F$^3$NO: Frequency-Decomposed Finite-Time Flow-map Neural Operators with Cross-Scale Conditioning

Fan Wu; Cheng Jing; Kookjin Lee

cs.LGOct 7, 2026

F $^3$ NO: Frequency-Decomposed Finite-Time Flow-map Neural Operators with Cross-Scale Conditioning

Authors: Fan Wu, Cheng Jing, Kookjin Lee

Organizations: Arizona State University Tempe, AZ

Abstract

Neural operators enable fast PDE forecasting, but repeated predictions accumulate errors and fine-scale structures remain difficult to resolve. We introduce a frequency-decomposed finite-time flow-map neural operator (F $^3$ NO) that leverages updated low-frequency features to guide nonlinear refinement of high-frequency information. Within each layer, this cross-scale conditioning connects global spectral processing with local detail refinement. The model directly predicts states at specified future times and adjusts the contributions of the two branches according to the prediction interval. For longer trajectories, it combines parallel predictions within short temporal segments with recursive propagation between segments. Experiments on five PDE benchmarks demonstrate improved forecasting accuracy over autoregressive and direct-prediction baselines. Ablations show that frequency-decomposed refinement can improve accuracy with fewer parameters, while the benefits of segmentation depend on spatial resolution and dynamical regime.

Figures & tables

Figure 1 : Overall architecture. (a) Given a source field $u(\mathbf{x},t_{0})$ and query time offsets $\{\Delta t_{i}\}_{i=1}^{T}$ , the model predicts future states $\hat{u}(\mathbf{x},t_{0}+\Delta t_{i})$ . (b) Each layer combines a pointwise linear map with FDSR, followed by activation and FiLM conditioning. (c) FDSR separates low- and high-frequency features, uses the updated low-frequency features to condition local detail refinement, and combines the branches through time-dependent gates. (d) FiLM applies channel-wise scale and shift parameters conditioned on the prediction interval.

Figure 2 : [KSE, $\nu=0.075$ , $S=128$ ] Frequency-resolved prediction error as a function of prediction horizon. Each cell reports the time-instantaneous (non-cumulative) per-mode relative $L^{2}$ error (%).

Figure 3 : [KSE,

\nu=0.075

S=64

] Prediction error measured as a function of prediction horizon. The prediction error is reported as the time-instantaneous (non-cumulative) relative

L^{2}

error at each time step.

	BE	KSE			NSE			SWE	RTI
	$\nu=0.001$	$\nu=0.075$	$\nu=0.1$	$\nu=0.125$	$\nu=10^{-5}$	$\nu=10^{-4}$	$\nu=10^{-3}$	$r\sim\mathcal{U}(0.3,0.7)$	$A_{t}=0.25$
Rollout
FNO	7.64 $\pm$ 1.09	33.23 $\pm$ 0.76	16.21 $\pm$ 1.89	12.45 $\pm$ 1.93	2.27 $\pm$ 0.12	2.61 $\pm$ 0.20	1.42 $\pm$ 0.32	3.67 $\pm$ 0.26	6.00 $\pm$ 0.18
FFNO	4.02 $\pm$ 0.43	8.02 $\pm$ 0.85	3.42 $\pm$ 0.28	2.10 $\pm$ 0.55	1.53 $\pm$ 0.03	0.76 $\pm$ 0.06	0.52 $\pm$ 0.02	2.52 $\pm$ 0.39	5.73 $\pm$ 0.31
FNO-TF	4.72 $\pm$ 0.01	3.64 $\pm$ 0.08	1.48 $\pm$ 0.10	0.80 $\pm$ 0.04	1.96 $\pm$ 0.03	1.02 $\pm$ 0.03	0.67 $\pm$ 0.03	2.02 $\pm$ 0.11	5.93 $\pm$ 0.11
FFNO-TF	4.26 $\pm$ 0.24	1.26 $\pm$ 0.06	0.47 $\pm$ 0.04	0.35 $\pm$ 0.01	1.60 $\pm$ 0.01	1.23 $\pm$ 0.06	0.45 $\pm$ 0.02	0.56 $\pm$ 0.11	5.34 $\pm$ 0.13

Table 1 : Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying datasets and varying PDE parameters, with a resolution

S=128

for BE/KSE, a resolution

S=64\times 64

for NSE and SWE, and a resolution

S=32\times 32\times 32

for RTI. Best and second-best performance are bolded and underlined, respectively.

