cs.LGOct 7, 2026

F3^3NO: 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

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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 (F3^3NO) 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(x,t0)u(\mathbf{x},t_{0}) and query time offsets {Δti}i=1T\{\Delta t_{i}\}_{i=1}^{T} , the model predicts future states u^(x,t0+Δti)\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, ν=0.075\nu=0.075 , S=128S=128 ] Frequency-resolved prediction error as a function of prediction horizon. Each cell reports the time-instantaneous (non-cumulative) per-mode relative L2L^{2} error (%).
Figure 3 : [KSE, ν=0.075\nu=0.075 , S=64S=64 ] Prediction error measured as a function of prediction horizon. The prediction error is reported as the time-instantaneous (non-cumulative) relative L2L^{2} error at each time step.
 BEKSENSESWERTI
 ν=0.001\nu=0.001ν=0.075\nu=0.075ν=0.1\nu=0.1ν=0.125\nu=0.125ν=10−5\nu=10^{-5}ν=10−4\nu=10^{-4}ν=10−3\nu=10^{-3}r∼U(0.3,0.7)r\sim\mathcal{U}(0.3,0.7)At=0.25A_{t}=0.25
Rollout  
FNO7.64 ±\pm 1.0933.23 ±\pm 0.7616.21 ±\pm 1.8912.45 ±\pm 1.932.27 ±\pm 0.122.61 ±\pm 0.201.42 ±\pm 0.323.67 ±\pm 0.266.00 ±\pm 0.18
FFNO4.02 ±\pm 0.438.02 ±\pm 0.853.42 ±\pm 0.282.10 ±\pm 0.551.53 ±\pm 0.030.76 ±\pm 0.060.52 ±\pm 0.022.52 ±\pm 0.395.73 ±\pm 0.31
FNO-TF4.72 ±\pm 0.013.64 ±\pm 0.081.48 ±\pm 0.100.80 ±\pm 0.041.96 ±\pm 0.031.02 ±\pm 0.030.67 ±\pm 0.032.02 ±\pm 0.115.93 ±\pm 0.11
FFNO-TF4.26 ±\pm 0.241.26 ±\pm 0.060.47 ±\pm 0.040.35 ±\pm 0.011.60 ±\pm 0.011.23 ±\pm 0.060.45 ±\pm 0.020.56 ±\pm 0.115.34 ±\pm 0.13
Table 1 : Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying datasets and varying PDE parameters, with a resolution S=128S=128 for BE/KSE, a resolution S=64×64S=64\times 64 for NSE and SWE, and a resolution S=32×32×32S=32\times 32\times 32 for RTI. Best and second-best performance are bolded and underlined, respectively.
 3264128
Rollout
FNO78.96 ±\pm 6.0650.76 ±\pm 2.9133.23 ±\pm 0.76
FFNO47.89 ±\pm 1.2318.31 ±\pm 1.328.02 ±\pm 0.85
FNO-TF101.68 ±\pm 3.8537.24 ±\pm 0.793.64 ±\pm 0.08
FFNO-TF72.93 ±\pm 1.1413.20 ±\pm 0.201.26 ±\pm 0.06
UNET-TF87.02 ±\pm 3.4131.13 ±\pm 1.489.04 ±\pm 0.98
Table 2 : Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying spatial resolutions S∈{32,64,128}S\in\{32,64,128\} with the fixed ν=0.075\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) ∥ut−u0∥\|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

DatasetPDE ParameterSNtN_{t}Rollout BaselineOur ModelN_trainN_test
InputOutputInputOutput
1D BEν=0.001\nu=0.001128, 64, 3220utu_{t}ut+1u_{t+1}(ut,Δt)(u_{t},\Delta t)ut+Δtu_{t+\Delta t}20481000
1D KSEν∈{0.075,0.1,0.125}\nu\in\{0.075,0.1,0.125\}128, 64, 3225utu_{t}ut+1u_{t+1}(ut,Δt)(u_{t},\Delta t)ut+Δtu_{t+\Delta t}2048256
2D NSEν∈{10−5,10−4,10−3}\nu\in\{10^{-5},10^{-4},10^{-3}\}64×6464\times 6420, 30, 50utu_{t}ut+1u_{t+1}(ut,Δt)(u_{t},\Delta t)ut+Δtu_{t+\Delta t}1000200
2D SWEr∼U(0.3,0.7)r\sim\mathcal{U}(0.3,0.7)64×6464\times 64101utu_{t}ut+1u_{t+1}(ut,Δt)(u_{t},\Delta t)ut+Δtu_{t+\Delta t}900100
3D RTIAt=0.25A_{t}=0.2532×32×3232\times 32\times 3225utu_{t}ut+1u_{t+1}(ut,Δt)(u_{t},\Delta t)ut+Δtu_{t+\Delta t}92
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, ν=0.075\nu=0.075 , S=128S=128 ] Per-frequency Fourier coefficient change from the initial state.
