Adaptive Mamba Neural Operators
Authors: Zeyuan Song
Zheyu Jiang
Venue: ICLR 2026 Poster (2026)
Keywords: Neural operator
partial differential equation
adaptive Fourier decomposition
Mamba
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We incorporate adaptive Fourier decomposition in Mamba architecture to design a new neural operator for PDE learning.

Abstract: Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks. The code is available at https://github.com/checlams/AMO.

Published as a conference paper at ICLR 2026
ADAPTIVE MAMBA NEURAL OPERATORS
Zeyuan Song & Zheyu Jiang School of Chemical Engineering Oklahoma State University Stillwater, OK 74078, USA
(taekwon.sony zheyu. biangteokstatt..dd
ABSTRACT
Accurately solving partial differential equations (PDEs) on arbitrary geometries and variety of meshes is an important task in science and engineering applica- tions. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the ker- nel integral formulation of SSMs. This is achieved by constructing Takenaka- Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and approximate the solution manifold of PDEs on wide range of geometries and meshes. In sev- eral challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and ir- regular domains, AMO consistently outperforms state-of-the-an solvers in terms of relative 1 error. Overall, this work presents a new paradigm for design- ing explainable neural operator frameworks. The code is available at https: //github. hwwpittub.cmm/haa. wtt.co///c.ccc........ccomm
INTRODUCTION
A wide range of scientific and engineering phenomena, including fluid dynamics, heat and mass transport, structural mechanics, and cell growth, can be characterized and modeled by partial dif- ferential equations (PDEs). Most nonlinear PDEs do not have analytical solutions and need to be solved numerically. Traditional discretization-bass tionally expensive. To speed up the solution process, neural operators have recently been proposed as an extension of neural networks to learn the dnfinitedimeensioalll solution operators of various PDE problems. It has been proven that, with finite-dimensional solutions as training data, neural op- erators accurately learn the infinite-dimensional solution space. Once learned, neural operators are mesh-indeppendent, so neural operators trained on coarse grids can generalize to finer grids.
approaches for solving PDEs can be computa-
Frequency- based neural operators, such as Fourier neural operator (FNO) L al. 2020), wavelet neural operator (WNO) (Tripura & Chakraborty (202).. multiwavelet transform (MWT) Guupa
et al. 202T), , U-shaped neural operator (Rahman al. 2022), spectral neural operator (Fanaskov & Oseledets 2023), and latent spectral model (LSM) (Wu et al. (2023), are attractive since the solution space of many PDEs can be naturally expressed in spectral bases. Frequency-based neural operators approximate the PDE solutions by learning how frequencies evolve, and nonlinear terms become convolution in the associated frequency domain. However, the performance of existing frequency- based neural operators may deteriorate in irregular geometries (Li et al., 2023), as their associated bases could lose orthogonality and eigenfunction properties in irregular domains (Lingsch et al.) 2023 Chen al. 2024). As a result, retaining these important properties for kernels and bases for irregular domains is critical.
Along this line, a recently proposed neural operator solver, latent Mamba operator (LaMO) Ti- wari et al.) 2025), shows great promise in capturing PDE solutions on irregular domains. LaMO integrates the efficiency of state-space models (SSMs) in latent space with the expressive power of kernel integral formulations in neural operators. Although the selective convolution kernels utilized in LaMO can effectively capture PDE solutions on irregular domains, their lack of orthogonality property may lead to spectral mixing. Furthermore, the kernels in LaMO are finite-order linear dynamic filters (Gu & Dao 2023), which may introduce low-pass filtering bias, leading poor

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covery of hiih-ffreog figh-fre and singular features (Gu et al. 2021 Gu & Dao 2023 In the illustrative experiments discussed in Appendix [B] we show that LAMO suffers from deviation in the propaga- tion of high-frequency perturbations for 1-D advection PDE, and it fails to capture the singularities in 2-D Darcy flow equation with fractal noise as the permeability field.
Recognizing the fact that LaMO lacks frequency-domen implementation, here we propose Adaptive Mamba Neural Operator (AMO), a novel neural operator architecture that synergizes an adaptive
Fourier decomposition with the efficiency of structured SSMs in the frenquency domain (Gu & Parnichkun et al. 2024). AMO parameterizes the SSM transfer function in a Takenaka- Malmquist (TM) system in reproducing kernel Hilbert space (RKHS). thus allowing state-free kernel construction and inference directly on the spectrum. The Mamba blocks in AMO serve as rational filters while retaining linear-time selective scanning. Furthermore, turns out that AMO structure resembles adaptive Fourier decomposition (AFD), novel signal decomposition technique achieving higher accuracy and significant computational speedup compared to conventional signal decomposition methods (Qian 2010 Qian et al., 2012). The architecture and design of AMO is fully guided by the AFD theory, thereby improving the mathematical explainability and groundness of AMO,
Dao 2023
Overall, our key contributions are summarized as follows:
AMO is the first neural operator which explicitly incorporates TM systems and Fourier-based methods into the Mamba structure. AMO accurately solves the PDE problems diverse ge- ometries and effectively handles singularities and long-range dependencics of PDE solutions, Furthermore, we develop theoretical foundations for AMO and prove that AMO performs AFD approximation of PDE solutions.
2.
The design of every component of AMO is fully guided by the AFD theory, leading to a math- ematically interpretable and grounded architecture. Using a TM layer, AMO projects the input into TM systems in a Hardy space, and constructs the reproducing kernels from adaptively se- lected poles. These adaptive poles serve to construct the reproducing kernels adaptively. We demonstrate the importance of utilizing adaptive poles as opposed to fixed poles and investigate how the number of adaptive poles influences the performance of AMO.
3. AMO outperforms state-of-the-art neural operator solvers in terms of accuracy across a diverse set of benchmark PDE problems, including plasticity, elasticity, airfoil, pipe flow, Navier-Stokes, and Darcy flow on various geometries. It also achieves outstanding performance in financial ap- plications, such as solving the Black-Scholes equation for the European option pricing problem.
2 RELATED WORK
Frequency-based neural operators. Early advancements in operator learning exploited spectral decompositions to encode global information efficiently. A notable example is FNO (Li et al.
2020). which parameterizes integral kernels in the Fourier domain to enable resolution-invariance.
However, FNO does not generalize well to irregular geometries (Li et al. 2020 Later, -NO 2023 was proposed to solve PDEs on general geometries. U-FNO (Wen al.) 2022 [i et al.
introduced architectural modifications to better capture localized details while maintaining FNO's s
global properties. Meanwhile, F-FNO (Tran et al. 2021 generalizes the FNO architecture for more efficient spectral layers and deeper architectures. On the other hand, neural operators based on the wavelet transform include WNO (Tripura & Chakraborty, 2003). MWT (Gupta et al. [202), Padé (Gupta et al. (022). and CMWNO (Xiao et al.) 2023a).. Fourier and wavelet transforms are both special cases of spectral decomposition, and neural operators based on spectral decomposition has recently been proposed (Fanaskov & Oseledets) (203)..
Attention-based neural operators. Attention mechanisms have been widely studied in neural op- crator domain. Some of the notable works include orthogonal attention (Xiao et al. 2023b), physics- cross-attention WWagg & Wang 2024). and nonlocal attention (Yu et al.) 2024). The Transformer structure is also a promising building block for neural operators. Some of the related works in- clude OFormer (Li et al. 2022). LSM (Wu et al. 2023), and Transolver (Wu et al. 2024). However, Transformers struggle to capture kernel integral transforms efficiently in complex, high- high-dimensional continuous PDEs (Guibas et al. 202T).