Rollout
	32	64	128
FNO	78.96 $\pm$ 6.06	50.76 $\pm$ 2.91	33.23 $\pm$ 0.76
FFNO	47.89 $\pm$ 1.23	18.31 $\pm$ 1.32	8.02 $\pm$ 0.85
FNO-TF	101.68 $\pm$ 3.85	37.24 $\pm$ 0.79	3.64 $\pm$ 0.08
FFNO-TF	72.93 $\pm$ 1.14	13.20 $\pm$ 0.20	1.26 $\pm$ 0.06
UNET-TF	87.02 $\pm$ 3.41	31.13 $\pm$ 1.48	9.04 $\pm$ 0.98

Table 2 : Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying spatial resolutions

S\in\{32,64,128\}

with the fixed

\nu=0.075

Figure 4 : (a,b,c) prediction error for the proposed model as a function of segmentation length over three spatial resolutions. (d)

\|u_{t}-u_{0}\|

quantifies how rapidly the system evolves away from its initial condition.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	PDE Parameter	S	$N_{t}$	Rollout Baseline		Our Model		N_train	N_test
Dataset	PDE Parameter	S	$N_{t}$	Input	Output	Input	Output	N_train	N_test
1D BE	$\nu=0.001$	128, 64, 32	20	$u_{t}$	$u_{t+1}$	$(u_{t},\Delta t)$	$u_{t+\Delta t}$	2048	1000
1D KSE	$\nu\in\{0.075,0.1,0.125\}$	128, 64, 32	25	$u_{t}$	$u_{t+1}$	$(u_{t},\Delta t)$	$u_{t+\Delta t}$	2048	256
2D NSE	$\nu\in\{10^{-5},10^{-4},10^{-3}\}$	$64\times 64$	20, 30, 50	$u_{t}$	$u_{t+1}$	$(u_{t},\Delta t)$	$u_{t+\Delta t}$	1000	200
2D SWE	$r\sim\mathcal{U}(0.3,0.7)$	$64\times 64$	101	$u_{t}$	$u_{t+1}$	$(u_{t},\Delta t)$	$u_{t+\Delta t}$	900	100
3D RTI	$A_{t}=0.25$	$32\times 32\times 32$	25	$u_{t}$	$u_{t+1}$	$(u_{t},\Delta t)$	$u_{t+\Delta t}$	9	2

Appendix

Table 3 : Overview of the datasets and prediction settings. Autoregressive baselines learn a one-step transition, while our model learns a time-conditioned flow-map mapping.

Figure 5 : [KSE,

\nu=0.075

S=128

] Per-frequency Fourier coefficient change from the initial state.

Figure 6 : [KSE,

\nu=0.075

S=128

] Mean per-mode relative

L^{2}

error as a function of Fourier mode

k

at the final prediction step

t=24

. The curves correspond to the final column of each heatmap in Figure 2 and report time-instantaneous (non-cumulative) errors.

