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

1 Introduction

• Autoregressive rollout can accumulate prediction errors into long-horizon drift and requires training through a costly, difficult-to-optimize unrolled computation graph.Finite-time flow-map methods instead predict a future state from a source state and elapsed time, but must still represent evolving spatial structures.Source: §1 · p. 1
• F3NO uses updated low-frequency features to condition nonlinear high-frequency refinement, with lead-time-dependent fusion of global spectral and local branches.The local branch processes high-frequency information without separate spectral weights for those modes.Source: §1 · p. 2
• Longer prediction intervals reduce recursive transitions but make each mapping harder to approximate; shorter intervals simplify mappings but require more frequent propagation.Segmentation combines parallel predictions within bounded intervals with recursive propagation across segment boundaries.Source: §1 · p. 2
• F3NO is evaluated on five PDE benchmarks, with ablations examining when frequency refinement and segmentation affect accuracy and efficiency.The benchmarks include KSE, Burgers’, Navier–Stokes, shallow-water, and Rayleigh–Taylor instability equations.Source: §1 · p. 2

2.1 Neural Operators

• Neural operators approximate mappings between function spaces, and Fourier neural operators parameterize an integral kernel in the frequency domain [30].FNO layers combine a global integral operator with a local linear transformation, with Fourier transforms enabling spectral convolution.Source: §2.1 · pp. 2–3

2.2 Flow-map Learning

• Flow-map learning models finite-time evolution directly from a state at time s to a state at time t, without explicitly representing instantaneous dynamics [7] [8].The learned operator can support direct long-horizon prediction without recursive rollout.Source: §2.2 · p. 3
• Time-conditioned FNOs use the prediction interval to modulate intermediate features, allowing one operator to predict at different lead times [22].For autonomous dynamics, the operator can be evaluated at a desired lead time or composed over shorter intervals.Source: §2.2 · p. 3

3.1 Finite-Time Prediction Parameterization

• The flow-map parameterization predicts a future state from a source state and time displacement, with zero displacement reproducing the source state by construction.The displacement is expressed in units of dataset snapshot spacing, and the operator represents normalized finite-time displacement rather than an instantaneous derivative.Source: §3.1 · pp. 3–4
• F3NO uses an FNO-based flow-map whose hidden features are conditioned on the requested time displacement through FiLM modulation.Unlike TCIN, this modulation is applied without instance normalization.Source: §3.1 · p. 4

3.2 FDSR: Frequency-Decomposed Spectral Refinement

• FDSR updates low-frequency features globally, then uses them to condition a local nonlinear correction of high-frequency features within the same layer.The local branch’s parameter count does not grow with the number of high-frequency modes.Source: §3.2 · pp. 4–5
• Frequency decomposition uses a Fourier low-pass projector with cutoff m to form low- and high-frequency feature components.The high-frequency component is formed by applying the complement of the low-pass projector.Source: §3.2 · p. 4
• A lead-time-dependent gate combines the spectral update and local correction, while FiLM scales and shifts channels of the combined representation.The gate learns how branch contributions vary with prediction interval without prescribing which branch dominates at longer lead times.Source: §3.2 · p. 5

3.3 Segmented Training and Prediction

• Segmentation trades the difficulty of approximating longer finite-time mappings against more frequent propagation of predicted states.Intermediate choices restrict recursive state propagation to segment boundaries; their accuracy depends on interval approximation error and boundary-error amplification.Source: §3.3 · pp. 5–6
• Hybrid-TF trains on bounded intervals using ground-truth states at segment boundaries and predicts the observed states within each segment directly.This avoids propagating prediction errors between training segments; Hybrid-RO instead feeds a predicted terminal state into the next training segment.Source: §3.3 · p. 5
• At inference, states within a segment share an input and can be evaluated in parallel, while each terminal prediction initializes the next segment.A one-interval segment recurs at each interval, whereas a segment spanning the full horizon predicts directly from the initial state.Source: §3.3 · pp. 5–6

4.1 Experimental Setup

• The benchmark suite covers Burgers, Kuramoto–Sivashinsky, Navier–Stokes, shallow-water, and Rayleigh–Taylor equations, with rollout and flow-map baselines.The rollout baselines include FNO, FFNO, UNet, S4FFNO, PDE-Refiner, and FreqMoE; flow-map baselines include MLP, DeepONets, TCIN, and HyperFNO.Source: §4.1 · p. 6