Figure 6 : [KSE, ν=0.075\nu=0.075 , S=128S=128 ] Mean per-mode relative L2L^{2} error as a function of Fourier mode kk at the final prediction step t=24t=24 . The curves correspond to the final column of each heatmap in Figure 2 and report time-instantaneous (non-cumulative) errors.
ModelModesParams.ν\nu
0.0750.10.125
Flow-map
FNO-Flow-map654,361,6013.38 ±\pm 0.060.90 ±\pm 0.040.43 ±\pm 0.00
FNO-Flow-map+FDSR483,447,1772.01 ±\pm 0.050.63 ±\pm 0.010.34 ±\pm 0.00
Hybrid-TF, τ=4\tau=4
FNO-Flow-map+FDSR (High-freq only)483,447,1771.25 ±\pm 0.050.36 ±\pm 0.010.19 ±\pm 0.00
Appendix
Table 4 : [KSE, S=128S=128 ] Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying ν\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.
ModelModesParams.ν\nu
ν=10−5\nu=10^{-5}ν=10−4\nu=10^{-4}ν=10−3\nu=10^{-3}
Flow-map
FNO-Flow-map168,414,9775.77 ±\pm 0.364.90 ±\pm 0.091.22 ±\pm 0.07
FNO-Flow-map+FDSR168,466,6974.88 ±\pm 0.114.79 ±\pm 0.501.08 ±\pm 0.08
Hybrid-TF, τ=4\tau=4
FNO-Flow-map+FDSR (High-freq only)168,466,6971.67 ±\pm 0.000.61 ±\pm 0.020.26 ±\pm 0.02
Appendix
Table 5 : [NSE, S=64×64S=64\times 64 ] Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying ν\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.
ModelModesNumber of Parametersν\nu
0.0750.10.125
Rollout
FNO-TF654,359,5533.64 ±\pm 0.081.48 ±\pm 0.100.80 ±\pm 0.04
FFNO-TF659,080,1931.26 ±\pm 0.060.47 ±\pm 0.040.35 ±\pm 0.01
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF322,394,5056.18 ±\pm 0.061.13 ±\pm 0.030.51 ±\pm 0.04
Appendix
Table 6 : [KSE] Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying ν\nu with Frequency-Decomposed Spectral Refinement and a fixed resolution S=128S=128 .
ModelModesNumber of Parametersν\nu
0.0750.10.125
Rollout
FNO-TF332,262,40137.24 ±\pm 0.793.30 ±\pm 0.071.10 ±\pm 0.07
FFNO-TF334,885,88913.20 ±\pm 0.201.55 ±\pm 0.070.46 ±\pm 0.03
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF161,345,92937.05 ±\pm 1.176.80 ±\pm 0.372.28 ±\pm 0.02
Appendix
Table 7 : [KSE] Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying ν\nu with Frequency-Decomposed Spectral Refinement and a fixed resolution S=64S=64 .
