analysis of FNO by Kovachki et al. (2021) is the effect of the wide lift-up and projection networks. For the generalization guarantee arguments of this work, the upper bound analysis is still sound, yet it should be clarified that the theoretical analysis on the role of the lift-up and projection networks is yet an open research question. Recent studies by Lanthaler et al. 2025); Lanthaler 2024) provide better view it with lower bound analysis.
KOLMOCOROV-ARNOLL NEURAL OPERATOR
Motivated by the pure-spectral bottleneck of FNO, we propose the Kolmogorov-Amold Neural Operator (KANO), an operator network capable of converging closely as desired on a generic pesition-dependent dynamics with practical model size. We first introduce the KANO architecture, and provide theoretical analysis on its dual-domain expressivity the following section.
4.1 KANO ARCHITECTURE
KANO utilizes an iterative structure of KANO layers LRAND to learn the unknown operator, akin to FNO. However, KANO excludes the wide lift-up and projection networks to maximize tractability
since it is known that wide KANs are fragile to symbolic recovery (Noorizadegan al. 2025):
g
(E) RAND .... (1) KANO-
(14)
LxANo(a)(x) = Do, F- Pop
(x,((x (x, * Fm)()(()](x) a(x)).
(15)
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where & is KAN sub-network for learnable non-linear activation; p(x. E) is another KAN sub- network, a pseudo-differentia symbol jointly parametrized by both spatial x and spectral & bases Note the "*" notation in CANO (15) instead of the block-diagonal multiplication notation *." in Leo (4). The spatial basis x of the symbol pxx, {) is convolution (differential) in spectral domain by the dual relationship (9) Therefore, the pseudo-differential symbol calculus of p(x, E) needs to be done by quantizing on both spatial and spectral domain (Hörmander) (007). and we choose
Kohn-Nirenberg quantization (Kohn & Nirenberg. 1965) compute the symbol calculus in CRAOO:
Fm Fmx((( Fm. (a)( = 1/2
ΣΣ BEEVEY
e (x v) p(x,))
(16)
where for a periodic domain D = (- )))
1ym,.....ym C is a uniform discretization with spacing h and D is an evaluation point. We denote Kohn-Nirenberg quantization (T6) as an
operator Op (p) == F p(x, {) * Fm defined by the symbol p(x. f). In the operator network
formulation introduced in Section 2.1 KANO's projection IIAAO then defined as:
IT KAND (9) == (Pm))s):
Pg E arg min g - Opm.(P)) (p)]
(17)
Symbolic Interpretability of KANO. By using compact KANS each for the symbol p(x, { and
non-linear activation in every KANO layer LANO (5)) KANO network 93 GEANO (14) is fully inspectable by visualizing the learned edges of all its KANs, potentially allowing closed-form symbolic formula of the learned operator with the manual provided by [iu al. (2024b a). In
addition, recent endeavors have greatly expanded KAN's symbolic recovery capacity to non-smooth,
discontinuous targets with high irregularities ( (Yu et al. 2024 Aghaei [024] Lei et al. 2025 Yang eet al.. 2025 Shiraishi et al. 2025). All of such advancements are easily and directly applicable in our KANO framework as well, when facing an operator with high irregularity coefficients.
Remark (Complexity analyses of KANO). As apparent in Eq. (T6), KANO layer must perform
double sum which can be computationally heavy. However, for the target operator class of variable-
coefficient PDEs such as position-dependee
dynamics, we show this can be compensated in principle
by the parameter efficiency we prove in the following Section 4.2 See Appendix
4.2 KANO'S 'S DUAL-DOMAIN EXPRESSIVITY
In contrast to FNO, KANO exploits sparse representations in both the spatial and spectral domains, hence decoupling the scaling of Epeoj and Eee by never letting the latent map be dense convolution. For instance, for the quantum harmonic oscillator in Eq. (8), a KANO layer LAANO (15) can parametrize H by taking p(x,E) x² + E2, both -d.. and x terms are each represented where they are
sparse, both leveraging the shift form S!?" 10p. By jointly parameterizing the operator in both spatial and frequency domains, KANO cherry-picks the sparse representation for every term in position-dependent dynamics, building the right inductive bias well-known to be essential for out-of- distribution generalization and model efficiency (Goyal & Bengio 2022
Trask et al. 2018).
This dual-domain expressivity of KANO first alleviates the input constraint; we first explain that Eproj of KANO scales practically by its width m for any physical input. Then we provide Theorem 2 as long the KANO projection (7) of operator generates smooth symbols KAN easily approximate, Eee scales practically by compact KAN sub-networks independent of Croj- In conclusion, generic cosition-depender dynamics with practical
KANO can converge closely as desired model size using any physical input, robustly generalizing outside the train subspace.
KANO practically has no input constraint. According to the quadrature bound from Demanet & Ying [201] Thm. 1&2), the error estimate of Nirn-beg quantization (16) obeys
|| - Op. (po))l < C Cm=
(18)
given norm-bound (finite-energy input of AB = {u ||u|| S B} where 8, C are geometric constants. Hence KANO width m scales polynomially by the desired Eproj given any physical data.
SShin al. 2022 first employed pseudo-differental operator framework for neural operator. They presumed the symbol pp(x x, E) to he separable as p(x, {) = Px(x) Pp (E), and used ML. sub-networks while retaining the lift-up and projection networks of a generic neural operator architecture (Kovachki et al. (023).
'Norm here and Equation Eq. 18 is the Sobolev norm

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Published as a conference paper at ICLR 2026
Table 1: Relative &y losses (x10 and parameter counts.
G2
Go
Model (params)
A
B
B/A
B
B/A
B
B/A
FNO (566k)
6.36
98.8
15.53
10.6
87.0
8.21
11.4
81.4
7.14
U-FNO (579k)
2.79
22.9
8.21
8.14
339
41.65
92.4
292
3.16
AM-FNO (548k)
20.9
19.35
1.20
16.5
13.75
1.16
29.8
25.69
PDNO (538k)
1.41
6.31
4.5
1.92
12.1
6.3
4.03
27.2
6.7
KANO (152)
1.04
1.44
1,38
0.629
0.749
1.19
0.716
0.737
1.03
KANO_ME LLP (2k)
3.37
6.59
1.96
4.49
8.07
1.80
3.59
6.87
1.91
0.512
0.526
1.03
0.498
0.500
1.00
0.520
0.536
1.03
(a)
(b)
(c)
KANO
Unseen
10"2
KANO
Unseen
KANO
Unsee
Rel 6 Losss
FNO
Rel 6 Loss
FNO
C10
Rei 6
10
Size
Beath
carped Colo
Fortade
Navo Parkl
stee
Pode
Gossion Heraie
10-1
Site Besth
Csuppe
Feradds
Ware
Facket
Palee
Gommla Herwlte
Slde
Beats Eatyrd
Codig Frndh
Wer
Gramin
Bermm
Figure 2: Loss test results. (a) 91 (b) G2 (e) G3. Note the logarithmic scale.
Theorem 2 (KANO stays practical for smooth symbol). If the KANO projection of an operator 9, II I (9) 17). generates a finite composition of smooth symbols Po E) and finite-degree non-linearities, the model size of KANO scales polynomially by the desired accuracy
Sketch of proof. Choosing m~ BB/))/> scales projection error down to 2/2 by Eq. (8) A fixed- width KAN then approximates the symbols to accuracy s/2 with O( 0(e-4/(2sp) parameters (Wang Corol. 3.4) (sp is a geometric constant). The finite-degree non- non-linearities add only
al.
2024
constant-size weights by the activation KAN, so the total parameter count ((--d/(2e>)) See
Appendix C.4 for restatement and full proof.
L
Corollary 1 (KANO is practical for generic desedien dynamics). For a finite com- position of spatial and spectral multipliers of maximum r-differentiable symbols with finite-degree non-linearity, Theorem 2 yields [0] = O
d/(2r)). (e--//2rr).