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SSM-based neural operators. To address the computational inefficiency of Transformer-basee neural operators, SSM and Mamba emerge as promising architectures for neural operator designs (Tiwari et al. 2025). Previous studies of SSM-based neural operators (Zheng al.) 2024 Cheng et al. 2024] Hu et al. 2024 Tiwari et al. 2025) have been applied to nonlinear PDEs on irregular geometries and dynamical systems. These works incorporate traditional SSMs with different scan strategies without considering the information in the frequency domain. On the other hand, our AMO considers the frequency information via its explicit kernel and SSMs from transfer function perspective (Parmichkun et al. 2024).
3
ADAPTIVE FOURIER MAMBA OPERATOR
3.1 PROBLEM STATEMENT
We frame our task as learning a solution operator for family of parametric PDEs. In general, consider a PDE defined on spatial domain Q C Rd and a time interval (O,
La fa(u(r, t)] = t), v(z,t) E x (0,T),
(1)
which subject to a set of initial and boundary conditions. Here, the parameter function a A specifies the coefficients and initial and boundary conditions of Equation [ In operator learning, our goal is to construct an accurate approximation for A F(D x [0, T),, which maps the parameter function to the corresponding solution function u(x; t) E F, via a parametric mapping
Go. The aim is to learn such that Go Rs G from a set of training data {aaj u;)}
3.2 AMO ARCHITECTURE
AMO is novel neural operator architecture that synergizes the mathematical groundness of AFD theory with the efficiency of structured SSMs in the frenquency domain (Gu & Dao 2023 Par- nichkun et al. 2024). Different from LAMO (Tiwari al. 2025). which compresses the physical tokens into a lixed-size latent representation, AMO utilizes a multi-layer fully-connected feedfor- ward neural network (MLP) to first map the encoded tokens to their counterparts on the reproducing kernel Hilbert space (RKHS), and then iteratively refine them by series of processing blocks. Each block uniquely integrates two components: (i) a TM layer containing global spectral transform via data-dependent TM bases, and (ii) bidirectional SSM (Gu et al., 2021 Gu & Dao (023) parame- terized by transfer functions in the frequency domain (Parnichkun et al. 2024) to efficiently capture long-range dependencies within the RKHS.
Neural architecture. is:
Given the parameter function (input) a, the output of AMO, denoted as âN,0.
NN.0 (a (2 a SN o L o o L' o R P) (a),
where is the function composition, N is the number of processing blocks, P the lifting operator which encodes into a lower-dimensional space (maps the input to the first latent representation zn) (Tiwari et al.) 2025 Li al. 2020). Q is the corresponding projection operator mapping the lower- dimensional space back to the original space (maps the final latent representation ZN+1 to the output) (Twarii et al. 2025 et al. 2020), is multi-layer neural network mapping the physical token I,..., N) is the processing block of AMO (which consists
an RKHS, L SSM' TM' of a TM layer and bidirectional SSM), and S (i = 1...., N) are aggregation layers with skip connections. These aggregation layers not only receive the final output from the layer sequence but also have access to the intermediate outputs from each of the preceding layers.
The lifting operator, P. projects the N₂ physical token inputs into a compressed set of M encoded tokens, where M < Ng. This projection is achieved via a cross-attentic mechanism. A learnable query array, gg Deseed, acts as the query. The key and value pairs are constructed by combining a linear projection of the input features Xphys with a positional embedding of their coordinates Sphys generated by positional encoding network PEN. Here, Xphys E N, Di stacks the feature vectors (x:).N, and Sphys N.x. stacks the coordinates {g. N. and the physical token is essentially
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pair (g:xx)) x;). The process for generating the initial representation Zu is formally defined as:
kv Linear Xphys) + (gphyy)
Z = CrossAttt(query= = L, key = kv, value kv),
(3)
= z + FFN(Z)),
where the output of the cross-ates cross-attention module is processed through a residual connection and a standard feed-forward network FFN.
The mapping operator, denoted by R, acts on the encoded representation produced by the lifting operator P, which transforms this discrete encoded tokens into representation within a continuous function space. Let Zu I Mx Dannsi be the set of encoded tokens generated by P, the operator R RR x - H maps this representation to its counterpart an RKHS H. This mapping is typically implemented as a multi-layer fully-connected feedforward network MLP, which processes each token independently as:
Z1 R(zo) MLP((z),
(4)
where denotes the projected tokens in the RKHS. We remark that, the mapping operator R maps the encoded tokens 2g to the new tokens 21 in H without knowing the physical information Xphys and Sphys-
The TM layer, denoted by TM' I,..,, performs global convolution via spectral transform, where the reproducing kernels and TM bases are constructed from data-dependent poles. To define the reproducing kernels, we parameterize a small MLP to predict a set of complex values called "poles" {ax [==) (denoted ans)) located in the unit disk D {= C: |=| < 1} from tokens 2, Once we have the set poles, we can explicitly define the reproducing kernel (z) as:
1
K=(2)
(5)
a2
where 2 E H. and a is a single pole satisfying |a] < Intuitively, we remark that each pole can be viewed "tuning knob" that selects a particular spatial pattern in the solution, with its location in the complex plane controlling how localized that pattern is. Adaptive poles allow AMO to survey more heavily in regions where the parameters change rapidly, while using fewer poles in smooth regions. Across layers, the poles evolve from broad, coarse patterns in early layers to more refined, problem--pecific patterns in deeper layers.
To generalize on irregular geometries, the kernels in Equation 5 need to be modified to become orthonormal. These modified kemnels are also known as the TM bases due their deep connection to TM systems. The first basis, denoted as B1, is simply the normalized kernel of Equation
/1-1e.12 Then, start with /1- but it is not orthogonal to 1-ags
with pole a (=;a))
1-ar=
B1. We reach the orthogonality by subtracting its projection onto B1, and we get B2(2; a1:2)
/1-1e2
(--01_ (1-770)
after normalization. This way, the bases Bi are finally formulated as:
1-022
(2;001)) /1 |a
0j azz'
(6)
04
where H and an are poles learned by the small MLP satisfying |avl < 1 for k 1, i. Overall, the i-th TM layer TM' applies small MLP z + a1s. and then construct the TM bases B, according to 6 We remark that, the tokens z; will be kept as the input of SSM' along with the TM bases D..
Bidirectional SSM block is effective in solving PDEs irregular geometries (Tiwari al. 2025
and employs inherent kernel integrals. However, this inherent kermel does not contain information in the frequency domain, thereby falling short in capturing high- figgufrry and singular features. To address this limitation, we utilize the transfer function in training SSMs in the frequency domain (Parnichkun et al. 2024). The SSM block SSM' generates the spectrum of output in the frequency domain (e(e")) as the product of the spectrum of input Z(e(w) and the transfer function He(e")).
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Le., Z(e")),,( )e We point out that the output is essentially the coefficient of discrete AFD operation with the form (z). Bi) (Qian 2010; Qian et al., 2011), where the inner product is defined
as (x, - - Σ - x x[n]// (eelnn/N Here, N denotes the length of signal æ {a(nx)) )
Let us consider the impulse response h2 of SSM block SSM' (linear time-invariant system) as:
1 2m 2x
h:[n]] Ju
Bi
(etw;ara)) du.
Then, the corresponding transfer function H2 be obtained as:
H:(e"")= 3; (etw; 01:;)-
(8)
By setting the transfer function of SSM be Equation g the SSM block computes correlation of the input z; and B;:
Yi(e")) He (((e)) = Be a1:i) ((e"))
(9)
in the frequency domain. In the time domain, Equation 9 leads to the update of zj:
M-1
2 (h+ 21{
Σ
z [n|
B ( (cl2x(n
a1:));
(10)
where l denotes the time shift in the correlation operations. output:
The zero-lag sample gives the final
M-1
21+100 = (h, 20] = Σ 2: B, (ei2an/M; ;11s) (21. Bi).