4.2 In-Depth Analysis of KSE Forecasting

• On KSE at S = 128, FDSR makes errors more even across Fourier modes and weakens dominant error bands; F3NO further reduces late-horizon errors.The analysis compares FNO-TF, direct flow-map prediction, flow-map plus FDSR, and F3NO.Source: §4.2 · p. 7, Figure 2 · p. 7
• On KSE at S = 64, F3NO maintains controlled error growth over longer horizons and is faster with lower error than rollout-based baselines.The flow-map formulation supports batched inference over multiple time gaps in one forward pass, unlike sequential autoregressive rollouts.Source: §4.2 · p. 7, Figure 3 · p. 7
• With other settings fixed, FDSR has the lowest error among the tested KSE variants across viscosities, and NSE ablations show the same ranking.The comparisons include high-frequency-only refinement, conditioning on original low-frequency features, and a parameter-matched pointwise MLP.Source: §4.2 · p. 7

4.3 Performance across PDE benchmarks.

• Across all benchmarks, F3NO achieves the lowest error, with approximately 6%-46% relative error reduction over baselines.Table 1 reports relative L2 error across five datasets and varying PDE parameters.Source: §4.3 · p. 8, Table 1 · p. 8
• As viscosity decreases, all methods’ errors rise, while F3NO achieves approximately 6%-46% relative improvement across viscosity settings.The authors attribute the method’s advantage to bounded flow-map intervals alongside global efficiency within segments.Source: §4.3 · p. 8
• At resolutions S ∈ {32, 64, 128}, the segmented flow-map remains lowest-error, improving 9%–42% over the second-best models.Rollout-based models narrow the gap as resolution increases but remain less accurate at the highest resolution.Source: §4.3 · pp. 8–9, Table 2 · p. 8
• Smaller segments tend to work better at higher resolutions and in more chaotic regimes; lower-resolution or milder regimes favor longer segments, sometimes τ = T.The reported state displacement helps contextualize why more chaotic regimes make direct long-range flow-map learning harder.Source: §4.3 · p. 9

Related neural operators

• Neural-operator variants modify FNOs through factorized spectral layers [51], temporal memory [11], or Fourier-domain channel mixing and soft-thresholding.The passage also identifies the FNO frequency-domain kernel parameterization [30].Source: §5 · p. 9

Multiscale and parameter-conditioned methods

• U-NO [46], U-FNO [52], and U-Net variants [20] use hierarchical or complementary local processing, while HyperFNO conditions generated weights on PDE parameters.Source: §5 · p. 10

Frequency-aware methods

• Unlike output-state refinement [32], spectral-layer fine-tuning [13], or frequency-expert weight transfer [15], FDSR conditions local nonlinear refinement on updated low-frequency features.The passage presents this as feature-level conditioning within each operator layer.Source: §5 · p. 10

Temporal prediction methods

• Autoregressive deployment can accumulate errors and increase long-horizon latency; F3NO combines finite-time mappings with segmented execution to balance mapping difficulty against recursive propagation.The passage contrasts this approach with Neural ODE integration and identifies neural flows [7] [9] and time-conditioned operators [22] as finite-time mapping methods.Source: §5 · p. 10

6 Conclusion

• Across five PDE benchmarks, F3NO achieves the lowest mean prediction error across all evaluated settings.The conclusion attributes this result to combining explicit cross-scale conditioning with flexible temporal composition.Source: §6 · p. 10
• FDSR uses updated low-frequency features to guide nonlinear high-frequency refinement, connecting global spectral processing with local detail correction.The conclusion describes this as a parameter-efficient way to process high-frequency information; lead-time-dependent fusion adjusts the branches’ contributions to the prediction interval.Source: §6 · p. 10
• Segmented training and prediction balance long-range mapping difficulty against recursive error accumulation by propagating predictions only across segment boundaries.States are predicted directly within each segment, reducing sequential dependence and potentially improving long-horizon accuracy.Source: §6 · p. 10

A Benchmark Problems

• The cited list includes 2D Navier–Stokes, 2D Shallow–Water, and 3D Rayleigh Taylor Instability; Table 3 summarizes their prediction settings, with predicted states of vorticity, water depth, and density, respectively.Source: A Benchmark Problems · p. 14, Table 3 · p. 14

1D Burgers’ Equation

• Burgers experiments use the PDEBench data at viscosity 0.001, test resolutions 128, 64, and 32, and final time T = 20.The benchmark is provided at spatial resolution 1024.Source: A Benchmark Problems · p. 14

1D Kuramoto–Sivashinsky Equation

• Kuramoto–Sivashinsky exhibits spatiotemporal chaos and sensitivity to initial conditions, making autoregressive rollouts challenging as local errors compound over time.The experiments span increasingly chaotic viscosity regimes and evaluate downsampled resolutions 32, 64, and 128.Source: A Benchmark Problems · p. 14