Remark 3 (Scope of KANO). Recent studies demonstrate that wide lift-up and projection networks are essential for strong performance on digeesiinni benchmarks (Diab Al Kobaisi 2025 Eker 2024 Liu al. 2023). In contrast, KANO is designed prioritize symbolic recovery with robust generalization, complementary to the scope of FNO. Because the core dual domain expressivity mathematically agnostic dimensionality, extending KANO to conventional high-dimensional use cases is a natural and promising direction future work.
5
EXPERIMENTAL RESULTS
5.1
SYNTHETIC- OPPRRARI GENERALIZATION BENCHMARKS
We benchmark FNO-based models and KANO on three position-dependeen operators:
f - 22. SunJ, Gzf - - 0.f + 0rzf, Gaf 5 x Oz + 2zn.f.
Our goal is quantify and compare the generalization of each model. We train the models only with Group dataset and evaluate them on the unseen Group B dataset.
Group A (Training families): Periodic, Chirped Cosine, Sine Beats.
Group B (Testing families): Sinc Pulse, Gaussian x Hermite, Wave Packet.
For each operator, we generate 2000 train pairs from Group A and 400 test pairs from Group B to evaluate the generalization by comparing the ratio between the average relative & loss over each group (Loss Test). We also interpolate the Group A and B function samples in 100 steps, apply ground-trutt truth operators in each step to build the interpolated dataset, and evaluate the loss ratio to that of the Group A samples (Interpolation Test). We trained FNO models of 2 layers, 64 width with no
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(a)
(b)
(c)
14
KANO
KANO FNO
KANO
FNO
FNO
Loss Ratio
10
Loss Ratio
Loss 1
2
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Interpolation Ratio
Interpolation Ratio
Interpolation Ratio
Figure 3: Interpolation test results. (a) 91 (b) 92 (c) 93.
p(a.()
I' (p(,, {)
Re(p(z.e)
p(a, E
Re(p(z.E)
P(x. E)
Output
Op(
Residual
(a)
(b)
Figure 4: (a) p(z, {) of 91. The middle edge does not contribute to the output. (b) p(x, E) of G2. (c) p(a, E) of G3. (d) of 93- Edge the residual in (d) looks linear, so we compared two scenarios, linear and cubic, which the latter achieved lower loss and better generalization.
mode truncation, and used one-layer KANs of grid 10 cubic B-splines edges for the KANO model. For other FNO variant baselines, used 2-layer models of similar size FNO. Lastly, we trained KANO variant for an ablation study, which we replaced the KAN subnetworks with compact MLPs of 32 hidden width and 2 hidden layers (KANO_MLP See Appendix B.1 experiment details. We used Adam optimizer and relative &g loss for training.
Results. As shown in Table and Figure 2 KANO shows consistent losses over Group and operators. KANO_MLP
Group B, validating its robust generalization ability for also shows comparable out-of-distribution performance, which suggests that the generalization ability of KANO stems from its dual-domain architecture apart from KAN. In contrast, FNO shows fragile cattof-diisribution behavior on Group dataset with the significant loss increases. U-FNO (Wen et al. 2022) and AM-FNO (Xiao al. 2024) show
worse results. On the other hand, PDNO (Shin 2022) shows the most stable generalization among FNO families, although not as robust as KANO and KANO_M M.... Along with the ablation study, this confirms that the pseudo-differential operators, while primarily relying on only spectral kernels makes the model fragile out of train distribution even with localized enhancements. Interpolation test results in Figure Sngureefurtin further empirically validate our theories. The FNO curves (red) of Figure show slow increases on early and mid- iiddinterpoolaton, suggesting that the FNO's learned in-sample mappings are yet close to the ground-truth operators. However, the FNO curves abruptly soar up in the latter ratio, suggesting that the interpolated functions are now far outside the train distribution. These results, together with KANO's ofee-order-o--maniine lower loss at just 0.03% of FNO's size, are consistent with our claims in Theorem and Theorem 2
et al.
operator framework is judicious for robust generalization on position-depender
After convergence, visualized the embedded KANs (Figure 4h. We then froze these learned symbols and continued training, referring to this variant as KANO symbolic. KANO symbolic recovered the exact symbolic coefficients of the ground-truth operator within the fourth decimal place (Table 2 KANO's loss matches KANO sonbolicooli'ss 's loss in Table confirming that KANO converged close to the ground-truth operator.
5.2 LONG-HORIZON QUANTUM DYNAMICS BENCIIARK
We provide this benchmark on two desition-dependent quantum dynamics: the quartic Houble-welll Hamiltonian (DW) and the nonlinear Schrödinger equation with cubic nonlinearity (NLSE):
i0.0 -m. w(xx) 2/ (DW), i0 -m w(xx) + + 10 & (NLSE),
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Table 2: Ground-truth vs. learned operators (coefficients rounded to 4th decimal place).
Ground-truth operator
Learned KANO operator
G = x² d2rl
f = (x² + 0.0003)
dzr
Gaf 2 0zf + S
Gaf 0.9996 x 0zf + Ouaf 0.0003 f
Gaf 1.0001 f + 0.99997 fz + 0.99997 8..
f = f + x Szf + Sraf
0.0002 f 0.0003 f 0.0001
State Fidelity
(b)
State Fidelity
1.0000
0.99
0,9950
0.95
ideiity
fideliyy
0.9900
0.95
0.9075
0.95
0,9050
0.94
0.9825
KANO pendavee
PNO Jui
0.98
KANO pendauu
ENO Jull
0.92
20
60
nn
80
300
20
60
10
100
Step
Step
Figure 5: State fidelity over 100 time steps. (a) DW (b) NLSE
where w(x) = x.
(x
32 2 + 0.295.
We generate 200 initial states and yield the state trajectories by the Hamiltonians, sampling momen- tum/position probability mass functions (PMFs) every 0. 1m for 100 time steps. The first 10 time steps are used for training, and the rest are used evaluate the long-horizon prediction.
We modify KANO to capture the quantum state evolution: Q-KANO. Symbol is parametrized expl expl-iAT -iA (x, <) where AT 0.1 ms. The adaptive activation is also defined as a complex exponential with learned phase & os([Op,, (Pe) 20pm. (pe (p)) for input wave function (xx):
g KAKO
[bb] Op.. (exp[ - iAT (o {)]) )bb
(19)
We investigate three supervision scenarios: Full-type training with full wave function, idealistic yet physically unattainable, Pos-type training with only position PMF, physically realistic yet the least informative, and & training with both position and momentum PMFs, which remains physically attainable while providing richer information although not full. We use Adam optimizer for all trainings. See Appendix B.2 for experiment details.
Results. We evaluate state infidelity infidelitybetteenn Tetween ground-truth evolution and model prediction at each time step (Table Figure 5). In case of KANO, the pos & mom- type training achieves indistinguishable infi- delity from the ideal full-type training base- line. The ablation study with MLP variant of Q-KANO achieved comparably low state infidelity by pos & mom-type training as well. Meanwhile, the pos-type training displays clear increase in infidelity, especially the NLSE.
Table 3: State infidelity after 90 additional time-evolution steps.
State Infidelity
Model & Train Type
Double- Well
NL.SE
FNO (full)
1.5 x 10
1.6 10-2
Q-KANO (full)
6.3 x 10
6.8 10 6
Q-KANO (pos & mom)
10
5,6 10 6
Q- Q-KANO (pos)
4.7 x 10-3
6,1 x 10
Q- Q-KANO MLP (pos & mom)
7.7 X 10
8.5 x 10-6
O-KANO.SYMBRR SYMBOLIC (full)
2.0 x 10 8
2.0 10
Q- O-KANO.SYAA SYMBOLIC (pos mom)
10 8
3.0 10
Q- D-KANO.SYYBLLLL (pos)
5.3 x 10
6,1 x 10
For predicted state + and ground-truth state 4, the state fidelity F is defined as the inner product between them (F i sp >), and the infidelity is defined as (1 F), hence shows how distant two states are.

