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DU-NO: A Parameter-Efficient Double U-Shaped Neural Operator for Phase-Resolving Wave Modeling

Enrique Hernandez Noguera, Md Meftahul Ferdaus, Nathan Cooper, Elias Ioup, Mahdi Abdelguerfi

arXiv:2609.12115v1cs.AI

TL;DR

Phase-resolving wave solvers capture nearshore dynamics but are costly, and accurate neural-operator hybrids can require tens of millions of parameters. DU-NO places compact convolutional branches only at shallow levels of a multiscale spectral operator; it improves on U-FNO on FUNWAVE-TVD waves and remains competitive or better on other fluid benchmarks.

  • Problem

    Phase-resolving nearshore wave models resolve individual waves but their computational cost limits operational uses, while accurate hybrid neural operators can be large.

    Source: §1 · p. 1

  • Method

    DU-NO combines a multiscale spectral pyramid with compact convolutional U-Net branches at its two shallowest encoder and decoder levels.

    Source: §1 · p. 1

  • Results

    DU-NO improves over U-FNO by 14.9% in relL2 on FUNWAVE-TVD waves with 10.8× fewer parameters, stays competitive on 2D Navier–Stokes, and surpasses baselines on shallow-water equations.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The results support selectively placing convolutional capacity at fine scales as a parameter-efficient approach for phase-resolving wave modeling and other evaluated fluid benchmarks.

    Source: §1 · p. 2, §6 · p. 8

    The study covers 2D regular grids, and its FUNWAVE surrogate is trained on a single bathymetry, leaving cross-bathymetry and real-coastline transfer untested.

    Source: §6 · p. 8

Abstract

from arXiv · show

Phase-resolving wave models such as FUNWAVE-TVD are the accuracy standard for nearshore dynamics, resolving the shoaling, refraction, and breaking of individual waves, but their cost rules them out for the ensembles, uncertainty quantification, and real-time warning that operational forecasting demands. Neural operators promise solver-level accuracy at a fraction of that cost, yet on wave-dominated fields the accurate ones are large: hybrid spectral-convolutional operators such as U-FNO (the strongest baseline in our study after DU-NO) buy their fidelity with tens of millions of parameters. We introduce DU-NO (Double U-shaped Neural Operator), a multiscale U-shaped spectral operator that attaches lightweight convolutional U-Net branches only at its two shallowest encoder and decoder levels. The placement follows a sampling argument: high-wavenumber content exists only on fine grids, so the local, full-band pathways go where that content lives, while the coarse, band-limited levels stay purely spectral. A depth-decaying mode schedule holds the model to 3.64M parameters, an order of magnitude below U-FNO. On our publicly released FUNWAVE-TVD benchmark, DU-NO attains the best autoregressive rollout error of six identically trained architectures, improving on U-FNO by 14.9% with 10.8x fewer parameters, and a frequency-band analysis shows the gain holds across all bands, including the high-wavenumber band where truncated-spectral operators collapse. Parameter-matched controls confirm the gain is architectural: rescaled to the same 3.6M budget, the best baseline still trails DU-NO by 28.6%. The advantage carries beyond nearshore waves: DU-NO matches the strongest baselines on 2D Navier-Stokes and wins clearly on PDEBench shallow-water rollouts. Code, trained models, and evaluation artifacts are available at https://anonymous.4open.science/r/duno-code-5A7B/.

1 Introduction

Phase-resolving models capture nearshore wave dynamics but are too costly for many operational uses. DU-NO places local convolutional branches only at fine scales, aiming to retain fidelity with fewer parameters.

Motivation

  • Nearshore flooding, overtopping, erosion, navigation safety, and coastal-defense design depend on wave dynamics, while the phase-resolving models that capture them are computationally costly.Neural operators offer solver surrogates that learn mappings between function spaces and can run orders of magnitude faster than data-generating solvers.Source: §1 · p. 1

Contributions

  • DU-NO combines a multiscale spectral pyramid with compact parallel U-Net branches at only the two shallowest encoder and decoder levels.A depth-decaying mode budget and capped channel width keep it to 3.64M parameters versus U-FNO’s 39.4M; bottleneck branches hurt accuracy at higher cost after downsampling removes high-wavenumber content.Source: §1 · pp. 1–2
  • The scale–frequency placement rule locates useful convolutional capacity in the gap between retained Fourier modes and the grid’s Nyquist limit, which shrinks with depth.The measured gap falls from 68% of field energy at the finest level to 17% at the coarsest.Source: §1 · p. 2
  • The FUNWAVE-TVD benchmark is a controlled 34-factorial dataset designed to stress steep, breaking, high-wavenumber wave dynamics underrepresented in common smooth, diffusion-dominated evaluations.The six architectures were trained with one identical configuration for three seeds.Source: §1 · p. 2

Empirical validation

  • DU-NO is the most accurate of six architectures on FUNWAVE-TVD waves, improving on U-FNO by 14.9% in relL2 with 10.8× fewer parameters.It also uses 4.4× less memory at comparable latency; parameter-matched controls and band-resolved analysis support the architectural advantage.Source: §1 · p. 2
  • The study evaluates DU-NO beyond nearshore waves, where it remains competitive on 2D Navier–Stokes and surpasses baselines on PDEBench shallow-water equations.Band-resolved analysis finds DU-NO’s gain across all bands, while truncated spectral operators concentrate error in the high-wavenumber band.Source: §1 · p. 2

2 Related work

Prior operator work spans spectral, convolutional, and attention-based designs, while hybrid U-shaped operators combine multiscale structure or local pathways in different ways. DU-NO targets high-frequency errors and phase-resolving wave fields that remain comparatively underrepresented in evaluation.