Flow-map
Model	Modes	Params.	$\nu$
Model	Modes	Params.	0.075	0.1	0.125
FNO-Flow-map	65	4,361,601	3.38 $\pm$ 0.06	0.90 $\pm$ 0.04	0.43 $\pm$ 0.00
FNO-Flow-map+FDSR	48	3,447,177	2.01 $\pm$ 0.05	0.63 $\pm$ 0.01	0.34 $\pm$ 0.00
Hybrid-TF, $\tau=4$
FNO-Flow-map+FDSR (High-freq only)	48	3,447,177	1.25 $\pm$ 0.05	0.36 $\pm$ 0.01	0.19 $\pm$ 0.00

Appendix

Table 4 : [KSE,

S=128

] Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying

\nu

to isolate the effect of low-frequency conditioning and nonlinear local branch. We isolate the effect of low-frequency conditioning by comparing the high-frequency branch that uses high-frequency features only, high-frequency features concatenated with the original low-frequency feature, and high-frequency features concatenated with the spectrally updated low-frequency feature used in FDSR. We also replace the high-frequency branch in FDSR with a parameter-matched pointwise MLP applied to the full undecomposed latent feature, while keeping the low-frequency spectral branch and all remaining components unchanged.

Flow-map
Model	Modes	Params.	$\nu$
Model	Modes	Params.	$\nu=10^{-5}$	$\nu=10^{-4}$	$\nu=10^{-3}$
FNO-Flow-map	16	8,414,977	5.77 $\pm$ 0.36	4.90 $\pm$ 0.09	1.22 $\pm$ 0.07
FNO-Flow-map+FDSR	16	8,466,697	4.88 $\pm$ 0.11	4.79 $\pm$ 0.50	1.08 $\pm$ 0.08
Hybrid-TF, $\tau=4$
FNO-Flow-map+FDSR (High-freq only)	16	8,466,697	1.67 $\pm$ 0.00	0.61 $\pm$ 0.02	0.26 $\pm$ 0.02

Appendix

Table 5 : [NSE,

S=64\times 64

] Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying

\nu

Rollout
Model	Modes	Number of Parameters	$\nu$
Model	Modes	Number of Parameters	0.075	0.1	0.125
FNO-TF	65	4,359,553	3.64 $\pm$ 0.08	1.48 $\pm$ 0.10	0.80 $\pm$ 0.04
FFNO-TF	65	9,080,193	1.26 $\pm$ 0.06	0.47 $\pm$ 0.04	0.35 $\pm$ 0.01
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF	32	2,394,505	6.18 $\pm$ 0.06	1.13 $\pm$ 0.03	0.51 $\pm$ 0.04

Appendix

Table 6 : [KSE] Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying

\nu

with Frequency-Decomposed Spectral Refinement and a fixed resolution

S=128

Rollout
Model	Modes	Number of Parameters	$\nu$
Model	Modes	Number of Parameters	0.075	0.1	0.125
FNO-TF	33	2,262,401	37.24 $\pm$ 0.79	3.30 $\pm$ 0.07	1.10 $\pm$ 0.07
FFNO-TF	33	4,885,889	13.20 $\pm$ 0.20	1.55 $\pm$ 0.07	0.46 $\pm$ 0.03
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF	16	1,345,929	37.05 $\pm$ 1.17	6.80 $\pm$ 0.37	2.28 $\pm$ 0.02

Appendix

Table 7 : [KSE] Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying

\nu

with Frequency-Decomposed Spectral Refinement and a fixed resolution

S=64

Rollout
Model	Modes	Number of Parameters	$\nu$
Model	Modes	Number of Parameters	0.075	0.1	0.125
FNO-TF	17	1,213,825	101.68 $\pm$ 3.85	49.83 $\pm$ 0.59	29.43 $\pm$ 0.39
FFNO-TF	17	2,788,737	72.93 $\pm$ 1.14	26.11 $\pm$ 0.77	13.58 $\pm$ 0.29
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF	8	821,641	84.36 $\pm$ 0.62	34.58 $\pm$ 0.37	18.65 $\pm$ 1.34

Appendix

Table 8 : [KSE] Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying

\nu

with Frequency-Decomposed Spectral Refinement and a fixed resolution

S=32

Intervention	KSE	NSE
F 3 NO	$1.0$ $\times$	$1.0$ $\times$
no high branch, layer 0	$15.3$ $\times$	$6.6$ $\times$
no high branch, layer 1	$10.4$ $\times$	$4.5$ $\times$
no high branch, layer 2	$23.5$ $\times$	$13.0$ $\times$
no high branch, layer 3	$3.4$ $\times$	$9.7$ $\times$
no high branch, all layers	$31.5$ $\times$	$17.0$ $\times$

Appendix

Table 9 : Rollout error growth when the high-frequency branch is removed at inference (

\tau=4

; KSE

\nu=0.075

S=128

; NSE

\nu=10^{-5}

S=64\times 64

). Factors are relative to F 3 NO.