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Table 4: Ground truth vs, learned symbols. Coefficients rounded to 4th decimal place.
Hamiltonian
Train Type
Learned symbolic structure
ground truth
x² x² 0.0625 + 0.295 0.5
DW
full
1.0004 4 + 0.0001 x² 1.0013 2² + 0.0625 + 0.2955 0.5
pos mom
1.0003 + 0.0001 x² 1.0008 x² + 0.0623 + 0.0001 + 0.5
ground truth
x² _² + 0.0625 x: + 0.295 0.5 |0]
NLSE
full
1.0005 x² 0.0001 x 1.0014 2 0.0626 0.2942 0.5
0.9815 y 2 + 0.0110
pos mom
0.9999 x 0.0003 x 1.0001 2² 0.0630 0.1141 + 0.5
0.9514 4 1//2 0.5504|pl
Φ ar E)
Φ(I, E)
O 01-1,, 2-)
w(x)
|-|
7.
(a)
(b)
(c)
(d)
Figure 6: pos&mom type training results. (a) Structure of the potential w(x) (b) p(x; {) of DW. (c)
pl,,{) {) of NLSE. (d) ((I-), <-) NLSE, Potential w(x) structure is clearly reconstructed.
In contrast, even with full type training, FNO fails to maintain low state infidelity after the long- horizon propagation as expected. Iterative time evolution pushes the wave function far outside the train convex hull, and FNO's learned in-sample mapping deviates from the ground-truth evolution rapidly, leading to four orders of infidelity increase compared to KANO.
Table 4 juxtaposes the learned symbols with that of the moound-trutt Hamiltonians and Figure shows the KAN visualizations from pos & mom-type training. With full-type training coefficients are recovered to the fourth decimal place, vindicating the ideal capacity of KANO when the information is fully provided. Under the realistic pos & mom-type training, the reconstruction remains accurate except for two terms: the constant (global phase) and the NLSE's s cubic coefficient. Both discrepancies predicted by quantum observability: global phases cancel in all PMFs, and the Kerr coefficient enters only through higher-order correlations that become harder to estimate from finite-shot statistics. Q- Q-KANO faithfully reveals what the data support and nothing more.
CONCLUSION
We have presented the Kolmogorov-Amole Neural Operator, an interpretable neural operator expres- sive on a generic -eittinn-deeendett dynamics. KANO cherry-picks sparse representations of each term via jointly parametrizing on both spectral and spatial bases, and achieves robust generalization outside train distribution while exposing clear tractable representation via its KAN sub-networks. In all our benchmarks, KANO has successfully recovered the ground-truth operators accurately to the fourth decimal place in coefficients. In addition to the superior out-of-distribution generalization, KANO has also achieved orders of magnitude lower losses with less than 0.03% of the model size compared to the FNO baseline. KANO shifts operator learning from an opaque, surrogate- based paradigm towards interpretable data-driven scientific modeling, and provides robust empirical evidence supporting its enhanced dual-domain expressivity and interpretability.
REPRODUCIBILITY STATEMENT
All code, configurations, and dataset artifacts required to reproduce the experiments are provided in wtthttps:///e ://ithuub com/ www.:ooonnnaa.ee jinucsb//////// KANO, along with step-by-step instructions for rerunning training and evaluation. Details in the data generation and implementation provided in Appendix B
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and the full proofs of Theorems and Lemma are provided in Appendix Calong with their mathematical restatements.
ETHICS STATEMENT
All authors this work sincerely adhered to the ICLR Code of Ethics. We do not expect any potential violation to the best of our knowledge.
ACKNOWLEDGMENTS
We thank Nina Miolane, Adele Myers, Louisa Cornelis for helpful early discussions and Anima Anandkumar for her feedback on our paper draft. Jin Lee and Zheng Zhang were supported by NSF ECCS-2328288 23282811 Murphy Yuezhen Niu was supported by the U.S. National Science Foundation grant CCF-2441912 (NSF CAREER), Air Force Office of Scientific Research under award number FA9550-25- 25--0146,, and the U.S. Department Energy, Office of Advanced Scientific Computing Research under Award Number DE DE-SC0025430.
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APPENDIX
A TABLE OF NOTATION
Table Main symbols and notation used in the paper.
Symbol
Meaning Definition
Domains, spaces, and operators
== L-12,
Periodic d-dimensional spatial box of side length L Ground-truth solution operator to be learned
g
Spectral spatial sampling
Em
Retained Fourier modes (truncated spectrum); m Fourier transform
F
Fm
Fm
Truncated Fourier transform by Uniform spatial grid
inverse
y (y Ym C
h
Grid spacing of )
Fourier Neural Operator (FNO)
CYNO
Single FNO layer
g
FNO network
Ro({)
Learnable block-diagonal spectral multiplier
Wo
Point-wise learnahle linear map
(-)
Point-wise non-linear activation
KANO layer, symbol calculus, and projection
LRANO
Single KANO layer
p(x, {)
Learnalle pseudo-differental symbol
Op,.. (p)
Kohn-Nirenberg quantization of width m defined by p
XxAND
KANO projection of g
g KANO
KANO network
to
Learnable activation
Kollogonon-Arool Network (KAN) primitives
ID edge function on layer f, connecting nth node of layer to dhh node of layer (& + 1)
b(t), (B((t)}
Base function and B-spline basis used to parametrize (12
Q-KANO (quantum dynamics) notation
((z)
Input wave function
w(x)
Quartic double-well potential
AT
Time step of propagation
(x( E)
Parametrized phase for symbol p(x, of Q-KANO
(..11.2)
Parametrized phase for non-linear activation of Q-KANO
go-KANO
Q-KANO network
Function spaces
(D(
Sguare-integrabl function space on domain D Sobolev function space order s 2 on domain D
HD)
B EXPERIMENT DETAILS
B. SYNTHETIC OPERATOR BENCHMARK
All experiments are carried out periodic functions / T + IR with T (-T,, π| and uniform trigonometric grid
2x
= Emin + Ar,
Ar
N ......,N N
N
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with N 128. Unless noted otherwise every random quantity is drawn independently for every sample.
Outer envelope. To avoid the Gibbs phenomenon all basis functions are multiplied by a smooth taper that decays zero in "/e-wide buffer the periodic boundary:
1,
|o| I 5.T
|
5T/66
A(x)
cOs
5 6
|a|| n
T/6
2
0,
|x| n.
The full "base" function is always feuse (I) = (xx).
Spectral derivatives and ground-truth operator. stencil:
Derivatives are computed with an exact Fourier
(x) F a
f"(x) = F 1--2
Ula, b] denotes random digit drawn from range [a, b).
Training families (Group A)
A1. sine beats:
g(x) sin(wi Φ1) sin(w22 d2), w = -U d: 20].
A2. .hirped_cosine:
cosine:
g(x) = cos(aa²)
a 12U
A3. periodic (random harmonic series):
(() = (we + 61) + cos(wa + 62),
w 8 U 3). 61,2 = U(0,2TT]
Unseen families (Group B)
B1. wave-packet:
g(x) exp (0-u)²
sin(wa++o) = U[-,,),, o= 1/2 U[0.5, 12U/2, 6), & U[0.2T]
B2. sinc pulse:
sin(ar)
[x| 10-12
g(x)
ax
a= 12U[0,5333
[ S 10-12 12
B3, gaussian hermite:
g(x) = Hn I
exp
(w-u)² 2n
n E (1,2,3) uniform, = == U[-2,2), a = U 2),
where Hn is the degree-n Hermite polynomial.
NORMALIZATION
Each realization is divided by its maximum absolute value, || obtain || 1. The envelope guarantees periodicity and keeps the numerical spectrum sharply band-limited.
Sample counts. #train = 2000 samples from the three Group A families for train data and #test 400 samples each from the Group A and Group B families for generalization tests.