Neural operators for PDE surrogacy

  • FNO-based operator research includes factorized, tensorized, multiwavelet, convolutional, and learned-kernel variants, alongside attention over latent physics tokens.DU-NO keeps the vanilla spectral block and changes where a complementary convolutional pathway sits in a multiscale operator.Source: §2 · p. 2

U-shaped and hybrid spectral–convolutional operators

  • U-FNO pairs a full-resolution spectral stack with U-Net branches, whereas U-NO uses a multiscale encoder–decoder that remains purely spectral.DU-NO combines a U-NO-style pyramid with U-FNO-style branches only at its two shallowest levels, reaching 3.64M parameters against U-FNO’s 39.4M at better accuracy.Source: §2 · p. 2

Spectral bias and high-frequency error

  • Neural-network spectral bias and FNO truncation constrain fine-scale fidelity, and spectral surrogates can accumulate error over long autoregressive rollouts.The paper contributes cross-architecture error analysis by wavenumber and connects high-frequency fidelity with long-horizon rollout stability.Source: §2 · pp. 2–3

Machine learning for ocean-wave forecasting

  • Prior wave-forecasting studies target integrated or point-wise quantities, while DU-NO addresses autoregressive reconstruction of a phase-resolving solver’s full spatial field.The cited examples include buoy time series, data-assimilated forecasts, differential-structure losses, and extreme-event classifiers.Source: §2 · p. 3

Wave dynamics as an operator-learning benchmark

  • Phase-resolving nearshore waves are comparatively underrepresented in operator-learning evaluations, which often center on smooth, diffusion-governed problems such as Darcy flow.The paper uses FUNWAVE-TVD waves as its primary benchmark and adds PDEBench shallow-water equations and 2D Navier–Stokes.Source: §2 · p. 3

3 The Double U-shaped Neural Operator (DU-NO)

DU-NO combines a multiscale spectral encoder–decoder with convolutional U-Net branches only at its two shallowest encoder and decoder levels. Its placement follows the shrinking spectral gap across pyramid levels, while a depth-decaying mode schedule keeps the model compact.

3.1 Problem setup

  • DU-NO uses an autoregressive one-frame map: a window of solution frames predicts the next-frame increment, and longer trajectories reapply the map to its own predictions.Training is teacher-forced for 10 successive increments, whereas evaluation uses free rollout.Source: §3.1 · p. 3

3.2 A multiscale spectral backbone with depth-decaying mode budget

  • The encoder–decoder combines retained-mode spectral mixing, a full-band 1 × 1 path, and shallow-level convolutional branches, with skip connections carrying encoder features into the decoder.The bottleneck has no convolutional branch, and the MLP head emits the increment added to the last frame.Source: §3.2 · pp. 3–4, Figure 1 · p. 4
  • Retained modes halve with depth from 12 to 6 to 3, with 2 at the bottleneck, while channels grow from 32 to 64 to 128 and are capped at 128.Halving modes as resolution falls keeps per-level spectral cost roughly constant; the complete model has 3.64M parameters.Source: §3.2 · p. 4

3.3 Selectively-placed Convolutional U-Net branches

  • The spectral gap holds 68% of field energy at the finest level and 17% at the coarsest, while branch cost grows 16× across the same span.High-wavenumber structure exists only on fine grids; DU-NO therefore places branches where the gap is large and keeps coarse levels purely spectral.Source: §3.3 · p. 5
  • Shallow-only branches match or improve on branches at every level at under half the parameters; bottleneck branches hurt accuracy at higher cost.Removing shallow branches degrades accuracy; doubling branch width keeps error within seed noise while increasing branch parameters to 8.08M (2.2×).Source: §3.3 · p. 5, Figure 2 · p. 5

3.4 Cost analysis

  • 3.64M parameters—10.8× fewer than U-FNO—with 4.4× lower peak inference memory and comparable latency.On the same CPU, DU-NO advances a nearshore forecast a median 173× faster than FUNWAVE-TVD.Source: §3.4 · p. 5

4 Experimental setup

The evaluation covers nearshore waves and two additional fluid-dynamics benchmarks, using shared data splits and a common training protocol across six architectures. Models are compared through autoregressive rollouts, with a longer horizon for shallow water because its 10-step transient is degenerate.