(11)
n=0
S has N neural layers and combines the skip connection zi with the inter- mediate outputs 2 [0] = L (z:) and Bi = TM (z:):
Aggregation layers
Z2 5" (z1, 22 (0) 9 2ュ 00]
for i == 3 3
(12)
Zi+1 = (2z 2i+ [0]. ,,) = zi + (2:+1 [0] O B,)
for 1,
where @ denotes the element-wiss (Hadamard) product.
Output. Finally, the output of uN.e is the projection of ZN+1 by the local transformation Q
Li
et al. 2020
N+1
M-1
uN.0 = Q
Σ n=0
( (e2an/MM and))
2 in
B,
3 Bi
(13)
4 PROPERTIES OF AMO
Connections to AFD theory. Adaptive Fourier decomposition (AFD) is novel signal decom- position technique that leverages the Takenaka-Malmquiss system and adaptive orthogonal bases Qian 2010 Qian al. (012)) It admits proved convergence of any signal E H such that 20, QQin et al. 2011 Wang et al. 2022 for the chosen orthonormal bases D;
Saitoh et al. 2016). Thus, the output of Equation 12 Zi+1, equivalent the AFD operation, Dx. Furthermore, the output in Equation 13 can be approximated as E ( B)) where ú;-1, Q(zi). This is also
i.e. 7 = Σ1_
(Zk
equivalent to the AFD operation. Thus, several theoretical properties of AMO, including conver- gence and error bound (see theorems and proofs in Appendix[D]. can be guaranteed with efficiently large layers, thanks AMO's deep connections with AFD theory.
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Connections to Parnichkun et al. (2024). Parnichkun al. (2024) proposed state-free inference of SSMs by learning the coefficients of the rational transfer function H instead of the traditional
state-space matrices A, B, and C' (Gu & Dao 2023). which is called rational transfer function (RTF)
approach. Specifically, the RTF learns H as:
b + 22z 2
b
(2) = ho +
(14)
1 012 a22
Og
where aj, b,, and hg are denominator coefficients, numerator coefficients, and feedthrough term, respectively. When it comes to AMO, we push the formulation of transfer function in Equation g and learn the rational transfer function by learning show that our way of learning poles leads to similar form of Equation with learned parameters (poles) as opposed to learning 2n + 1 parameters in RTF.
poles 1:m (for n terms). In Appendix E we
Computational complexity. In terms of computational complexity, AMO has an overall compu- tational complexity of C(N(M log M + D)) + O(N.MD). The former is from the processing block, whereas the latter comes from P and Q. When M treated as a constant with M < N, and a local decoder is used, the dominant cost reduces to O(N, + O(N M log M). Consequently, the complexity grows linearly with the number of mesh points N₂- With mesh size fixed, it approximately linear in the number of latent tokens M and the number of blocks N.
5 NUMERICAL EXPERIMENTS
To illustrate the effectiveness of AMO, conduct numerical experiments with multiple baseline neural operators on diverse datasets including three categories: (i) regular grids: 2-D Darey flow equation and 2-D Navier-Stoker equation (Li et al. 2020). (ii) irregular geometries: plasticity, air- foil, pipe, and elasticity (Li et al.] (023). (in) PDEs with singularities: European option pricing under the Black-Scholes equation, and 3-D Brusselator (reaction-diffusion) equation from Cao al. 2024) (see Appendix
Metric. In the training and evaluation stage, we utilize relative L error as the metric for accuracy for all problems:
1 (1)((a:) I(a))laa N [|9(a=))==
Rel-1,2
(15)
where N' denotes the number of samples. We also consider training time, the number of parameters, and/or GPU memory usage as metrics for computational efficiency.
Implementation details. For baselines, we follow the implementation settings of their works. Note that the architecture of FNO (Li et al.) (020) has been updated after publication, we evaluate FNO using the newest architecture. For AMO, we train 500 epochs on all datasets. We use AdamW optimizer with decoupled weight decay 1 x 10-5 base learning rate 2 X 10-4, and cosine decay schedule (Loshchilov Hutter 2017) with a linear warm-up over the first 10% of total steps. The nonlinearity is GELU inside the processing blocks. We clip global grad-norm at 0.5 each step. Un- less stated otherwise, we use batch size 16, latent width 128, 64 latent tokens, 32 adaptive poles, and 4 processing blocks with SSM state size 16, depthwise 1-D convolution (per channel) of kernel size 4, channel expansion ratio 2. Experiments are conducted on a Linux workstation running Ubuntu (kernel 6. 14, glibe 2.39) with Python 3.13.5 (Anaconda), PyTorch 2.8.0 2.8.0+cu129 (CUDA 12.9), an AMD Ryzen 9950X (16-core) processor, and a single NVIDIA GeForce RTX 4090 (48 GB) GPU. CUDA is enabled.
5.1 NUMERICAL RESULTS OF BENCHMARK DATASETS
Table I shows the comprehensive comparison with various baselines the six benchmark problems.
Among those problems, N-S and Darcy flow datasets apply regular grids, elasticity dataset uses point clouds, whereas others are generated under structured meshes (Li et al. 2020 2023). AMO con- sistently outperforms existing SOTA models by an average improvement of 28. 42%. In particular, for airfoil, Darcy, and N-S datasets, the relative 1/2 error decreased more than 30% compared to

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the existing SOTA models, demonstrating the superior performance of AMO compared to existing frequency-, transformer- and Mamba-based models when solving complex dynamics and handling inregular geometries. To solve the complex dynamics, Tiwari et al. (2025) incorporates latent repre- sentations and SSMs, which can be considered as integral kemelss without orthogonality. Meanwhile, ONO (Xiao al. 2023b) irregular geometries, including elasticity (0.0050 + 0.0043), plasticity (0.0007 + 0.0006), airfoil (0.0041 - 0.0020), and pipe (0.0026 0.0023), show that the systematic integration of orthonor- mal kernels and SSMs leads to an exact AFD approximation and in turn improves PDE solution accuracy in irregular geometries.
orthogonal attention ensure orthogonality. Numerical results
Relative L error comparisons of AMO with baselines across six benchmark datasets. Lower relative L error is better. We quantify the improvement as the gain of AMO relative the L error of the second best model. Bold means the best model, underline means the second best model, red means the third best model, and blue means the fourth best model.
Table I:
Models
Elasticity
Plasticity
Airfoil
Pipe
N-S
Darcy
FNO (Li et al. 2020
0.0229
0.0074
0.0138
0.0067
0.0417
0.0052
U-FNO Wen et al. 2022
0.0239
0.0039
0.0269
0.0056
0.2231
0.0183
F-FNO
Tan et al. 2021
0.0263
0.0047
0.0078
0.0070
0.2322
0.0077
LNO
Wang & Wang
2024
0.0052
0.0029
0.0051
0.0026
0.0845
0.0049
ONO
Xıao et al. 2023b
0,0118
0.0048
0.0061
0.0052
0.1195
0.0076
WMT
Gupta ct al. 2021
0.0359
0.0076
0.0075
0.0077
0.1541
0.0082
Galerkin (Cao. 2021
0,0240
0.0120
0.0118
0.0098
0.1401
0.0084
LSM Wu ct al. 2023
0.0218
0.0025
0.0059
0.0050
0.1535
0.0065
OFormer (Li al. 2022
0.0183
0.0017
0.0183
0.0168
0.1705
0.0124
Transolver Wu ct al.
22024
0.0062
0.0013
0.0053
0.0047
0.0879
0.0059
Transol (Luo et al.