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B.2 QUANTUM DYNAMICS BENCHMARK
We model quantum apparatus with 200 state-preparation protocols each with perfect reprodueibility. capable of generating an identical initial state whose wave function is drawn from one of the three families: Periodic, Gaussian wave-packet, and Gaussian x Hermite. The prepared initial states evolve under one of unknown, ume-independent Hamiltonians, and two arrays of 128 detectors measure position and momentum on uniform grids, yielding probability mass functions (PMFs) every 0. 0.lms for 100 time steps. PMFs collected from the first time steps are used for training the models, and the rest of the PMFs collected from the remaining 90 time steps are used to evaluate the long- long-horizon fidelity drop beyond the train steps.
B.2.1 QUANTUM APPARATUS ASSUMPTIONS
State preparation. collection of calibrated protocols can each prepare a designated initial wave-function each of one of three real-vate valued families Periodic, Gaussian wave-packet, or Germiien-Hermite: 45("m (ar) E 1.2
m 1, ..... 200. Repeated shots under the same protocol : start from exactly the same 1b enabling trajectory-level reproducibility for every member of the ensemble.
2. Hamiltonian stability. The (unknown) Hamiltonian is time-independent, so trajectories are perfectly repeatable once 4/n fixed.
3. Dual-basis detection. Two 128-grid projective detectors measure the position basis {|x,)) and the momentum basis {[s)), J. yielding empirical probability mass functions (PMFs) fz (1) = [<x));
and (5) = (x()) on common torus grid Tz, I. == 4.
B.2.2 DATA GENERATION DETAILS
For each of 200 distinct sample trajectories we
1. draw the initial wave function and propagate on the Hamiltonian with high-resolution Strang split:
81 jus for 10000 micro-steps, producing coarse snapshots every 100 steps (AT 0.1 ms,
T = 1,..., 100);
2. store (th,, PT p where (i) We (2:)) and p ())
(E,)1.
Only the first 10 coarse steps are used for training; the remaining 90 steps test fidelity drop on long-horizon. All simulations employ an 128-point FFT grid to match the detectors.
Spatial discretization. We place the problem on a periodic box of length = with N = 128 grid points æ, Tmin + jAr, Ar L./. Periods suppress wrap-around artifacts because every initial state is tapered by the smooth envelope A(x) defined Appendix B.1 Spatial derivatives are taken spectrally: let &m 2mm/ for m -N/2, ...., N/2 1. Writing vm F (Sm))
0,4 = [immmm
Oza = F 1- Fm].
Strang-splitiing time integrator. Let K. ==
== -rz (kinetic), V + + w(xx) & (potential) and W + + |0122 & (cubic nonlinearity). With time step At the second-order Strang factorization reads
(K+V+N)A = e (C+V) ANN e e
( + O(At").
Because IC is diagonal in Fourier space and V in real space we implement each half step explicitly:
3 V/2
2 At bb,
$
X/2
om t e s Sm: F(w),
N
v t - 110/00 d. F F--(
repeat KC/2 and V/2.
iStuare-integrable function space.
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The L norm is renormalized after every macro step to compensate numerical drift. For DW Hamiltonian, non- non-linearit time evolution is omitted.
Integrator parameters. We inner step öt 10-6. A macro step of AT 100 öt 10-4 is recorded and the sequence is propagated for Taxx = 100 AT' = 10-2
Initial-state families (real wave-functions). Each sample draws a real profile f(x) from {A1, As} below, multiplies it by the envelope A and normalizes it in L
A(x) f(x)
v0(xx)
|
A1. random harmonic series
f(z) Σ Sm cos(2 + c(t) sin(2mmE), { {= +2 el)) (O,
Sm
A2. Gaussian wave packet f(x) = expl-(xx xo)) (2]0)] with vp ~ 07.40 0.4x] and 0~U
A3. Gaussian-Hermite mode f(x) Hn (=( uniform, Zo. a as above, and H,, the Hermite polynomial.
xo)²2/(2²)) with E {0,1,2)
Stored quantities. For every sample index and every snapshot E {0, 1,..., 100) we save
(x, (I,T), (2, T) (E.))))
wavefunc, pos pdf, mom pdf.
All arrays written in float except the complex wave-function, stored complex64. Alto- gether one call to generate dataset (num samples 200) produces 200 x 101 x 80,800 labeled records.
B.2.3
TRAIN TYPE DETAILS
Let v
(T)
be the T-step prediction of Q-KAÑO given vho-
Train Type
Train Dataset
Loss function
Dfull 1/
(T) (b(T))1
full
complex (T)
||b((T) ll2
Spos = DKI(D!") Il 1
pos
px
pos & mom
(T)
P
(T)
Spaskmoo DRI(P!")
(0 + (KI(P (T) || |/3b
C PROOFS
C.1 NOTATION AND PRELIMINARIES
Throughout, T : |- rtd denotes the flat d-torus and 2d the lattice of Fourier indices, For & E g let =x The Fourier coefficient of a square integrable function is
(2T) f f(x) -itxx dx.
f(E)
In Sobolev space H² with an order of smoothness I, the Sobolev norm of function is
(d
|
Σ (1+1 ((E))
2
SEE
Asymptotics. Write S B if < CB for constant depending only on fixed parameters (dimension, regularity exponents, etc.).
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Vector Notation Fix spatial dimension d 2 and index j [l....,d]. For & (EI.....: Ea) dd we write
=-1:=(E1,.... Eoth,... Ed) E zd-1
for the vector obtained by removing the j-th coordinate of &. Conversely, for a (0l1,...,,a-))) zd-1 and we define the insertion map
(a, n)j (a1, ................................ aj-1, n, aj...... ad-1) E 2d
We use |. loo for the max-norm 2d i.e. E maxigiea [E When we write ū(a,r), r), this is
shorthand for the d-dimensional coefficient r)))
C.2 PROOF OF LEMMA 1
Restatement of Lemma (with explicit notation). Let u E H*(TT) with s > 0, and assume that its Fourier coefficients are compactly supported:
i(E) = for all 161 > No.
and
ú O.
Fix j E (I,.... df and set v(x) :- Iy u(x). Then there exist
REZ' 2
{1, 2No 1},
<> O. REN,
and an infinite set the fiber at jth coordinate.
Ea C 15 E zd; E-j ==j=a, a, |6,1 2 R}
such that
(EE) 2 (1+
Ag Ea.n.
(20)
In partieular, if for some E Zd-1 one has
No ū(a,r) a in the shorthand above),
r((a,r),) + 0 (equivalently,
No
then the bound equation 20 holds with the sharper exponent 1.
Proof. We work with square-integrable function v E (TT)) where T [0,2m]
The periodic "coordinate" and its Fourier coefficients. The function + Iy is not periodic. Introduce the zero-mean, periodic ID sawtooth
Step -
(x) zz - n, x E [0, 2TT]
extended periodically to T A direct computation (factorization of the integral and one dimenniiiii integration by parts) shows that its Fourier coefficients are supported on the joo-coordiiate: for & E Zd.
if if and E # 0,
(E)
TE,
(21)
0, if 0 E-, # 0.
Moreover Vj 1/. + π, hence
VEE) F(b)ul(E) + क ū(E).
(22)
Step 2 - Exact coefficient formula outside the support of û. Since i(k) = 0 for [6] > No. the
second term in equation 22 vanishes whenever [Eloo > No. Using equation 21 and the convolution
theorem, we obtain for any & E Zd with 00 > No:
i(E)
V(E)
Σ 3 ū(( RE24
Σ
(23)
E 55-6
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Thus, along any fixed transverse index Ê-x Zd-1, the tail v(a, n) for large [n] is a finite sum of simple fractions in the single variable n.
Step 3 - Moments on a nontrivial fiber and the first non-vanishing moment. Because ñi # 0, there exists at least one Zd-1 for which the fiber
Fa fr E Z: ū(a,r) r) o 0}
is nonempty. Define the (finite) coefficients C, = u(ox r) for E [-Noo No. No). and their moments
NO
Hp
p Cr
(p 2 0).