2D Navier–Stokes Equations

• The 2D Navier–Stokes benchmark uses vorticity on a 64 × 64 grid, with viscosities 10^-3, 10^-4, and 10^-5 and viscosity-dependent final times.Data are generated on a 256 × 256 grid and recorded at intervals of 0.25 seconds.Source: A Benchmark Problems · pp. 14–15

2D Shallow Water Equations

• The 2D shallow-water benchmark is a radial dam-break problem, evaluated at 64 × 64 resolution using water depth as the predicted state.The public benchmark is provided at 128 × 128 resolution; h denotes water depth, u and v velocity components, and hu and hv directional momentum.Source: A Benchmark Problems · pp. 14–15

3D Rayleigh–Taylor Instability

• The 3D Rayleigh–Taylor experiments downsample the public dataset to 32 × 32 × 32 and predict density alone.They retain the first 100 simulation steps, subsample every four steps, and use trajectories of 25 states.Source: A Benchmark Problems · pp. 15–16

B.1 Rollout Baselines

• The rollout baselines include FNO, FFNO, U-Net, and S4FFNO, spanning spectral operators, a convolutional encoder-decoder, and a memory-augmented operator.FFNO factorizes spectral operations across spatial dimensions; S4FFNO adds a causal sequence memory module to FFNO.Source: §B.1 · p. 16
• PDE-Refiner iteratively refines the same next-state prediction to address low-amplitude frequency components that can accumulate errors during long rollouts.Its inference process reconstructs the next state from the refined residual, then propagates that state autoregressively.Source: §B.1 · p. 17
• FreqMoE uses low-frequency pretraining followed by high-frequency expert fine-tuning, with spectral regions assigned to experts and a learned gate controlling their contributions.The baseline uses the same FNO backbone and training budget as the FNO baselines.Source: §B.1 · p. 17

B.2 Flow-map Baselines

• Flow-map baselines predict future states from a source state and elapsed time, including time-conditioned HyperFNO and feature-modulated FNO-IN.The MLP baseline also takes the source state, spatial coordinates, and temporal displacement as input; DeepONet combines branch and trunk features.Source: §B.2 · p. 18

C Implementation Details

• The implementation section describes training strategies for autoregressive rollouts, model specifications, and hardware configurations.It distinguishes autoregressive rollout, which feeds back predicted states, from Markov teacher forcing, which trains on the ground-truth previous state but evaluates autoregressively.Source: C Implementation Details · p. 19, §C.1 · p. 19

C.1 Training Strategies for Autoregressive Rollouts

• The rollout training strategy uses first-order Markov conditioning, so the model conditions only on the immediately preceding state.In the rollout setting, predicted states are fed back as inputs for subsequent steps.Source: §C.1 · p. 19

C.2 Training Strategies for Hybrid Execution Modes

• Hybrid-TF re-grounds training at segment boundaries with ground-truth states, whereas Hybrid-RO feeds each predicted terminal state into the next segment.Re-grounding prevents early prediction errors from corrupting subsequent training segments; both modes use predicted boundary states at inference.Source: §C.2 · p. 19
• Shorter segments simplify each prediction but require more recursive transitions; longer segments improve computational efficiency while increasing the flow-map's burden.Segmented execution balances local stability and global efficiency between rollout and a single full-horizon flow-map.Source: §C.2 · p. 19

C.3 Training Details

• Models are trained for 200 epochs with AdamW, and reported test losses are averaged over three runs with varying random seeds.The setup uses a 0.001 initial learning rate and 0.0001 weight decay.Source: §C.3 · p. 20

C.4 Model Specification

• The shared FNO backbone uses four Fourier layers, with width 128 for KSE and BE and width 64 for NSE, SWE, and RTI.Time-conditioned variants use a FiLM hypernetwork to generate modulation parameters from the prediction interval.Source: §C.4 · pp. 20–21

D Frequency-Decomposed Spectral Refinement

• FDSR splits hidden features into low- and high-frequency components, processes them through spectral and local branches, then fuses their contributions according to the target time displacement.A low-pass mask selects the retained modes, while the local branch conditions high-frequency refinement on updated low-frequency features.Source: D Frequency-Decomposed Spectral… · p. 21

D.1 Frequency-Dependent Evolution and Prediction Errors

• Low-frequency changes remain relatively stable early, while changes become concentrated in mid-to-high frequencies and grow more pronounced at later snapshots.The comparison covers KSE at t = 6, 12, 18, and 24.Source: §D.1 · p. 22
• F3NO further suppresses high-frequency errors after FDSR, achieving the lowest error across displayed modes and the most consistent error curve.FDSR alone reduces error at every displayed mode, though appreciable errors remain at the highest frequencies.Source: §D.1 · p. 22, Figure 6 · p. 22