ModelModesNumber of Parametersν\nu
0.0750.10.125
Rollout
FNO-TF171,213,825101.68 ±\pm 3.8549.83 ±\pm 0.5929.43 ±\pm 0.39
FFNO-TF172,788,73772.93 ±\pm 1.1426.11 ±\pm 0.7713.58 ±\pm 0.29
Rollout with Frequency-Decomposed Spectral Refinement
FNO-TF8821,64184.36 ±\pm 0.6234.58 ±\pm 0.3718.65 ±\pm 1.34
Appendix
Table 8 : [KSE] Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying ν\nu with Frequency-Decomposed Spectral Refinement and a fixed resolution S=32S=32 .
InterventionKSENSE
F 3 NO1.01.0 ×\times1.01.0 ×\times
no high branch, layer 015.315.3 ×\times6.66.6 ×\times
no high branch, layer 110.410.4 ×\times4.54.5 ×\times
no high branch, layer 223.523.5 ×\times13.013.0 ×\times
no high branch, layer 33.43.4 ×\times9.79.7 ×\times
no high branch, all layers31.531.5 ×\times17.017.0 ×\times
Appendix
Table 9 : Rollout error growth when the high-frequency branch is removed at inference ( τ=4\tau=4 ; KSE ν=0.075\nu=0.075 , S=128S=128 ; NSE ν=10−5\nu=10^{-5} , S=64×64S=64\times 64 ). Factors are relative to F 3 NO.
 BEKSENSESWERTI
 ν=0.001\nu=0.001ν=0.075\nu=0.075ν=0.1\nu=0.1ν=0.125\nu=0.125ν=10−5\nu=10^{-5}ν=10−4\nu=10^{-4}ν=10−3\nu=10^{-3}r∼U(0.3,0.7)r\sim\mathcal{U}(0.3,0.7)At=0.25A_{t}=0.25
Flow-map 
FNO-TCIN91.68 ±\pm 0.628.24 ±\pm 0.053.38 ±\pm 0.032.00 ±\pm 0.019.72 ±\pm 0.527.71 ±\pm 0.172.19 ±\pm 0.090.25 ±\pm 0.055.15 ±\pm 0.03
FFNO-TCIN1.88 ±\pm 0.072.84 ±\pm 0.011.01 ±\pm 0.040.59 ±\pm 0.093.03 ±\pm 0.143.25 ±\pm 0.071.73 ±\pm 0.130.26 ±\pm 0.005.12 ±\pm 0.02
FNO with FiLM only2.24 ±\pm 0.053.38 ±\pm 0.060.90 ±\pm 0.040.43 ±\pm 0.005.77 ±\pm 0.364.90 ±\pm 0.091.22 ±\pm 0.070.26 ±\pm 0.015.06 ±\pm 0.01
F 3 NO (w/o hybrid)1.37 ±\pm 0.112.01 ±\pm 0.050.63 ±\pm 0.010.34 ±\pm 0.004.88 ±\pm 0.114.79 ±\pm 0.501.08 ±\pm 0.080.23 ±\pm 0.014.98 ±\pm 0.01
Appendix
Table 10 : Relative L2L^{2} error (%) with mean ±\pm standard deviation for varying datasets and varying PDE parameters with a resolution S=128S=128 for BE/KSE, a resolution S=64×64S=64\times 64 for NSE and SWE, and a resolution S=32×32×32S=32\times 32\times 32 for RTI. The hybrid segmented execution uses τ=4\tau=4 for KSE, NSE, and RTI, τ=19\tau=19 for BE, and τ=10\tau=10 for SWE.
Figure 7 : [BE] Prediction error for the proposed hybrid model as a function of segmentation length with ν=0.001\nu=0.001 across three spatial resolution, S=32S=32 , S=64S=64 , and S=128S=128 .
Figure 8 : [KSE] Prediction error for the proposed hybrid model as a function of segmentation length with ν=0.075\nu=0.075 , ν=0.1\nu=0.1 , and ν=0.125\nu=0.125 across three spatial resolution, S=32S=32 , S=64S=64 , and S=128S=128 .
