Fourier Neural Operators (FNOs) offer an efficient paradigm for solving partial differential equations (PDEs). However, FNOs rely on a fixed Fourier basis and hard frequency truncation, which inherently limit their ability to model non-periodic, localized, and fine-scale solution structures. We propose the Stochastic Adaptive Fourier Decomposition Neural Operator (SAFDNO), a spectral neural operator that replaces predefined Fourier modes with an adaptive Takenaka-Malmquist (TM) orthonormal system derived from the theory of Stochastic Adaptive Fourier Decomposition (SAFD). Instead of performing an expensive greedy pole search in classical SAFD, SAFDNO amortizes stochastic pole selection through a neural pole predictor and constructs the adaptive TM system directly from latent features. The resulting operator performs learned filtering on coefficients of analytic branches in the Hardy space with respect to the input-adaptive TM system, preserving the global receptive field of spectral operators while providing a more flexible representation than using fixed spectral bases. Across nine PDE benchmark problems, SAFDNO achieves the best performance on all six regular-grid problems among strong neural operator baselines, with especially notable gains on Darcy, Burgers, and Navier-Stokes, where fixed Fourier modes are often less effective at modeling localized oscillations, sharp transitions, and multiscale structures. SAFDNO also exhibits stronger zero-shot super-resolution performance and shows less performance degradation when deployed on finer discretizations. These results suggest that the input-adaptive TM system provides a promising alternative to fixed spectral representations for neural operator learning.
Figures & tables
Figure 1: Performance on Navier-Stokes. Highlighted region indicates a representative area where SAFDNO handles multiscale structures better than FNO. Bottom plots show lower prediction errors for SAFDNO, especially for longer rollouts and higher frequencies.
Figure 2: Learned poles on the Darcy flow. Representations for deeper layers are increasingly global, while earlier layers focus on localized and sharp structures.
Model
Structured Mesh
Regular Grid
Plasticity
Airfoil
Pipe
Navier-Stokes
Darcy
DeepONet
0.0079
0.0130
0.0097
0.4508
0.0411
FNO
/
/
/
0.1468
0.0108
MWT
0.0076
0.0075
0.0077
0.1541
0.0082
U-FNO
0.0039
0.0269
0.0056
0.2231
0.0183
Geo-FNO
0.0074
0.0138
0.0067
0.1556
0.0108
Table 1: Test relative ℓ2 error on five standard PDE benchmarks. † denotes ablation variants. / indicates that FNO is not applicable to this task.
Figure 3: Localized pattern on Darcy flow. The highlighted region indicates a representative localized pattern where SAFDNO better matches the ground truth.
Burgers
1D CFD
2D CFD
Reaction-Diffusion
DeepONet
2.62e - 1
1.99e - 2
2.76e - 2
5.08e - 4
FNO
1.12e - 2
8.25e - 3
1.66e - 2
1.17e - 4
U-FNO
1.70e - 2
8.27e - 3
1.17e - 2
1.75e - 4
F-FNO
1.55e - 2
8.80e - 3
1.69e - 2
8.47e - 5
U-NO
7.53e - 3
7.72e - 3
1.57e - 2
7.66e - 5
AM-FNO (MLP)
8.92e - 3
7.66e - 3
1.21e - 2
1.78e - 3
Table 2: Test relative ℓ2 error performance on additional regular-grid PDEs. † denotes ablation variants.
Params
32×32
64×64
128×128
FNO
701,281
0.0730
0.0832
0.1031
SAFDNO (Ours)
447,769
0.0413
0.0528
0.0643
Table 3: Zero-Shot Super-Resolution on 2D Darcy Flow. Operators are trained at 32×32 resolution and directly evaluated at all resolutions. FNO uses 12×12 retained Fourier modes, while SAFDNO has 12 poles along each spatial dimension, hence 24 poles in total.
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Learned poles of SAFDNO on the Darcy flow problem across four SAFDNO layers. Each marker denotes a predicted pole in the unit disk D .
Dataset
Resolution
No. train
No. test
Source
Darcy
85×85
1000
200
[ 2 , 32 ]
NS
64×64,T=20
1000
200
[ 2 ]
Pipe
129×129
1000
200
[ 8 ]
Airfoil
221×51
1000
200
[ 8 ]
Plasticity
101×31,T=20
900
80
[ 8 ]
Burgers
1024,T=20
1000
200
[ 33 ]
Appendix
Table 4: Description of benchmark datasets used in the main experiments.
FNO
SAFDNO
Dataset
Width
Layers
Modes
Width
Layers
Poles
Plasticity
/
/
/
64
4
(16,16)
Airfoil
/
/
/
64
4
(16,16)
Pipe
/
/
/
64
4
(16,16)
Navier-Stokes
64
4
(32,32)
64
4
(16,16)
Darcy
128
4
(12,12)
64
4
(32,32)
Appendix
Table 5: Hyperparameters for SAFDNO and FNO.
Metric
FNO
SAFDNO
Params
19.01M
1.56M
Train Time / Epoch (s)
5.79
25.49
Train Memory (MB)
741.76
541.11
Inference Time / Sample (ms)
1.07
8.66
Inference Memory (MB)
209.00
115.59
Inference FLOPs (GFLOPs)
2.20
7.98
Appendix
Table 6: Efficiency comparison on the Darcy benchmark with a batch size of 1 . Training time is the average epoch time. Inference time and memory are measured per sample.
Metric
FNO
SAFDNO
Params
33.59M
1.55M
Train Time / Epoch (s)
7.45
43.30
Train Memory (MB)
2187.64
11974.76
Inference Time / Sample (ms)
10.59
76.04
Inference Memory (MB)
312.39
56.72
Inference FLOPs (GFLOPs)
3.16
36.29
Appendix
Table 7: Efficiency comparison on the Navier-Stokes equation with a batch size of 6 . Training time is the average epoch time. Inference time and memory are measured per sample.
Model
poles/modes
Test rel. err. (%)
Params (M)
FNO
(8, 8)
1.10
8.52
FNO
(16, 16)
1.12
33.69
FNO
(32, 32)
1.14
134.35
SAFDNO
(8, 8)
0.81
1.54
SAFDNO
(16, 16)
0.64
1.55
SAFDNO
(32, 32)
0.54
1.56
Appendix
Table 8: Scaling effect of the number of Fourier modes and SAFD poles on the Darcy flow benchmark. We vary the number of modes or poles. Under the same setting, SAFDNO consistently achieves lower test error than FNO. Increasing the number of SAFDNO poles improves accuracy while keeping the trainable parameter count nearly unchanged, whereas increasing the number of FNO modes substantially increases the number of trainable parameters.
Figure 5: Zero-Shot Super-Resolution of SAFDNO on 2D Darcy Flow.
Figure 6: Performance of SAFDNO on Darcy
Figure 7: Performance of SAFDNO on Navier-Stokes
Figure 8: Performance of SAFDNO on Airfoil
Figure 9: Performance of SAFDNO on Pipe
Figure 10: Performance of SAFDNO on Plasticity
Figure 11: Test relative ℓ2 error - SAFDNO vs FNO
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.
Noam Koren, Rafael Moschopoulos, Kira Radinsky +1
Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University
Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining O(NlogN) complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.
Xuxiang Zhao, Angelica I. Aviles-Rivero
Qiuzhen College, Tsinghua University · YMSC, Tsinghua University
Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately 4 to 15 times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of 73 times fewer parameters across multiple PDE benchmarks.