Let m (1..... 2No + 1) be the smallest index for which 1mm11 o 0. Such exists since not all C, vanish.
Step - Asymptotics along a line and a polynomial lower bound. For n E Z with |n| > No. formula equation 23 along the line É-, reads
No Σ No
v(a, n)) =
C
Expanding - = 1 02) for |n| > 2No and collecting terms yields the asymptotic expansion
v(a, n) = Mu
μ1 n
pmm11
n +
+0
|n| + co.
n By the choice of m, the first nonzero term is Jmm- 14m-1/n" 1/n". Consequently, there exist R E N and e > such that
((a,))
for all n 2 R, n
m
Step 5 5- Conclusion and the special case = 1. Then equation holds for all & Ea.8 with the exponent m determined Step 3, by the definition Ea,R and that [n]] 2 R. If μ4o = Σ, c, + for the chosen fiber (equivalently, No _ No a(a,)) r) * 0), then m and we obtain the sharper [V(E) (1 (61) along the line {-j az.
Remark 4. The explicit one-dimensional tail equation 23 shows that multiplying a band-limited field by the coordinate æ; produces a polynomial Fourier tail decay along lines parallel to the jth coordinate, with rate (1 + [5,1)==m where is the first non-vanishing moment of the finitely many coefficients on the relevant fiber. In particular. when m = 1, the decay is exactly (1 + (5,1)=1 Such algebraic tails are consistent with the preudo-spectral projection error estimate quoted in Kovachki
el al. [222] Thm. 40).
C.3 PROOF OF THEOREM
Restatement of Theorem 1 Let (al,..., E Nd with total degree M == |oc > 1 and
define the position-multilier
M(x) == σ7 x22
For inputs band-limited to radius No and lying H (TV) with s > S. any Fourier Neural Operator g that achieves [[M(xx) - G - S s (0 1) must employ a spectral bandwidth
(FNO width) m Z eM/e and a parameter count |0] 2 exp(ce-Madl Mdd/s) for some > 0 depending only on (d, s, No).
Proof. Set s'
8 with d
Step 1 Algebraic tail produced by M(x). Applying Lemma once per factor of æy shows that for some constant Co > 0 and infinite set Exo
zd
Co
| [[M(()u](E)) >
AA E Ex.
(24)
(1 M+I
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Step 2 - Pseudo-spectral projection lower bound. For any H pseudo-spectral projection error estimate gives (Gottlieb & Orszag 1977)
)(())) 1/2
II - [f 2 Ci
Σ (1 (1
E
where I is an identity operator. Insert f = (M() and the tail bound equation 24 summing over Ego n (o > m] yields
I - (x)))l 2 m (M
Imposing this residual
< e forces
2 M/s
C2 -, 0.
(25)
Step Canonical neural scaling in the latent map. An FNO with spectral radius (width) m manipulates latent vector of dimension (2m + 1) ~ ma. Approximating generic Lipschitz map
G Cm Cm" accuracy s/2 with fully connected network requires Yarotsky 2017:
De Ryck
et al.
2021 at least
parameters 2 8
(Kovachi et al.
2021 Remark 22)
Step Substitute the bandwidth lower bound. Using equation 25 md -Md/s Hence the latent network must have at least
super--xxpnenttial curse of dimensionality in the target accuracy
C.4 PROOF OF THEOREM 2
Restatement of Theorem
2 Let & 2 8 2 0 and 8p > d/2. Consider any finite composition
g (st Ci) ....o (si Li), C: 2.=0pml Op:(
where each Kohn-Nirenberg symbol p; belongs to W (TT x T and each s; is point-wise non- linearity of uniformly bounded degree (so the number of such nonlinearities is O()) and independent of E). For every e > 0 there exists single-layer KANO KANO such that
KANO
HHSHH S
00| ()-0)(2))))
Proof. Step 1 Kohn quantization error. By the quadrature bound from Demanet & Ying (011). for each C; we have
C Op.. (p:) ( H> S C' C'
where Op,, (p:) keeps only frequencies El I m. Since each & is a bounded-degree pointwise map, its Nemytskii operator is Lipschitz range and set Lmax max; Li Let M := m= (
bounded sets; write L; Lip(s, 110 on the relevant
|p (1 }. Because Sp > 2 implies IIp; L. $ Ilp w both C, and Op,(P,) are bounded on H with bound independent of m. A telescoping estimate for the interleaved composition then yields
g [9) S C..C'Bm-(--)
C. S (Liaxx Tmaxx-
Standard square integrable periodic Sobolev space on the product torus T2d.
Lipschitz constant the point-wise nonlinearity & on the relevant value range, i.e. L == sup (((a)=s(b)) with a, b restricted to the compact interval attained the i-th preactivations.
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Pick m = (2C. C2B/e) C'B/e) 1/0-5) part
Then the full composition g deviates from its projected counter-
(9) == (s Opm. ... a (51 Opm (P:))
by at most e/2 H Hd operator norm.
Step 2 KAN approximation of both symbols and pointwise nonlinearities. By the width-fixed KAN scaling law from Wang et al. (2024), for any > there exists a Kolmogorov-Amold Network Pi. with ii,y On 4/((2)))) such that llp, Pi.y S 7 on T x Likewise, since each ç fixed-degree pointwise map, its action the compact value range visited by the projected flow can be uniformly approximated by width-fixed KAN L i, with size L O0nn111(20) and
||s: Vinll 17. Choose
E 4f
7ay =
Ma ==
4e
Define the single-layer KANO
G ( Em Opm 2 ((m)yy))
1)
(L 1.
Opm (LI.mym))) )),
where the same spectral radius m from Step is used in every Opm (.).
Step 3 Error accumulation beyond projection. Using linearity of the symbol-to-operan map and stability of Nemytskii (pointwise) maps under uniform approximation, the post-projection error splits into sum of the symbol parts and the nonlinearity parts:
IKANO
KANO
|H-H"
OP... (p:) -OPm o ( H
|s5
L i.ma 1
I l Taym + l enl = c/2.
Combining with Step yields || g gkANO|
"H I S.
Step 4 Parameter complexity. Summing the sizes of all KAN blocks gives
|0| Σ ( + Σ ( + 0(1) d(E/0)-d/(25)) + 1=1
Since l and the number/degree of the s; are fixed (do not scale with s), the dominating term is ((e-d/(2s))), establishing the claimed complexity.
D ADDITIONAL DETAILS ON THE SPREC-PECTRAL BOTTLENECK OF FNO
In Section 3.1 performed a linear analysis of single FNO layer via its Jacobian to illustrate the pure-spectral bottleneck. We showed that, in the first-order approximation, all spectral off-diagonals arise from the spectrum of the input- dependeen gate o o'(z(u, -)) (12). Hence, although single FNO layer is capable of generating spectral off dif-oiagsnals, they are tidd to the input distribution and leads to the structural fragility in out-of-distribution performance (generalization on the unseen input distribution). Experimental results in Section are aligned with our concerns.
In this Appendix, we show that the same phenomenon persists every order of the Fréchet expansion of single FNO layer, and composing deep layers does not remove the input distribution dependence of the spectral off-diagonals generated by the model well.
Throughout, we work on the flat torus T = [- m] and Sobolev spaces H as in Appendix C
D.1 HIGHER-ORDER SPECTRAL OFT-DIAGONALS OF FNO
Recall the FNO layer Lixo
(x)) = a [F Ra, (E) Fm(()(EE) + Wa, -(xx)
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For the analysis it is convenient to collect the linear terms into a linear operator
A = F m Ro(E)
o Fm + Wa.
(26)
so that
Crso (u) dz x)), z(u, = (Au)(x),
(27)
where is the non-linear activation acting point-wise.