	BE	KSE			NSE			SWE	RTI
	$\nu=0.001$	$\nu=0.075$	$\nu=0.1$	$\nu=0.125$	$\nu=10^{-5}$	$\nu=10^{-4}$	$\nu=10^{-3}$	$r\sim\mathcal{U}(0.3,0.7)$	$A_{t}=0.25$
Flow-map
FNO-TCIN	91.68 $\pm$ 0.62	8.24 $\pm$ 0.05	3.38 $\pm$ 0.03	2.00 $\pm$ 0.01	9.72 $\pm$ 0.52	7.71 $\pm$ 0.17	2.19 $\pm$ 0.09	0.25 $\pm$ 0.05	5.15 $\pm$ 0.03
FFNO-TCIN	1.88 $\pm$ 0.07	2.84 $\pm$ 0.01	1.01 $\pm$ 0.04	0.59 $\pm$ 0.09	3.03 $\pm$ 0.14	3.25 $\pm$ 0.07	1.73 $\pm$ 0.13	0.26 $\pm$ 0.00	5.12 $\pm$ 0.02
FNO with FiLM only	2.24 $\pm$ 0.05	3.38 $\pm$ 0.06	0.90 $\pm$ 0.04	0.43 $\pm$ 0.00	5.77 $\pm$ 0.36	4.90 $\pm$ 0.09	1.22 $\pm$ 0.07	0.26 $\pm$ 0.01	5.06 $\pm$ 0.01
F 3 NO (w/o hybrid)	1.37 $\pm$ 0.11	2.01 $\pm$ 0.05	0.63 $\pm$ 0.01	0.34 $\pm$ 0.00	4.88 $\pm$ 0.11	4.79 $\pm$ 0.50	1.08 $\pm$ 0.08	0.23 $\pm$ 0.01	4.98 $\pm$ 0.01

Appendix

Table 10 : Relative

L^{2}

error (%) with mean

\pm

standard deviation for varying datasets and varying PDE parameters with a resolution

S=128

for BE/KSE, a resolution

S=64\times 64

for NSE and SWE, and a resolution

S=32\times 32\times 32

for RTI. The hybrid segmented execution uses

\tau=4

for KSE, NSE, and RTI,

\tau=19

for BE, and

\tau=10

for SWE.

Figure 7 : [BE] Prediction error for the proposed hybrid model as a function of segmentation length with

\nu=0.001

across three spatial resolution,

S=32

S=64

, and

S=128

Figure 8 : [KSE] Prediction error for the proposed hybrid model as a function of segmentation length with

\nu=0.075

\nu=0.1

, and

\nu=0.125

across three spatial resolution,

S=32

S=64

, and

S=128

Figure 9 : [NSE] Prediction error for the proposed hybrid model as a function of segmentation length with

\nu=10^{-3}

\nu=10^{-4}

, and

\nu=10^{-5}

with a fixed resolution

S=64\times 64

Panel of Figure 10 — Figure 10 : [SWE/RTI] Prediction error for the proposed hybrid model as a function of segmentation length.