2025
0.0064
0.0014
0.0051
0.0027
0.1010
0.0089
LAMO Tiwari et al. 2025)
0.0050
0.0007
0.0041
0.0038
0.0460
0.0039
AMO (ours)
0.0043
0.0006
0.0020
0.0023
0.0278
0.0021
Improvement
14.0%
14.3%
51.2%
11.5%
33.3%
46.2%
Computational Efficiency. To explore the computational efficiency of AMO, we focus on Darcy and airfoil problems. On average, AMO reaches 46.2% and 51.2% reduction in training time over SOTA models in these two problems, shown in Figure [ With light architectures and small GPU memory, AMO achieves the best training speed. Compared to the SOTA neural operator, LaMO (Tiwari et al. 2025), AMO is 1.2x faster and ~ 2.5x lighter with similar GPU memory. Instead of using orthogonal attention as in ONO (Xiao et al. 2023b). AMO employs bases in the orthogonal form (Equation 66, which does not require orthogonalization process, thereby saving 2.7x in training time and 3x in GPU memory compared to ONO.
Scalability. We examine the computational scalability of AMO on 2-D Darcy flow problem. From Table we observe that, as the grid dimension changes from 64 to 128 (N, becomes 4 times larger), both training and inference times increase approximately linearly (by about times), which aligns with the computational complexity result mentioned earlier. The memory usage remains relatively constant with only a slight increase. This reflects the architectural characteristies of AMO, where the main computations (SSM blocks) are performed on M latent tokens rather than N, physical points, and thus the memory footprint is largely decoupled from the input resolution N..
Table 2: AMO computationally scalable with respect to input resolution N..
Grid dimensions
Grid size N,
Training time (seclepoch)
Inference time (sec/epoch)
GPU memory (GB)
64 64
4096
14.0
0.007
2.3
16384 256 256
128 128
52.5
0.28
2.4
65336
206.0
1.12
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Learned pole distributions across layers. To understand how the adaptive poles are selected and evolved, Figures showcase Showcass the distributions per layer for 2-D Darcy flow and 3-D Brusselator equations. The learned poles of AMO Darey flow problem tend approach to the boundary of the unit disk, while those on the Brusselator problem tend to be in the interior of the unit disk. The reason is that, the challenging characteristics and singularities of the Darcy flow problem at the boundaries, and then more adaptive poles would be put there. Meanwhile, the complexity of the Brusselator problem does not come from the boundaries. It comes from the local, non-linear reaction that happens at every single point inside the domain. Therefore, most of the learned poles should be put inside the unit disk.
located
5.2 EUROPEAN OPTIONS PRICING
To demonstrate the versatility of AMO in solving different PDEs in different contexts, consider the European calls/puts problem modeled using the Black-Scholes equation with continuous divi- dend yield q. For contract/market parameters (r. q, K, T, csaall,,,,,, the price V(S,t) satisfies
the Black-Scholes equation (Barles & Soner 1998):
a + j S2 SssV (r () rV =0, S [Smin [Smin:=maak Smax]:
& E [0, T),
(16)
(+ sign for calls, - for puts) and the linear boundary conditions t0 = 0 for calls, V(O, Ke-r(T-))) for puts, and controlled growth as S oc. This problem setting leads to two singular features: (i) the terminal payoff kink at S = K (jump in 0,V. concentration in 0ss V) as moomm + 1; and (ii) degeneracy near small $ as learn the operator that maps the parameters
with terminal payoff V(S,T)
a result of the 52 diffusion term. Our goal (r,a,g K, T, isscall) to the price V(S,t). t). By comparing AMO with set of top-performing solvers, we observe from Table 3 that average improvements of 25%, 4. 4.1%, and 52.7% have been achieved by AMO in terms of relative L error, training time, and parameter counts, respectively. This indicates that AMO can accurately and efficiently solve PDE problems with singular features.
(a) Darcy
(b) Airfoil
Parsns Count
Parame Count
5M
ONO
5M
OND
GPU Memory (GB)
GPU Memory (GB)
10M
10M
LND Former
Trassovvr
3
Transhwer
Qalerkin
Gakerkn
Oure
LINO
Lati
2
2
15
30
30
60
70
Training Time (s/epoch)
Training Time (s/epoch)
Figure 1: Comparisons of training time per epoch, number of parameters, and GPU memory among existing SOTA models on (a) Darcy and (b) airfoil, where AMO exhibits the strongest incremental gains.
5.3 ABLATION STUDIES
Adaptive kernels vs. static kernels. We now consider the need and benefits of using adaptive kernels. A kernel is adaptive when its parameterization (e.g., coefficients) varies with the input. In this work, the formulation of Equation 6 varies with the learned poles aı and thus is an adaptive kernel. We also randomly fix the value of α1:s for static kemnlss for comparison. Furthermore, although total of poles are needed for i-th processing block, one and select the best poles for implementation. Table 4 shows the relative L error results across six benchmark datasets and the European options (EO) dataset. We find that, using adaptive kernels, the
still identify more poles
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Table 3: European option pricing: relative I error and resource profle. Lower better for error, GPU memory, and training time. Parameter counts shown in millions. Bold best, underline second best, and red third best.
Models
Rel. L (4)
Training Time (sec/epoch, L)
Params (M, +)
FNO Li et al. 2020
0.0016
25.1
3.78
LNO Wang & Wang
2024
0.0010
21.7
2.56
Transolver (Wu et
2024
0.0012
22.3
5.91
LAMO (Tiwan et al.
2025
0.0008
22.5
3.52
AMO (ours)
0.0006
20.8
1.21
relative L errors reduce significantly compared to using static poles for all benchmark problems considered. In fact, the relative L errors when selecting only 4 poles are lower than those when selecting 32 static poles.
Table 4: Relative I,2 error comparisons for Static vs. Adaptive kernels across seven benchmarks.
Lower is better.
Models
Number poles
Elasticity
Plasticity
Airfoil
Pipe
N-S
Darcy
EO
AMO (static)
32
0.0097
0.0021
0.0067
0.0072
0.1103
0.0174
0.0035
0.0056
0.0012
0.0033
0.0029
0.0311
0.0057
0.0014
0.0051
0.0010
0.0031
0.0027
0.0298
0.0047
0.0010
0.0049
0.0008
0.0027
0.0025
0.0281
0.0036
0.0009
AMO (adaptive)
0.0046
0.0008
0.0023
0.0028
0.0290
0.0029
0.0008
0.0043
0.0006
0.0020
0.0023
0.0278
0.0021
0.0006
0.0048
0.0007
0.0036
0.0031
0.0372
0.0046
0.0009
Need for ensuring orthogonality. To understand how orthogonal kernels affect AMO perfor- mance, we conduct another ablation study by using non-orthogonal kernels (i.e., Equation 5 in the AMO framework. In this case, the transfer functions used in SSMs H.(e
(1 - las12) En=o(a))" to match the output of AFD operation. Without orthogonality. AMO experiences higher relative L error, especially for problems with irregular geometries (e.g.. airfoil 0.0020 0.0083 and elasticity 0.0043 0.0094). At the same time, the training time also in- creases by- ~ 50.3% per epoch on average across all six benchmark datasets. This shows that the use of orthogonal kernels (i.e., TM systems) helps improve both accuracy and computational efficiency of AMO solver.
Choice of SSMs. Finally, evaluate the choice of bidirectional SSMs in AMO compared to unidirectional SSMs and multidirectional SSMs. Results Figure 2 indicate that the choice of bidirectional SSMs in AMO consistently outperforms other two SSMs in all datasets.
5.4 EXPERIMENT USING REAL-WORLD NOISY DATASET
To validate AMO's performance on noisy real-world datasets, we perform experiments using the latex glove DIC (Digital Image Correlation) original dataset Yoou et al., (002)). The goal to learn the mechanical response of a nitrile glove sample directly from experimental data, without assuming known constitutive law. The goal is predict the displacement field at the current loading step. The input includes the spatial coordinates, the displacement field from the previous step, and the current boundary displacement. We compare the performance AMO to the current SOTA of this dataset, IFNO, well as FNO as follows. To ensure fair comparison, we conduct experiments using the same settings as IFNO with the number hidden layers ranging from 3 to 12.