We first compute the k-th Fréchet derivative (Fréchet 1906) of (u)o
Proposition (Structure of higher-order derivatives
single Fourier layer). Let E H
and k be the order of the Freéchet derivative. Also let h1..... hy E H be the arbitrary direction functions of each order. Then for every integer k 2 1, the Freéchet derivative of (u)(()(x)) is:
D L hx](x) = (x(U, x)) (A,;))
(28)
j-1
where a (8) is the usual scalar k-th derivative. More explicitly, for each channel g.
D (u)( hell (x) = (z, x))
II Ab (x))
(29)
Proof. Since A 26 is linear, Dz(u) = A and D'z(u)= = 0 for all 2 2. By the chain rule, D (x) o'( d((z(u, x)) (Ah,)(x),
where of acts point-wise on each channel. For higher order derivatives, we apply the Faa di Bruno formula (Faa di Bruno, 1855). Because all higher derivatives of vanish, every term involving D D'z(w z()) with r > drops out, leaving
D (iso h&] =k( (2(u)) Ah ...,, AbR]
(30)
Since a acts point-wise, this reduces to Eq. 28). and to Eq. (equation 29 on each channel.
Thus, as apparent in Eq. every Fréchet derivative is point-wiss wise product of
an -nput-de gate σ (1)) x)) depending on the current input u, and
the product of fixed linear responses (AA;)(() incapable of generating spectral off- diagonals.
Therefore, the structure the high-order derivatives a single layer FNO is just as tied to the input distribution the first-order Jacobian shown in Section..11
D.2 SPECTRAL REPRESENTATION HIGH-ORDER DERIVATIVES AND MULTI-LAYER FNO
Fix 2 and define
(k, := (1) (2(1,))))
b,(x) = (Ah;)(x), i= I=1,..,,k
(31)
Then Eq.
28
reads
k
D LoNo (nso( hx](x) = (u x) II bx(x).
(32)
The Fourier transform of product of (k+1) functions is (k+ 1) fold convolution, so for each output frequency & E 2d we have
D L hi....... (E) (#x (u) by bz)(5)
(33)
Spectrally, the linear operator A (26) acts diagonally:
Ah;(E) A k,(g),
(34)
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where A(E) ((() + W in Eq.
12
Expanding the convolution in Eq. (33 yields
&
D LINO [hi...... hxl(E)
Σ
Sk u1) (5 -
Ex
II ((E)) (35)
E Ex.2.
depends inputu
where {E,} } are the frequency variable for each direction {br}. Eq. could be read as multi-linear operator whose spectral kernel
(35 shows that the k-tth derivative
(u Ei...... Ex) == (u) - -1
E6)
II (
(36)
connects (E1....,E. Ex) to the output function frequency &. Two structural facts follow immediately from Eq. (36):
1. Input function u decides the value (u) = F ao (0() (z(u,
The spectral multipliers ((;) depend only on the learned weights and remain diagonal in Fourier: they are independent of input u but does not contribute spectral off off-diagonals.
Consequently, for any 1:
Every non-diagonal spectral coupling is induced by Fourier coefficients of non- linear activation derivatives o (z)(z((,)) evaluated on the current input u.
Higher orders & 2 2 introduce convolutions higher derivatives g(k), ; but they do not mitigate the input dependence argument of spectral off-diagonals provided in Section 3.1
Deep FNO networks. full FNO is a composition multiple FNO layers:
END
(37)
Applying the Faà di Bruno formula (Faa di Bruno 1855) this composition, every term in CFND
'D DD g (u) becomes product of:
spectral multipliers (Ay(E)} from the linear parts of the layers, and
Fourier transforms of activation derivatives (() (2 (u, from intermediate layers j and derivative orders r 2 1.
Thus, deep FNO, all spectral off-diagonals any order still factor through Fourier transforms of activation derivatives evaluated on intermediate pre-activations governed by the input Adding layers introduces more such gates but not mitigate the input dependence on spectral off-diagonals introduced in Section 3.1
D.3 THE MEANING OF INPUT DEPENDENCE OF SPECTRAL OFF-DIAGONALS WHEN LEARNING SPECTRALLY DENSE GROUND-TRUTH OPERATORS
Let T be fixed linear operator such the position multiplier a(x) x r²((z)) in Section 3.1 In Fourier basis, T is represented by dense Toeplitz matrix such as T,, [x22 as in Eq. [1 and its Fréchet derivatives
(T(u) T for all D²T(u) = for all k 2,
i.e, the spectral off off-diagonals of T are completely independent of the input u.
Suppose we wish to learn with robust ofitt--diittiiuttin performance over a Sobolev ball
BB = uu H |||||и~ S B}-
natural notion of generalization is small error in operator norm for an unseen Sobolev ball Bp: g (u) Tu H²
sup ue 5g
< E.
(38)
11 H
This requires that Bg
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the Jacobian (g
(u) stays close to T, and
* all higher-order derivatives Di (M)), k 2 2, remain uniformly small (negligible).
Combining Eq. (36) with the composition structure above reveals a paradox:
1. If the gates ((k) (z()) (u. -)) and their Fourier transforms Sk (u) vary significantly with u E Bg, then both the Jacobian and all higher-order kernels K2 (u: E.,....., Ex) the input. In that case the effective spectral off-diagonals of 9 cannot coincide with a single, input-independent Toeplitz kernel across all E Bn. Any dense off-diagonal pattern learned from the train distribution will be tied to the train subspace and becomes fragile under distribution shift, as observed in our experiments in Section
vary with
2. If, on the other hand, we try to make these kernels effectively independent of u on B3, then Eq. (28) and (q.(36) force all activation derivatives be nearly constant (for 1) or nearly zero (for k 2 2) on the relevant pre-activation range. In this regime, the network is forced into an almost linear operating region where
a' is approximately constant, 81 (u) is concentrated near zero frequency and D ( becomes (block-)diagonal in spectral basis; and
((k) = 0 for k 2 2, so higher-order terms vanish.
The resulting layer effectively reduces to a spectral multiplier A(E) and cannot represent a dense Toeplitz map such as T [xr²] whose off-diagonals are non-trivial and fixed independent inputs.
In other words, for spectrally dense, position- operators, FNO faces a fundamental trade-off:
it can use strongly input-dependent gates to synthesize spectral off-diagonals, but then those off-diagonals are necessarily tied the input distribution,
it can suppress the input dependence of the gates to emulate a fixed operator, but then the spectral kernel collapses towards a diagonal (or nearly diagonal) multiplier.
In neither case does a practical-size FNO realize fixed, input-independent dense Toeplitz kernel with robust nut-of-distributio generalization for spectrally dense operators, even though sufficiently large FNO can parametrize the off-diagonals of the in-sample mapping on the training distribution.
What we do (and do not) claim about FNO. The generalized analysis in this Appendix refines the statement of Section 3.1
We do not claim that FNO cannot generate spectral off-diagonals. Eq. 12 and Eq. (36)
show that at first and higher order, off off-diagonals appear whenever the activation derivatives have non-trie trivillvial Fourier coefficients.
We do claim that for spectrally dense operators, these off-diagonals always tied to nepunddepeneent gates. As a result, a large FNO can fit the off-diagonals of the in-sample mapping on the training subspace, but it cannot efficiently learn fixed dense off-diagonals with robust out-of-distribution generalization on unseen function spaces which is exactly what we observe our experiments in Section
Stacking more layers introduces more input-dependent gates but does not create an input-independent spectral mixing mechanism, so depth does not mitigate this bottleneck.