Trained $\tau$	$j{=}1$	2	4	6	8	10	12	16	24
4	1.60	1.18	1.00	–	–	–	–	–	–
8	2.35	1.31	1.00	1.25	2.10	–	–	–	–
10	2.40	1.32	1.00	1.02	1.15	2.43	–	–	–
12	2.47	1.48	1.00	1.03	1.27	2.58	3.22	–	–
24	2.70	1.62	1.00	1.02	1.26	1.67	2.17	2.75	2.62

Appendix

Table 11 : [KSE,

\nu=0.075

S=128

] Direct error per snapshot interval,

\eta(j)

, normalized by the minimum error within each trained-

\tau

model. For all models, the minimum error occurs at

j=4

, consistent with the empirical choice to run KSE

S=128

\tau=4

Trained $\tau$	$j{=}1$	2	4	6	8	10	12	16	19
2	1.30	1.00	–	–	–	–	–	–	–
4	1.76	1.19	1.00	–	–	–	–	–	–
8	2.86	1.95	1.32	1.24	1.00	–	–	–	–
10	3.78	2.87	1.73	1.47	1.12	1.00	–	–	–
12	4.55	2.86	1.86	1.69	1.42	1.23	1.00	–	–
16	5.67	3.62	2.81	2.96	3.28	2.99	2.25	1.00	–

Appendix

Table 12 : [BE,

\nu=0.001

S=128

] Direct error per snapshot interval,

\eta(j)

, normalized by the minimum error within each trained-

\tau

model. For each trained-

\tau

model, the minimum of

\eta(j)

occurs at its longest supported lead; the full-horizon model attains its minimum at j=19, consistent with the empirical choice

\tau=19

Trained $\tau$	$j{=}1$	2	4	6	8	10	12	16	19
4	1.45	1.07	1.00	–	–	–	–	–	–
8	1.63	1.21	1.00	1.04	1.20	–	–	–	–
12	1.64	1.27	1.00	1.03	1.03	1.10	1.20	–	–
16	1.64	1.20	1.00	1.02	1.10	1.21	1.33	1.68	–
19	1.59	1.24	1.00	1.01	1.06	1.15	1.24	1.46	1.58

Appendix

Table 13 : [NSE,

\nu=10^{-5}

S=64\times 64

] Direct error per snapshot interval,

\eta(j)

, normalized by the minimum error within each trained-

\tau

model. For all models, the minimum error occurs at

j=4

, consistent with the empirical choice to run NSE

S=64\times 64

\tau=4

Trained $\tau$	$j{=}1$	2	4	5	10	20	25	50
10	1.90	1.47	1.18	1.11	1.00	–	–	–
20	4.46	1.36	1.09	1.05	1.00	1.08	–	–
50	2.55	1.56	1.07	1.03	1.00	1.02	1.03	1.06

Appendix

Table 14 : [SWE,

r\sim\mathcal{U}(0.3,0.7)

S=64\times 64

] Direct error per snapshot interval,

\eta(j)

, normalized by the minimum error within each trained-

\tau

model. For all models, the minimum error occurs at

j=10

, consistent with the empirical choice to run SWE

S=64\times 64

\tau=10

Trained $\tau$	$j{=}1$	2	4	6	8	10	12	16	24
4	1.42	1.04	1.00	–	–	–	–	–	–
10	1.37	1.04	1.00	1.08	1.36	2.11	–	–	–
16	1.99	1.12	1.00	1.14	1.39	1.98	2.78	4.13	–
24	2.40	1.22	1.00	1.48	2.07	3.41	4.97	7.25	7.39

Appendix

Table 15 : [RTI,

A_{t}=0.25

S=32\times 32\times 32

] Direct error per snapshot interval,

\eta(j)

, normalized by the minimum error within each trained-

\tau

model. For all models, the minimum error occurs at

j=4

, consistent with the empirical choice to run RTI

S=32\times 32\times 32

\tau=4

Figure 11 : [KSE,

\nu=0.075

S=64

] Relative

L^{2}

error versus missing time-step percentage under irregular temporal sampling. Missing masks are sampled independently per test trajectory, with

t=0

and the final prediction time retained. Ours refers to our main FNO-Hybrid-TF model.

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