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1.3c
10
Unidirectional SSMs
5. 5.3c-02
Bidirectional SSMs
Relative L2 Error
84-4
Multidirectional SSMs
10
-2
9.6e-03
9.76-03
7.6e-05
Be-03
4.9e-03
1e-43
2:1e-33
1.5e-03
1.26-03
10 -3
Elasticity
Plasticity
Airfoil
Pipe
N-S
Darcy
EO
Dataset
Figure 2: Contribution of three SSMs across seven benchmark datasets. Note that we do not apply weights shared for all experiments. Lower is better.
Table 5: Relative L error AMO and other baselines using the latex glove DIC (Digital Image Correlation) original dataset.
Number of hidden layers
AMO
IFNO
FNO
2.87E-02 + 4.29E-04
3.43E-02 + 4.96E-04
3,40E-02 + 4.09E-04
2.50E-02 + 3.28E-04
3.34E-02 4.53E-04
3.84E-02 + 4.21E-04
12
2.32E-02 + 4.20E-04
3.32E-02 + 4.41E-04
4.66E-02 + 1.47E-03
In addition, You et al. (2022) also reported the results of generalized Mooney-Rivinn (GMR) model in two settings. The relative L errors GMR model fitting and GMR inverse analysis are 3.30E-01 and 2.91E-01, respectively. We observe that AMO consistently outperforms other models in every L. Finally, the best reported result of IFNO is 3.30E-02 4.63 4.63E-04 when I. = 24 Yoou et al, 2022]. Although do not conduct the experiment T. 24 due the limited time, our AMO still performs better than the best result of IFNO.
6 CONCLUSIONS
In this paper, we propose a novel neural operator AMO for solving nonlinear PDEs on irregular geometries and singularities. AMO maps the physical tokens in RKHS where the global spectral transform and data data-dependent orthogonal kernels are incorporated. By conducting a tailored design of the TM layer and SSM block fully guided by the AFD theory, we show that the output of AMO exactly matches with AFD oepration, hence offering rigorous convergence guarantee and other de- sirable properties. We show that the novel architecture of AMO enables its outstanding performance compared to existing SOTA neural operators in series of physical and financial benchmark prob- lems.
REPRODUCIBILITY STATEMENT
All code and datasets have been either made publicly available an anonymous repository or as part of supplementary material to facilitate replication and verification. The experimental setup, including training steps, model configurations. and hardware details, is described in detail in the paper. We have also provided full description of implementation details, assist others in repro- ducing our experiments. Additionally. six benchmark datasets, such as pipe, are publicly available, ensuring consistent and reproducible evaluation results.
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A
NOTATION LIST
a
Parameter function (input)
uNo0
Output of AMO with N blocks and parameters
N
Number of processing blocks
N,
Number of input physical tokens
M
Number of encoded latent tokens (M < N₂)
Demhed
Embedding dimension of latent tokens Input physical features
Xlays
8phys
Positional embedding coordinates
Token representation after the i-th block
Zo
Encoded tokens produced by the lifting operator
21
Tokens mapped into RKHS by operator
P
Lifting operator mapping physical tokens to encoded tokens
Q
Projection operator mapping latent tokens back output space Mapping operator from latent tokens to RKHS
R
C
Processing block at layer i (SSM' TM')
Si
Aggregation operator with skip connections at block i TM layer performing spectral transform via TM bases Bidirectional SSM block parameterized by transfer function i-th TM basis generated by poles 01:i
TM
SSM
Bi(2;a1s)
Set of learned poles {a)....,a:) a in the unit disk D
Ka(z)
Reproducing kernel T-az
(e""
Transfer function of the i-th SSM block
h4(n] [n]
Impulse response of the i-th SSM block Inner product 1 -ーー x (ee2(m N) N
(x, f)
Element-wise (Hadamard) product
H
Reproducing Kernel Hilbert Space (RKHS) Length of signal inner product definition
Ñ
B ILLUSTRATIVE EXAMPLES
1-D advection PDE with high-frequency perturbation. vection benchmark governed by
We evaluate LaMO 1-D linear ad-
U, + cu = 0
(17)
on a periodic unit interval. Initial conditions va(x) are synthesized as smooth Fourier mixtures 04 sin(2 + Φu) with amplitudes decaying ag (1 A:) to which we add a weak ggh-frequency spike at wavenumber kыi to probe aliasing and phase accuracy. Trajectories are advanced to time T with a conservative first-order upwind scheme Courant number CFL At/Azz < 0.5, ensuring stability while preserving sharp phase relationships; the target is the advected field u(:,))
Aemax
Jam
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draw P points Published as a conference paper at ICLR 2026
3
2
phase error (rad)
-1-
-3
50
100
150
200
250
mode k
Figure 3: Phase enror of solutions predicted by LaMO.
Figure 3 visualizes the phase error of LaMO's predictions, revealing a pronounced degradation for high-frequency modes (approximately & [140, 250]). This suggests that LaMO struggles to faithfully capture phase at the upper end of the spectrum.
2-D Darcy flow equation with fractal noise. We construct challenging 2-D) Darcy dataset by solving
Vk((k( = f(x y)
(18)
on [0, 1|2 with homogeneous Dirichlet boundaries, where the permeability & is positive, highly het- erogeneous, and fractal-like. Specifically, & is generated by exponentiating band-limited fractional Gaussian field (small Hurst parameter for roughness) and then modulating it with narrow channel masks and inclusions to induce strong anisotropy and high contrast. The forcing f combines weak background term with several randomized Gaussian sources/sinks, which produce near-singular be- havior in the solution. The variable- coefficient elliptic problem is discretized using a flux-conservative 5-point stencil with harmonic averaging of k, and solved to tight tolerance via conjugate gradients. For learning, each sample is subsampled irregularly: {(7x, }:)} and record u(z,, yi). yielding pairs (XY, U' without exposing & or f-
variable-coefffcceent
Cartesian grid
To visualize and stress singular structures, show in Figure 4 (a) and (c): (i) contours of the potential highlighting global flow topology, and (ii) a logarithmic map of the gradient magnitude, log VVu, computed capture the singularities of and log Vul. Furthermore, once the complex singularities appear, the performance of LAMO will be affected.
reconstructed dense grid via triangulation. Figure shows LAMO cannot
3-D Brusselator problem. We introduce new 3-D Brusselator duffuusion-reaction equation) problem using the dataset from Laplace neural operator (LNO) (Cao et al. 2024). The Brussela- tor problem is formulated as:
02y dr
Oy
D
+
ky²
f(x,)),
(19)
at.
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20
1.0
0.8
0.6
10
0.4
5
0.2
0.0
-
-10
-0.4
0.66
-55
(a) Ground truth
(b) Predicted by LaMO
10a
10
10-1
4
110
10-7
-0011
10-13
10-18
10-15
10-4
(c) Ground truth log Vvl
(d) Predicted by LaMO
Figure 4: The predicted results produced by LaMO compared to the ground truth.
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Layer
1.00
Layer
Layer
0.75
Layer
Layer
0.50
0.25
Image
0.00
-0.25
-0.50
-0.75
-1.00
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Real
Figure 5: Learned poles distribution for the 2-D Darcy flow equation.
where y(x, t) represents the concentration of chemical substances or particles at location and time (xx t) is the source term and A is the amplitude the source term. In this problem, the diffusion coefficient, D 0.01, and the reaction rate, k = 0.01.
C DISTRIBUTION OF SELECTED POLES REFLECTS PROBLEM CHARACTERISTICS
To understand how AMO's pole selection process adaptive to the characteristics and nature of the problem, we illustrate the learned pole distributions for the 2-D Darcy flow problem and 3-D Brusselator problem in Figures To clarify, here give brief overview of the visualization results: The distribution of selected poles for the 2-D Darcy flow problem is shown in Figures 5ln and 6 respectively.