D.4 WHY KANO DOES NOT SUFFER THE SAME PROBLEM
By contrast, KANO directly learns an input-independent pseudo-differentio symbol (x E) in the dual bases via Eq. (15). KANO has spectral kernel capable of mode mixing as it is governed by spatial basis x (which is convolution in spectral basis) as well as the frequency mode & via symbol px(x, {), not only by the non-linear activation gates tied to the input function Once P& is learned, it is shared across all input functions in the ball Bg, including even the unseen function subspace. This dual-domain, symbol-based parameterization allows KANO to learn the fixed -ff-diagonals of spectrally dense, position-dependen operators with robust out-of di-dtrtiibttion generalization, as confirmed by experiments in Section
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E COMPUTATION AND MEMORY COMPLEXITY OF KANO
In this Appendix we compare the memory and computation complexity of KANO compared to FNO. We first quantify the per-layer costs of single FNO layer 4 and single KANO layer [1] and then combine them with the parameter complexity results of Section 3.2 and Section 4.2 to argue that, the target class of variable-coefficientt PDE and position-dependent dynamies, the higher per- cost of KANO is compensated by better model size scaling.
Throughout this appendix work on the discrete torus T with uniform spatial grid y (y1,...., J and truncated Fourier set = {E1..... Em } as in Section 2. and Section
E.1 PER-LAYER FLOPS AND ACTIVATION MEMORY
We measure complexity in floating-poing point operations (FLOPs) and activation memory per forward pass for single layer of each model. Backward passes in standard automatic differentiation assumed to be within a constant factor of the forward cost and do not change the asymptotic conclusions.
FNO layer. Consider single FNO layer 44
(a)) o F Ro (E) Fm Fm)(a)())(x)) + Wa, a(x)
with Cj input channels and Cou output channels. On d-dimensional grid y with m coordinates, single forward application of LANo has the following costs:
FFT and inverse FFT: F' and F are applied channel-wise and cost
FFT cost d(Cinn + Cou) m² log mo). Spectral multiplier: for each retained mode & E Ro(E) CCaxCh is dense matrix; multiplying by i({)= E CCo costs (Ciin Cout) per mode, hence
spectral block
Om" Cin Cour).
We is applied at each spatial point E 3 as a dense matrix in Point-wise linear map: channel space, costing
spatial linear map Am" ConCout.
Nonlinearity: the point-wise nonlinearity Om Cou)).
Collecting terms, for fixed channel counts we obtain the per-layer forward cost:
FLOPs(Crs)) Am" logm" + +m Ammlog m²).
(39)
The activation memory footprint is dominated by storing u, F. u, the pre-activation z(u,)) -) and the post-activation:
memory (Froo) Am" Cin + md Cout) O(m").
(40)
The parameter memory is Omm² Cou for the spectral multipliers plus (Cin Cou) for We.
KANO layer. Now consider a KANO layer 15
(RANo(a)
De, F
(po (a)))) a(x)).
where Po, (x, E) is implemented by a width fixed KAN symbol network and o is another width-fixe KAN activation network. Using Kohn-Nirenbery quantization (16):
Fx(x)( F.n (a)(()))) = 11
(xx x) ((y)))
CEEYEY
on a d-dimensional grid y with md points and a retained frequency set with md modes, a single forward application of LKANO has the following costs:
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KN quantization (double sum): for each output location evaluate the double & E and y E EV, i.e. m terms per x. Each term involves a dense matrix vector product p(x, E) a(y) of cost CaminCon). Summed over all x y this yields
KN operator
Am CouCo
Symbol network evaluation: the symbol KAN Po, is evaluated once per pair (x, <) E v xxxE. E, i.e. at mdd points. Let Pp denote the cost of single forward evaluation of pö,- Then
symbol KAN ~ Am Pp).
Activation KAN: the activation network De, is applied point-wise at each spatial point x E y. Let P denote the cost one forward pass of Φ0 Then
activation KAN
Am" Po).
Collecting terms, for fixed channel counts and fixed KAN architectures we obtain the per-layer forward cost:
FLOPs(CxANO) Am Cont + m PPp + mPP).
(41)
The activation memory footprint is dominated by storing the feature maps and, if materialized, the symbol grid po. (x: memory (CAAN)) Om" Cin + md Cou + m²d Cin ~ Om2² CinCou):
(42)
in addition the parameter size of the KAN subnetworks.
Inference cost of compact KAN subnetworks. For completeness, we quantify the inference cost P, and P& of the KAN subnetworks. Consider fully connected KAN layer with input width din- output width dowt, and G' basis functions per edge. Each edge (j i) carries learnable univariate function Sij + IR represented as
G (6(()) = Σ
Wiig Og (x),
where 1G are fixed basis functions (e.g. B-splines or rational functions) and Wijg are learned coefficients. Evaluating (1, for a given scalar input Ty requires computing the active basis functions 6g (2₁) and dot product over G elements.
single KAN layer thus computes. for each output coordinate
dn
y:
fo((,))
and the total cost of one forward pass through this layer would be (dour di G), up to lower order terms from basis evaluation. The parameter count of this layer is of the same order, O d, G). Under the assumption
compact KAN subnetworks, widths, depths, and the number basis functions G are all small and independent to the operator resolution m. If we denote by N an upper bound on their layer widths and by LKAN their depth, then their total inference costs satisfy
Pp. Pe (TRAN N2 2²G). G)-
In the per-layer KANO complexity (41) the KAN subnetworks contribute only a constant-factor overhead that does not grow with the spatial or spectral resolution, hence for large resolution m >> N, LKaN the dominant term would be the KN quantization term 3
E.2 COMPUTATION COMPLEXITY AS FUNCTION OF ACCURACY FOR POSITION-DEPENDENT DEPENDENT DYNAMICS
As apparent in Eq. 39 and Eq. KANO requires orders heavier FLOPs compared to FNO due to the expensive double sum nature of its KN quantization, when two models are of similar size. For small resolution m, the inference cost P, and Po still set hard lower bound on the computation
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complexity well. However, one of the main arguments make in this work is that KANO enjoys incomparable parameter efficiency compared to FNO on the target class of spectrally dense operators such position-dependem dynamics: parameter complexity of KANO scales polynomially where that of FNO can scale super-exponentialy on spectrally dense ground-truth operator. Therefore, KANO's expensive FL.OPs requirement can be compensated by parameter efficiency where FNO suffers the curse of dimensionality.
For position-dependent dynamics 9 and accuracy s > write [e)x and (e)) for the smallest parameter counts required by FNO and KANO respectively to achieve |g - Gal S e, following Theorem and Theorem 2 Also, let (E) and (E) denote the cost of one forward (or forward+backwaan
pass of an FNO or KANO architecture chosen to achieve accuracy on the ground-truth operator. We not attempt to model the number of optimization steps; instead, we focus on how the cost of a single training or inference step scales with E. Assuming model must perform least constant number of FLOPss per parameter in each step, lower bound for computation complexity can be estimated as:
(E) 2 Co [Orooe)),
(E)
(E) Z d (0AAoo)))
(43)
for constants co. c1 O. Such lower bound assumption is not universal for any neural network, but in FNO architecture all its parameters equally appear in the matrix computation via the dense parameter block and W; especially for implementations that evaluate and update all parameters in every step. Hence, computation complexity lower bound assumption by its model size is reasonable for FNO.
Computation complexity of FNO by accuracy for position operators from Theorem 1 For the Theorem in Appendix C.3 shows that FNO g that achieves
M(x) == x ....', do of total degree M := ad] 2 1, the restatement of
|M g H Se
on Lund-limited inputs can suffer curse of dimensionality:
[+)) ~ explee-Ma/0)
(44)
for some constant > 0 depending only on geometric constants of Sobolev space H From Eq. 43
the computation complexity of FNO by accuracy on M can be written as:
TE) 2 co 10mo(=)] ~ ((oxp(e-Md/o))
(45)
Computation complexity of KANO by accuracy for position operators from Theorem 2 the other hand, Theorem 2 and Corollary show that as the KANO projection of M yields symbols (x {) of sufficient smoothness, KANO can achieve ||MM
On
g GRAND
|| I with
|0kNooE)| O(=
(46)
where = =d (2s,) or 3 d/(2r) geometric exponent determined by the symbol regular- ity (Wang et al.) (024). From Eq. (43). the computation complexity of KANO by accuracy on M can be written as:
TE)) Z dg [#))No(s)] ~ ((epp(=
(47)
Therefore, when the ground-truth operator is spectrally dense yet its KANO projection give smooth enough symbol so that both Theorem [ and Theorem 2 are effective (for instance, -eeeedetd
dynamics), lower bound of computation complexity of FNO can prevail over that of KANO because of the curse of dimensionality discussed in Section 3.2
We emphasize that this is an asymptotic statement under the specific operator class where both Theorem | and Theorem 2 Phold. For operators that are nearly spectrally diagonal (e.g. standard FNO benchmarks), FNO is theoretically much faster than KANO when the two models are in similar size. However, when learning a spectrally dense, cowabbe-oofffciintt operators to high precision, KANO's parameter efficiency can, in theory, compensate for its expensive double sum KN quantization while FNO faces a super- exponential growth in model size hence per-step FLOPs to achieve the same level of accuracy.