Figure 9 : [NSE] Prediction error for the proposed hybrid model as a function of segmentation length with ν=10−3\nu=10^{-3} , ν=10−4\nu=10^{-4} , and ν=10−5\nu=10^{-5} with a fixed resolution S=64×64S=64\times 64 .
Figure 10 : [SWE/RTI] Prediction error for the proposed hybrid model as a function of segmentation length.
Trained τ\tauj=1j{=}1246810121624
41.601.181.00––––––
82.351.311.001.252.10––––
102.401.321.001.021.152.43–––
122.471.481.001.031.272.583.22––
242.701.621.001.021.261.672.172.752.62
Appendix
Table 11 : [KSE, ν=0.075\nu=0.075 , S=128S=128 ] Direct error per snapshot interval, η(j)\eta(j) , normalized by the minimum error within each trained- τ\tau model. For all models, the minimum error occurs at j=4j=4 , consistent with the empirical choice to run KSE S=128S=128 at τ=4\tau=4 .
Trained τ\tauj=1j{=}1246810121619
21.301.00–––––––
41.761.191.00––––––
82.861.951.321.241.00––––
103.782.871.731.471.121.00–––
124.552.861.861.691.421.231.00––
165.673.622.812.963.282.992.251.00–
Appendix
Table 12 : [BE, ν=0.001\nu=0.001 , S=128S=128 ] Direct error per snapshot interval, η(j)\eta(j) , normalized by the minimum error within each trained- τ\tau model. For each trained- τ\tau model, the minimum of η(j)\eta(j) occurs at its longest supported lead; the full-horizon model attains its minimum at j=19, consistent with the empirical choice τ=19\tau=19 .
Trained τ\tauj=1j{=}1246810121619
41.451.071.00––––––
81.631.211.001.041.20––––
121.641.271.001.031.031.101.20––
161.641.201.001.021.101.211.331.68–
191.591.241.001.011.061.151.241.461.58
Appendix
Table 13 : [NSE, ν=10−5\nu=10^{-5} , S=64×64S=64\times 64 ] Direct error per snapshot interval, η(j)\eta(j) , normalized by the minimum error within each trained- τ\tau model. For all models, the minimum error occurs at j=4j=4 , consistent with the empirical choice to run NSE S=64×64S=64\times 64 at τ=4\tau=4 .
Trained τ\tauj=1j{=}124510202550
101.901.471.181.111.00–––
204.461.361.091.051.001.08––
502.551.561.071.031.001.021.031.06
Appendix
Table 14 : [SWE, r∼U(0.3,0.7)r\sim\mathcal{U}(0.3,0.7) , S=64×64S=64\times 64 ] Direct error per snapshot interval, η(j)\eta(j) , normalized by the minimum error within each trained- τ\tau model. For all models, the minimum error occurs at j=10j=10 , consistent with the empirical choice to run SWE S=64×64S=64\times 64 at τ=10\tau=10 .
Trained τ\tauj=1j{=}1246810121624
41.421.041.00––––––
101.371.041.001.081.362.11–––
161.991.121.001.141.391.982.784.13–
242.401.221.001.482.073.414.977.257.39
Appendix
Table 15 : [RTI, At=0.25A_{t}=0.25 , S=32×32×32S=32\times 32\times 32 ] Direct error per snapshot interval, η(j)\eta(j) , normalized by the minimum error within each trained- τ\tau model. For all models, the minimum error occurs at j=4j=4 , consistent with the empirical choice to run RTI S=32×32×32S=32\times 32\times 32 at τ=4\tau=4 .
Figure 11 : [KSE, ν=0.075\nu=0.075 , S=64S=64 ] Relative L2L^{2} error versus missing time-step percentage under irregular temporal sampling. Missing masks are sampled independently per test trajectory, with t=0t=0 and the final prediction time retained. Ours refers to our main FNO-Hybrid-TF model.

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