We observe that, across the layers, the learned poles of AMO on Darcy flow problem tend to ap- proach the boundary of the unit disk, while those on the Brusselator problem tend be in the interior of the unit disk. The reason is that, Darcy flow problem is elliptic equation, which is a smoothing operator. Thus, even though the input coefficient (the permeability) is very rough and discontinuous, the solution inside the domain will be well-behaved. Therefore, the challenging char- acteristics and singularities adaptive poles would be put there. Meanwhile, the complexity of the Brusselator problem does not come from the boundaries. It comes from the local, non-line -ineer reaction that happens at every single point inside the domain. Therefore, most of the learned poles should be put inside the unit disk.
the Darcy flow problem are located at the boundaries, and then more
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1.00
Layer
Layer 1 (All Heads)
Layer Heads)
0.75
Layer 3 (All Heads)
Layer (All Heads)
0.50
0.25
Image
0.00
-0.55
-050
-0.5
-1.00-
-1.00
-0.75
0..50
-0.25
0.00
0.25
0.50
0.75
1.00
Real
Figure 6: Learned poles distribution for the 3-D Brusselator equation.
D THEORETICAL RESULTS OF AMO
Basic settings. LDt = (= C: |z| < 1}. Consider reproducing kernel Hilbert space (RKHS) (H, (5-))) of complex-valued functions on D with the following properties.
Assumption D.I. There is family of normalized reproducing kernels (ea a D} C H such that
v1 la
(z)
E H.
(freahH = f la|2 Vf DED.
(20)
1 ōz
Given a pole sequence d1:00 = (a1, (01,a2,-. .2....) C D, define the Takenaka-Malmquist (TM) system by
i-1 2 - aj a
(2) = Cai (2),
(x) Ca. (2)
II
(i 2).
(21)
Assume (Bi)21 is an orthonormal system H. and its closed linear span equals the model space
KB == span 0. 2 13 C H,
(22)
where B is the Blaschke product with zeros }
AMO notation. Let E H. be the latent target representation and u²* = Q(s). where Q H. + U a Lipschitz decoder with constant T.g. Define the ideal TM coefficients and partial sums
(8,
D.n,
B.
(23)
AMO learns estimates èn of of (via an SSM in the frequency domain) and aggregates them through the skip connection:
4+ zi Br
21 ==
(24)
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D.1 AGGREGATION IDENTITY AND FREQUENCY-DOMAIN COEFFICIENT EXTRACTION
Lemma D.2. Under 24 one hus, every N E,
24+1
Σ à B.
(25)
Proof. The proof is by induction. For N = 1, 22 21 + 11 = 1. 80 25
holds. Assume
holds for N. i.e. 2N+1 Em C Bg. Then
N+1
N4 2N+1 + CN+1
Σ
4 Bi,
which establishes the claim for N + 1.
Lemma D.3. Suppose the i-th SSM has transfer function
H Betw))
(26)
so that the block multiplies the input spectrum by Di and outputs the zero-lag correlation. If the discrete inner product used by AMO is a consistent quadrature for (r on the class {s} { then
G - (8, B.HH ot as the quadrature is refined.
(27)
Proof. By 26 the block forms (pointwise on the grid) Y, = B. in the transform domain; the zero- lag correlation is the discretized inner product (s, 3i disc. Consistency of the quadrature implies (s, Bi) + (s, Di)H as the grid refined. Hence 2 + C
D.2 CONVERGENCE IN THE MODEL SPACE AND PROJECTION ERROR
Theorem D.4. Under Assumption D.
AMO recovers the exact coefficients of = (8, Bi) then
S
B
1K
(28)
the orthogonal projection onto KB. Consequently,
||u Q(sn)l < Lo ||s TKgg8HH + Le
sNIW N
Lg dist(s, KB). (29)
Proof. Because { is an orthonormal basis (ONB) of KB. the Fourier expansion of Ik s in this ONB has coefficients (s, B,),, and the N-th partial sum equals sx. Convergence in norm to the projection is standard for orthogonal series in Hilbert space, giving The bound 29 follows from Lipschitz continuity of Q:
||u* - 2(*x) (ss) - ((xx)ll < Lolls sN s Le(Is 11 *l1 + [ - 8N ll)-
Remark. No greedy maximal selection is used. The MLP MLP-generated poles determine K3; AMO converges to Ix, s, and to s whenever s E KB-
D.3 BEST N-TERM ERROR AND RATES WITHOUT GREEDY SELECTION
Definition D.5. Let D = (BB(; 0ns E D', E N} be the TM dictionary. Define the best N-term error
N
E (s) = inf CUN
Σ
ca B,((;aaa))
(30)
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Theorem D.6. Let 01:N be the poles output by the MLP and set of (s, (s, AMO
learns C₁ then
Σ 23 Bi 01:))
S
EN((s)
+ (N) +
C - c3 12)
(31)
H
where
pote (N) = inf
Σ cs Bi (:ā(1x))))
(s)(s) 2
(32)
Ca:N
Proof. Choose
@ 11NN CIN
best best that
attain
(or e-attain)
Ex(() and
denote best
Σ N
best
Bi
best ) Then
1ie
N
||8 AN
best
8N A
best
Σ
+ Σ (c (:)
< (8) + (N) +
(2
let - 2313)
1/2
im
The last inequality uses the definition of pole and orthonormality of (B d1s))
L
Corollary D.7. Assume for the fixed MLP-produced poles u1s that the exact TM coefficients satisfy the weak-ll decay
lo2l" S sCi 1/1 O<p<2
where (lc71*) is the nonincreasing rearrangement. Then
inf
B,(+ūaa)) =(NN
(33)
CIN
If. addition, Apole (N) o(1) and (Ee [a - 2/2) o(1). then the AMO error in 31 O(N+
Proof. For an orthonormal system, the best N-term error equals the g tail of the rearranged coeff- cients. With |e]l" < Ci-1/p and p < 2,
Σ ((gr) E
S C Σ 12/2
-(N1
ON
hence the norm error (square root) is O(N
D.4 LEARNING AND DISCRETIZATION ERRORS
Assumption D.8. Each 2; obtained by ERM over m i.i.d. frequency samples using a hypothesis class with effective capacity der under sub-Gaussian noise, so that
E B -- = =
duft
(34)
Lemma D.9. Let ((--) N be discrete inner product (e.g., uniform frequency grid) that a consistent
quadrature for {:5 )u on the class generated by {8} U {..). Then there exists saise & 0 such that
Kf. (f.g) sl S (N) for all f {s}, (Bi121
(35)
Proof. Since point evaluations are continuous linear functionals an RKHS and the involved func- tions continuous compact subsets, standard quadrature consistency yields 35 (If f.g analytic in an annulus around the unit circle, one gets exponential rates; under Sobolev regularity, algebraic rates.)
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Theorem D.10. Under Assumptions D.1 and D.8 and Lemma D.9 the AMO output after N blocks and N grid points satisfies
N
l1u uN.oll & Le En(() (E((s)) + polc +
Σ
C ct1 1/2)
)213)11)
+ Edisc (N),
(36)
i=1
with Elic --,l = O( ( m) and Ediso + as Ñ + D0.
Proof. Apply Theorem D.6 to bound the latent H-error. Then use Lipschitz continuity of Q to transfer the bound to the output space. The discretization error adds (N) due to 35
D.5 STABILITY TO POLE PERTURBATIONS Lemma D.11. For a, b E D and 2 E ID,
1
|a b
(37)
az
1 bz
|a| (1 [b])
la - bl
1
|a/2
v1
6//
S
(38)
max (a), 1b|1?'