28

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Published as a conference paper at ICLR 2026
F USE OF LARGE LANGUAGE MODEL
Large Language Model (LLM) is used to polish the writing in this paper, such as detecting grammar errors and typos. LLM is also used search for appropriate references for correct citations, and all the proposed references are fully inspected and verified before citing in the paper.
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Reviews:
[{"review_id": "rjhL6AMOF8", "reviewer": "Reviewer_Nuuh", "summary": "The authors consider the neural operator learning setup and consider a class of operators which involves spatial multipication and term it as position dependent dynamics to highlight the shortcoming of the existing fourier neural operator framework. Since the multiplication and differentiation have a dual relationship under the fourier transform, the authors argue that the considered class defined by the multiplication map is just a two step up-shift sparse matrix in the spatial basis, however, takes the form of a dense Teoplitz matrix in the spectral basis and the fourier operator cannot efficiently approaimte the off-diagonal elements of this matrix. The authors first show that this position dependent dynamics will induce a super-exponential scaling in the fourier operator. To adress this issue the authors propose the KANO framework which replaces the nodes of standard MLP with simple sum operations and learn the univariate 1D functions. This reparameterization is similar in expressivity to MLPs upto constant depth and width empirically. Also, they propose using a KAN sub-network jointly parameterized by both spatial and spectral basis and enjoys sparse representations in both spatial and spectral domains. They also discuss that the projection error scales efficieitnyl by its width and the latent network error also scales efficiently independent of the projection error and thus, it is more efficient as compared to the FNO for the considered class. Furthermore, they also verify this empirically by considering a dataset of the considered class where they observe that KANO outperforms FNO with much lesser size required.", "questions": "Since the FNO based layers have been also used for a lot of other tasks like weather forecasting [1] or like token mixers for transformers [2], do the authors have any comments on the broader applicability of this framework?\n\n[1] Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators.\n[2] Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers. ICLR 2022.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Investigating the bottlenecks of the FNO by considering this position dependent dynamics class and then drawing the inference that this will incur dense representation in one basis and sparse in another is really interesting. Further utilising the KA network based parameterization and resolving this issue is also interesting and provides new insights for various scenarios of this operator learning framework. Further verification by empirically investigating this under different testing /training families and demonstrating that KANO can outperform FNO with much lesser paramter. Also the section 5.2 further including two-position dependent quantum dynamics benchmark further strenghten their claims.", "weaknesses": "No major weaknesses but the authors could have also compared the FNO and proposed KANO on the original equations considered in the FNO paper to see how does it perform in that setup to understand its overall use case as against this specific class. It is a bit unclear how this new framework will do on the  other tasks that FNO can perform very efficiently.", "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": "GCvWqHeazs", "reviewer": "Reviewer_bFzm", "summary": "The authors proposed the Kolmogorov-Arnold Neural Operator KANO that uses the KAN to construct the neural operator, which has very solid advantages over FNO both from theory and experiment.", "questions": "1. Are there other results that demonstrate the advantages of this method?\n\n2. For the demonstration of the spectrum limitation of FNO, the author uses the linear approximation. Will the off-diagonal contribution arise in higher orders? What will that affect in the argument in this paper?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The explanation of why the FNO have problems is very clear. Strongly support the contribution and advantages of the proposed KANO.", "weaknesses": "1. The idea is quite natural, if picky. But it does not affect the contribution.\n2. The experiment is quite limited. I think this method should benefit a lot from the flexibility of the KAN. Extra experiments of diverse operators can make this paper attractive to a broader audience.", "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": "8ZSQQJ3HsK", "reviewer": "Reviewer_mZHA", "summary": "The paper introduces the Kolmogorov–Arnold Neural Operator (KANO), a novel neural operator that learns pseudo-differential symbols in both spatial and spectral domains using Kolmogorov–Arnold Networks (KANs). The authors identify a theoretical limitation in Fourier Neural Operators (FNOs), the \"pure-spectral bottleneck\", which restricts their expressivity for position-dependent dynamics. They provide a rigorous theoretical analysis showing that FNOs suffer from super-exponential scaling in model size for such operators, while KANO achieves polynomial scaling. Empirical validation includes synthetic PDE operator learning and quantum Hamiltonian learning, where KANO demonstrates strong out-of-distribution generalization, high parameter efficiency, and symbolic recovery of operator forms.", "questions": "How does KANO perform on widely used PDE benchmarks such as 2D Navier–Stokes or heterogeneous material modeling?\n\nCould you include comparisons with stronger baselines like U-FNO, PDNO, or adaptive FNO variants?\n\nWhat are the runtime and memory costs of KANO compared to FNO during training and inference? Given the dual-domain computations, how does training time scale?\n\nHow robust is symbolic recovery when operator coefficients are non-smooth or discontinuous?\n\nWhat optimization challenges or instabilities were encountered when training the KAN sub-networks within the operator framework, and how were they addressed?\n\nDid you perform ablation studies to understand the individual contributions of the dual-domain design versus the KAN sub-networks? For instance, what is the performance of a KANO variant that uses MLPs instead of KANs?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This paper makes a significant and well-structured contribution to SciML. It identifies a fundamental flaw in a popular architecture, proposes a principled solution, and provides convincing evidence of its superiority. The authors present a rigorous theoretical diagnosis of FNOs, exposing the “pure-spectral bottleneck.” They show that for position-dependent dynamics, including quantum mechanics and fluid flow, FNOs suffer from super-exponential scaling. This is framed not as a minor drawback but as a deep architectural flaw.\n\nIn response, the KANO is both novel and well aligned with the mathematics of the problem. By jointly parameterizing operators in spatial and spectral domains through a pseudo-differential framework, KANO represents each term in its naturally sparse basis. This dual-domain approach provides the right inductive bias for robust learning.\n\nThe empirical results reinforce the theory with striking effect. KANO does not merely improve modestly over FNO but shows dramatic gains: flawless out-of-distribution generalization, 0.03% of the parameters, and symbolic operator recovery accurate to the fourth decimal. This interpretability moves the model from black-box approximation toward genuine scientific discovery.", "weaknesses": "The main weakness is the narrow experimental validation. Results are confined to 1D synthetic operators and quantum systems, leaving scalability to high-dimensional PDEs untested. Absent are standard benchmarks such as 2D Navier–Stokes or 3D elasticity, along with any runtime or stability analysis, making the practical utility unclear.\n\nComparisons are also limited. Most results benchmark only against vanilla FNO, whose flaws are already established, creating a straw-man dynamic. Stronger baselines like U-FNO, AM-FNO, PDNO, or multi-scale FNO variants are missing, making it hard to judge whether KANO’s advantage stems from its design or from outdated comparisons.\n\nFinally, while symbolic recovery is compelling, it remains demonstrated only in simple, smooth cases. Whether this interpretability extends to non-smooth, higher-dimensional, or more complex operators is left open. Overall, the idea is strong, but the evidence is not yet broad enough to confirm its readiness for widespread use.", "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)