Stochastic Adaptive Fourier Decomposition for Operator Learning

Pengqing Shi; Liming Zhang; Tao Qian; Stephen Tierney; Jie Yin; Junbin Gao

cs.LGOct 3, 2026

Stochastic Adaptive Fourier Decomposition for Operator Learning

Authors: Pengqing Shi, Liming Zhang, Tao Qian, Stephen Tierney, Jie Yin, Junbin Gao

Organizations: University of Sydney · University of Macau · Macau University of Science and Technology

Abstract

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

\ell_{2}

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

\ell_{2}

error performance on additional regular-grid PDEs. † denotes ablation variants.

	Params	$32\times 32$	$64\times 64$	$128\times 128$
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\times 32

resolution and directly evaluated at all resolutions. FNO uses

12\times 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

\mathbb{D}

Dataset	Resolution	No. train	No. test	Source
Darcy	$85\!\times\!85$	$1000$	$200$	[ 2 , 32 ]
NS	$64\!\times\!64,\;T=20$	$1000$	$200$	[ 2 ]
Pipe	$129\!\times\!129$	$1000$	$200$	[ 8 ]
Airfoil	$221\!\times\!51$	$1000$	$200$	[ 8 ]
Plasticity	$101\!\times\!31,\;T=20$	$900$	$80$	[ 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 10: Performance of SAFDNO on Plasticity

Panel of Figure 11 — Figure 11: Test relative $\ell_{2}$ error - SAFDNO vs FNO

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