2 2-6 a
for F(z;a)
1- āz=
4 4/a- - 6l
(F(z;)) F(z;b)] s (1 [al)(1 - [b])
[(z;a)] 1.
(39)
Proof. For 37
1
(77 b)z
1
(1 0z (1 bz)
and 1 - [2 2 1-11-1110 |a||==] 2 1 |a|. |2| < 1, yielding the bound. For 38 use the mean-value theorem
on + vI 1 with |a|?, [6|2 and ||a|2 16 s la bl (1aa b
I 2/a b). For 39 expand
(b + (a b) + (ab bT)z
a
(1 T2) (1 bz)
and bound the numerator by Claa bl for |z| while the denominator is bounded below by (1 - [a])(1 bl).
Theorem D.12. Let a1ss ai E D with -āy I 65. Then there exist constants C. (depending on a1:)) such that
dz
AA B((;a1a))l I C.
(40)
ag
1=1
Consequently, for any coefficients C.
N
83
Σ
(
2 C
(Σ)
lay1
(41)
1=1
Proof. Write
Bi(;a1a) = Ca. II P(sa;),
B:(;āna) = ca, II F(;ā))
Use the product telescoping identity
II Pa k=
II k=1
Q
Σ
[I] (Pr Qa) (II
jck
j>k
20

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Published as a conference paper at ICLR 2026
with P1 Cā.. Q1 Cass and Pk- = F(;āx=1)) Qs = F(; for k 2. Taking sup-norms and using F(::a)
i-1 [F((;ū)) - F(;
[; Bi (:aa))l & ||a,- - layllo
f=1
Apply Lemma D.11 to bound each term by constant times (5/(111a,)) la,l).
Since evaluation func-
tionals are continuous and the kernel is bounded compact subsets, there exists embedding constant Comb with ||/|| S /ombll/llo on the set considered; thus 40 follows with C. absorbing all constants. Finally,
N
Σ 1=1
a ( B:(aaai))
Σ ((āāa) - ((ja1))
giving
D.6 END-TO-END CONVERGENCE WITHOUT GREEDY SELECTION
Theorem D.13. Assume:
1. s KB:
2. D Bl c7/3]1/2 < oo (as sample size - 3o and model capacity increase);
3. Edisc (N) + 0 as Ñ x0
Then
lim I|u" uN 0.
Proof. Since KB and { an ONB of KB. Theorem D.4 gives sN s in H. In In for this fixed pole sequence one has EN(s) = Apole (N) = 0. Using (2) and (3), we obtain ||u" uN
D.7 CONNECTION SSM CORRELATION AND AMO OUTPUT
Proposition D.14. With He(e")) Beteww) the i-th SSM block computes 2 R (2i, Bi) Hence, by Lemma D.2 after N blocks
2N+1
uN.0 (==++1).
(42)
Proof. The coefficient claim follows from Lemma D.3 applied to 2; in place of s. The aggregation identity is Lemma D.2 The last equality the definition of Q.
Corollary D.15. All space error bounds transfer to the PDE output space via
||ue uN.ell _ Lo
Σ G.
Edisc
E TRANSFER FUNCTION
We consider a (finite) Blaschke product
H(2) II - Pj² 1=1
lp,l <
(43)
Pj
and convert it into a single ratio of polynomials whose coefficients match the parameterization used train SSMs.
21

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Published as a conference paper at ICLR 2026
Polynomial expansion and 2- form.
Denote numerator and denominator polynomials
Bpoly (2)
II py),
Apoly (2) - II ps²),
(44)
so that (z) constant term in the denominator, divide numerator and denominator by 2 and then normalize:
Booly (z) Let d deg Pply deg Apoly
Apaly (a)
To obtain the form with unit
(=)
El=o k= 04
normalize,
ho +
bk
k
/
Σ aa2 ) k=
(45)
Σx= BE
The SSM coefficients are then reduced as:
00 hg 3
3k
6
k = 1, I....,d.
30
Example (n = 2).
With P1, E C, expand
Bpoty (2) = (= P1) (= - -22 2² (pi+p2)= + P1P2.
Apoly(2) =1 (1--p==(( - -p==) = (p1+ + (2112)=²
Divide by 2 to get polynomials in = and normalize by the denominator's constant term (gg P1P2), yielding
1- (p1 (2)) (p1P2)=
hg bia=! 1 b22
H(z)
PIP2 - (pI+P2)= 2-2
1 + a-a22= + a22-2 -2'
with
pi + p2
P + +
ho
b
b2 =1
P1P2
P1P2
P1P2
P1P2
Efficient computation for large n. Direct symbolic expansion scales poorly. Instead, we multiply degree-1 polynomials using FFT-based convolution. Represent each factor by its coefficient vector:
(= P;) + 11, -Pjl.
(1-p}=) + [1, -Pyl:
and iteratively convolve to form Bpoly and Apoly- By the convolution theorem, polynomial multipli- cation is element- wise in the frequency domain, giving Odd log d) complexity. After both polyno- mials are assembled, convert to by dividing by z then normalize by the denominator's 's constant
term to obtain (ho. { {bx}) as in 45
22


Reviews:
[{"review_id": "VLaw6BMLyv", "reviewer": "Reviewer_ceLB", "summary": "This paper proposes the Adaptive Fourier Mamba Operator (AFMO), which is a neural operator that integrates Takenaka-Malmquist systems with state space models for solving PDEs on irregular geometries. The core novelty is in parameterizing SSM transfer functions using adaptive poles in a RKHS. AFMO constructs orthonormal bases from learned poles and proves that the output performs adaptive Fourier decomposition.  Experiments show performance gains on irregular geometries.", "questions": "1. When does s ∈ K_B hold for PDE solutions?  Characterize the class of PDEs where the convergence guarantees apply. \n\n2. Authors should provide architectural details of the small MLP predicting poles. You should show learned pole distributions for different problems. \n\n3. Resolution invariance experiments have to be conducted. \n\n4. Failure analysis of AFMO has to be described. \n\n5. How does AFMO scale to 3D problems? \n\n6. Please report time and memory as a function of grid size. \n\n7. How is orthogonality maintained during training?\n\n8. How does AFMO compare to explicit AFD+preprocessing+standard neural operators? \n\n9. You claim LaMO has low-pass filtering bias, but doesn't your method also have similar frequency characteristics since its also has an SSM transfer function, which is also a rational function.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Strong theoretical grounding with explicit connection to AFD theory with convergence guarantees. \n\n2. Integrating TM systems into mamba SSMs via transfer functions is well motivated. \n\n3. Experiments are comprehensive with thorough ablations. \n\n4. The method is computationally and seems exaplanability.", "weaknesses": "1. Limited baseline comparisons in that recent methods, such as BENO, UNO, and new transolver variants. \n\n2.   Several gaps exist in theoretical formulations: Theorems assume s ∈ K_B (target in model space), but this is rarely true for real PDEs. What happens when this fails? Theorem C.6's bound includes Δ_pole(N) (suboptimality gap), but no analysis of when and why this is small. Convergence rates (Corollary C.7) require weak-ℓ^p decay of the coefficient. When does this hold for PDE solutions?\n\n3. Illustrative examples seem cherry-picked. \n\n4. All experiments use relatively coarse grids. \n\n5. Computational complexity analysis is not convincing enough.", "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": "ruq2uMXk9X", "reviewer": "Reviewer_mMjt", "summary": "This paper introduces Adaptive Fourier Mamba Operator (AFMO), a novel neural operator architecture that combines Adaptive Fourier Decomposition (AFD) theory with the state-space efficiency of Mamba models. By leveraging Takenaka–Malmquist (TM) systems and reproducing kernels in the frequency domain, AFMO provides a mathematically interpretable, efficient, and accurate solver for PDEs across diverse geometries. The model demonstrates strong performance across physical and financial benchmarks, surpassing SOTA neural operators like FNO, Transolver, and LaMO in both accuracy and computational efficiency.", "questions": "1) How sensitive is the model’s performance to the number and spatial distribution of adaptive poles in the TM system? How do you determine the optimal number of poles for a given problem?\n\n2) Can AFMO be extended to time-dependent PDEs or coupled multi-physics systems? For example, in the Navier–Stokes equations, how do the adaptive poles evolve over time, and can this dynamic behavior be visualized?\n\n3) How does AFMO handle noisy or incomplete boundary conditions, especially when applied to real-world experimental or observational data?\n\n4) Would incorporating spectral regularization or physics-based constraints further stabilize training or improve interpretability?\n\n5) How does the model perform on very high-resolution 3D geometries, and what are the associated computational trade-offs? Have you considered performing individual ablation studies on the TM and SSM layers to quantify each component’s contribution?\n\n6) It would also be interesting to evaluate AFMO on the CARS dataset, as done in Transolver, to provide a more direct comparison of generalization across complex flow scenarios.