Benchmarks

  • The primary benchmark is a generated FUNWAVE-TVD nearshore-wave dataset of 81 simulations, with Navier–Stokes and PDEBench shallow water added to test generalization.The wave data cover four forcing parameters; the three datasets use 128 × 128, 64 × 64, and 128 × 128 grids, respectively.Source: §4 · p. 6

Metrics

  • Evaluation uses free autoregressive rollout, with 10 steps for FUNWAVE and Navier–Stokes and 90 for shallow water; the primary metric is free-rollout relative L2.The shallow-water horizon avoids its degenerate 10-step transient, where all models reach a floor near 10−4 relative L2; Figure 3 also decomposes fRMSE into low, mid, and high radial-wavenumber bands.Source: §4 · p. 6

Baselines

  • Six architectures are evaluated, and every baseline is also retrained at a parameter-matched configuration rescaled to DU-NO’s 3.6M budget.The baselines are U-FNO, U-NO, Transolver, FNO, and convolutional U-Net.Source: §4 · p. 6

Training protocol; Seeds, selection, and reporting

  • All six architectures use one frozen training protocol per dataset, and each model–dataset pair is trained with three seeds and selected by validation rollout error.Test numbers use the best-validation checkpoint and report mean ± standard deviation over seeds; the Table 2 ablation sweep is single-seed exploratory work.Source: §4 · p. 6

5 Results

On the FUNWAVE-TVD benchmark, DU-NO achieves the best rollout rel-L2 among the tested architectures and remains strongest across frequency bands and longer rollouts. Ablations support shallow branch placement and a depth-decaying mode schedule, while tests on Navier–Stokes and shallow water show transfer beyond FUNWAVE.

5 Results

  • 28.6% is the remaining U-FNO control gap: at DU-NO’s 3.6M parameter budget, the best matched baseline still trails DU-NO.The parameter-matched controls use the identical protocol.Source: §5 · p. 6
  • 0.0946 ± 0.0020 rel-L2 gives DU-NO the best FUNWAVE rollout accuracy, versus 0.1111 ± 0.0034 for U-FNO, a 14.9% improvement.The comparison is on the FUNWAVE-TVD benchmark; Table 1 also reports parameters, peak memory, and latency.Source: §5 · p. 6, Table 1 · p. 7
  • DU-NO has the lowest error in every frequency band, while pure-spectral FNO has the worst high-band error.The figure reports each band’s error relative to that band’s truth RMS; the prose identifies the high band as k ≥13.Source: Figure 3 · p. 7, §5 · p. 7

5.1 Ablations

  • 0.0937 rel-L2 is achieved with a second branch pair at the next-shallowest level; deepest-level placement loses most of the benefit, and bottleneck-only placement is worse than none.The placement ablation varies branch placement with the backbone and protocol held fixed; rows are single-seed results.Source: Table 2 · p. 7, §5.1 · p. 7

5 Results

  • At 90 rollout steps, DU-NO keeps the lowest error at every step and spatial correlation above ρ = 0.9 for 42 steps, versus 39 for U-FNO.The other models remain at or below 27 steps above that correlation threshold.Source: §5 · p. 7

5.1 Ablations

  • 16.22M parameters and 0.0967 rel-L2 result from holding modes constant (12/12/12/8), versus 0.0937 for the canonical 12/6/3/2 schedule.The constant-mode model uses 4.5× the parameters for no gain; the 2.66M 8/4/2/2 schedule is within seed noise of the canonical model at 0.0946 rel-L2.Source: §5.1 · pp. 7–8

5.2 Generalization beyond FUNWAVE-TVD waves

  • DU-NO ties U-FNO on 2D Navier–Stokes and is most accurate by a clear margin on 90-step PDEBench shallow-water rollouts, where Transolver diverges.The first-window training protocol is a beyond-horizon stress test; under the full-trajectory control, most architectures reach rel-L2 ≤0.002 and no longer separate.Source: Table 3 · p. 8, §5.2 · p. 8

6 Conclusion

DU-NO is the most accurate of six architectures on the FUNWAVE-TVD benchmark while using substantially fewer parameters and less memory than U-FNO. The study also identifies open boundaries in dimensionality, bathymetry transfer, predicted state variables, and long-rollout accuracy.

6 Conclusion

  • DU-NO is the most accurate of six architectures on FUNWAVE-TVD, with 10.8× fewer parameters and 4.4× less memory than U-FNO.Source: §6 · p. 8
  • DU-NO stays competitive on 2D Navier–Stokes and wins on the shallow-water benchmark by a clear margin.Source: §6 · p. 8
  • DU-NO places compact U-Net branches at the shallowest pyramid levels, where a sampling argument locates the spectral gap for local convolution.Source: §6 · p. 8
  • The study covers 2D regular grids and a FUNWAVE surrogate trained on one Duck, NC bathymetry; cross-bathymetry and synthetic-to-real coastline transfer remain untested.Source: §6 · p. 8
  • The model predicts surface elevation alone, leaving velocity-dependent quantities such as nearshore currents beyond its present reach.Source: §6 · p. 8
  • Analytical validation of the placement rule and control of long-rollout error accumulation remain open problems.Source: §6 · p. 8

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