D.2 Cross-Scale Conditioning and Local-Refinement

• Across KSE and NSE settings, the full FDSR block has the lowest mean error at every tested viscosity; updated low-frequency features improve refinement beyond original low-frequency features.The parameter-matched undecomposed MLP has higher mean errors, so additional nonlinear capacity alone does not explain the improvement.Source: Table 4 · p. 23, Table 5 · p. 23, §D.2 · p. 24

D.3 Spectral Cutoff, Parameter Efficiency, and Spatial Resolution

• FDSR improves prediction accuracy over standard counterparts in autoregressive rollout and improves accuracy while decreasing parameter counts across tested backbones and operation modes.The reported cutoff retains low-frequency modes for spectral processing and uses the remaining modes for high-frequency refinement.Source: §D.3 · p. 24, Table 6 · p. 24
• Hybrid execution outperforms both rollout and flow-map baselines, while FDSR remains useful at lower resolutions where direct prediction can already model the horizon effectively.At S = 64 and S = 32, direct flow-map+FDSR and F3NO results are nearly identical, so segmentation adds little there.Source: §D.3 · pp. 25–26, Table 7 · p. 25, Table 8 · p. 25

D.4 Layer-wise Contribution of High-Frequency Branch

• Removing the high-frequency branch from any single layer raises rollout error 3.4 to 23.5× on KSE and 4.5 to 13.0× on NSE.Replacing branch outputs with their test-set mean is as damaging; the most important layer is in the middle on KSE and deeper on NSE.Source: §D.4 · p. 26, Table 9 · p. 26

E Segmented Training and Prediction

• The segment length sets the trade-off between temporal span per prediction and recursive propagation, and Hybrid-TF re-grounds training at segment boundaries.Hybrid-RO instead propagates predicted boundary states during training; both modes use predicted boundary states at inference.Source: E Segmented Training and Prediction · p. 26

E.1 Effect of Segmentation across Time-Conditioned Operators

• Segmentation improves time-conditioned finite-time prediction when full-horizon mapping is difficult; F3NO has the lowest error in every reported multi-segment configuration.For F3NO, mean error falls from 2.01% to 1.15% on KSE with ν = 0.075 and from 4.88% to 1.44% on NSE with ν = 10−5.Source: §E.1 · p. 27, Table 10 · p. 27

E.2 Effect of Segment Length and Training Strategy

• In BE and KSE, pure rollout generally has the largest errors, while longer segments often improve performance, especially at lower resolutions and higher viscosity.In smoother regimes, the global flow-map can be competitive.Source: §E.2 · pp. 28–29
• For NSE, both pure rollout and full-horizon flow-map can be suboptimal, while intermediate segments balance recursive accumulation and difficult long-range prediction.Similar patterns are reported for SWE and RTI.Source: Figure 9 · p. 28, §E.2 · p. 29, Figure 10 · p. 29
• Hybrid-TF often yields lower or more stable errors in challenging settings, while Hybrid-RO is mainly competitive in easier cases where boundary-state drift is small.Re-grounding with ground-truth segment boundaries during training helps prevent early boundary errors from accumulating across later segments.Source: §E.2 · p. 29

E.3 Error Propagation in Autoregressive and Segmented Prediction

• The preferred interval from direct error per snapshot interval agrees with empirical segment-length sweeps across the benchmarks, without a full segmented rollout for each candidate interval.The reported minimizer is stable across tested training segment lengths for KSE, NSE, SWE, and RTI, while BE favors longer leads.Source: §E.3 · pp. 30–31
• A segment-wise error-propagation bound motivates selecting intervals using direct prediction error per snapshot interval, η(j) = ϵj/j.For KSE at S = 128, η is smallest at j = 4 across tested training segment lengths, matching the empirical sweep.Source: §E.3 · pp. 30–31, Table 11 · p. 30
• The experimental η(j) uses empirical mean relative L2 direct-prediction error rather than the absolute-error upper bound, making it a heuristic criterion.The bound assumes the learned predictor is Lipschitz on the relevant states and has bounded direct error.Source: §E.3 · p. 31

E.4 Irregularly Sampled Measurements

• As missing time steps increase, all methods degrade, but the proposed segmented flow-map has the lowest error across missing rates and degrades most gracefully.The KSE evaluation removes 20%, 40%, 60%, or 80% of time steps from trajectories.Source: §E.4 · p. 32, Figure 11 · p. 32

• The optimal segmentation length τ is user-defined and may vary with the underlying dynamics and spatial resolution.Source: F Limitations · p. 33
• Frequency decomposition uses a fixed spectral cutoff; adaptive partitioning may offer more flexibility across PDE regimes.Source: F Limitations · p. 33
• Further theoretical understanding of stability and error propagation remains valuable despite empirically stable and accurate long-horizon prediction.Source: F Limitations · p. 33