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1) Introduces a mathematically grounded operator framework combining AFD theory and Mamba-based SSMs.\n\n2) Provides strong empirical improvements across varied PDE benchmarks, including irregular and singular domains.\n\n3) Demonstrates computational efficiency, achieving faster training and lower GPU memory than existing models.", "weaknesses": "1) Ablation studies could be expanded to include more operator baselines (e.g., Laplace Neural Operator).\n\n2) Scalability to large 3D or time-dependent PDEs is not thoroughly evaluated.\n\n3) The sensitivity of performance to adaptive pole selection and hyperparameters is not deeply analyzed. It would be valuable to visualize how adaptive poles evolve across layers for different PDEs — this could provide strong insights into how the model captures changing dynamics.\n\n4) While the theoretical motivation is strong, a more intuitive discussion of adaptive pole behavior and dynamics would make the approach easier to interpret for a broader audience. Additionally, as shown in the appendix, the LaMO deviation in the propagation of high-frequency perturbations can also demonstrate the performance of AFMO on those datasets.", "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": "poJjcrjTQD", "reviewer": "Reviewer_P8Sy", "summary": "The AFMO paper presents an architecture that achieves state of the art results in solving partial differential equations (PDEs) on arbitrary geometries and diverse mesh types using neural operators. The authors propose Adaptive Fourier Mamba Operators (AFMO), which integrates Takenaka-Malmquist (TM) systems for constructing reproducing kernels within SSMs. The architecture follows a structured pipeline: a lifting operator compresses physical tokens via cross-attention, followed by mapping to a reproducing kernel Hilbert space (RKHS), then N processing blocks each containing a TM layer and bidirectional SSM, with aggregation layers incorporating skip connections, and finally a projection operator to output space. The key innovation lies in TM layers that construct orthogonal bases from adaptively learned poles, while SSMs operate in the frequency domain with transfer functions. The empirical evaluation demonstrates substantial improvements across diverse benchmarks. Compared against 11 baseline methods including FNO, LaMO, Transolver, and ONO on standard tasks spanning regular grids, irregular geometries, and problems with singularities. AFMO exhibits superior computational efficiency with approximately 46-51\\% training time reduction and is 2.5$\\times$ lighter than LaMO (ICML 2025).", "questions": "1. The claimed $O(N_s D) + O(NM \\log M)$ complexity is confusing. Is $M$ constant? What about cases where M less than $N_s$ may not hold? What about adaptive cases where M must change with problem complexity? Do you include the TM basis construction overhead in the computational cost?\n\n2. Can you report statistical significance of the results?\n\n3. What are the evaluation conditions? Why are reported numbers slightly different from the respective papers?\n\n4. How does memory scale with problem size during training vs. inference?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.  AFMO outperforms 11 baseline methods on all benchmarks with good margins. \n2. AFMO achieves better accuracy while being faster and more memory-efficient than competing methods\n3. First work to explicitly incorporate TM systems into Mamba architecture with frequency-domain SSMs", "weaknesses": "1. A significant weakness lies in the limited conceptual novelty, as the core components are largely borrowed from existing work. Takenaka-Malmquist systems are well-established in signal processing (Qian 2010, 2012), while frequency-domain SSMs were recently explored by Parnichkun et al. (2024). The main contribution is engineering, integrating these established techniques into a neural operator framework. Although this combination is novel, the paper overclaims its innovation by positioning it as a fundamental advance. The connection to Adaptive Fourier Decomposition is somewhat obvious given the explicit use of TM bases, and the theoretical results presented largely follow from known AFD convergence properties rather than providing genuinely new mathematical insights. The work represents a competent integration effort, but lacks the conceptual depth claimed.\n\n2. Some design choices appear empirically driven rather than theoretically motivated, contradicting claims of \"fully guided by AFD theory\". For example, bidirectional SSM (LaMO also uses this) seems to be empirically determined as AFD doesn't require bidirectionality. Another point is that aggregation by element wise product seems to be an arbitrary empirical choice, not theoretically grounded/motivated.\n\n3. The experimental design, although it follows other published work in PDE solving, is weak. No confidence intervals, error bars and statistical significance testing is reported. While FNO was modified, AFNO is not reported. The modifications of FNO is also not reported (readers would expect to see that in an appendix). European option pricing is a 1D test case with no relevance to the irregular geometry arguments raised to motivate the state space model usage. Instead a 3D test case would have conviced readers better.\n\n4. Crucial implementation and evaluation details are missing. I am listing a few here.\n(i) How is the \"small MLP\" structured? What initialization? How are poles constrained to $|a_k| < 1$ during training?\n(ii)  How are TM bases computed for poles near boundary? The required product terms seem numerically unstable.\n(iii) Hyperparameter selection is not reported\n(iv) For some test cases (such as pipe), the reported baseline numbers differ from those in the respective papers. The reason for this should be reported. What are the dimensions on which the evaluation is performed? Are these choices computationally driven?\n\n5. The theoretical details in Appendix C raises a lot of concerns. \n(i) Theorem C.4 requires exact coefficient recovery. How is the approximation error in practice handled?\n(ii) Theorem C.13 requires $s \\in K_B$, but no analysis or arguments on when PDE satisfies this is provided.\n(iii) Assumption C.1 about RKHS properties is strong and not verified for practical cases", "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": "m6pxRSH8tC", "reviewer": "Reviewer_HH9D", "summary": "The paper proposes Adaptive Fourier Mamba Operators (AFMO), a novel neural operator architecture that integrates:\n\nTakenaka–Malmquist (TM) systems and Adaptive Fourier Decomposition (AFD) theory,\n\nwith state-space models (SSMs) in the frequency domain.\n\nThe key idea is to parameterize the SSM transfer function using adaptive poles learned by a small MLP, allowing state-free inference and orthogonal rational basis construction directly in the spectral domain.\nAFMO is designed to combine orthogonality, adaptivity, and linear-time efficiency, aiming to outperform LaMO (Latent Mamba Operator) and FNO-family models on irregular geometries and singular PDEs.", "questions": "Please provide more motivations and ablation studies", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "To me it feels like blending method A to method B, which is not innovative enough", "weaknesses": "Empirical scope.\n\nAll benchmarks are 2D or low-dimensional. It would strengthen the paper to include a 3D PDE (e.g., Navier–Stokes 3D or Poisson 3D) or demonstrate scalability across resolution levels.\n\nNo robustness analysis to noise, parameter shifts, or out-of-distribution geometries.\n\nInterpretability of RKHS mapping.\nThe “mapping operator R” from latent tokens to an RKHS is implemented as an MLP without clear physical meaning. What does the learned RKHS correspond to in PDE solution space? Some visualization of learned poles or spectral responses would improve interpretability.\n\nJustification of using SSM is not grounded", "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)