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arxiv: 2604.06433 · v1 · submitted 2026-04-07 · ⚛️ physics.comp-ph · cs.LG· physics.flu-dyn

Recognition: no theorem link

Operator Learning for Surrogate Modeling of Wave-Induced Forces from Sea Surface Waves

Clint Dawson, Corey Trahan, Eirik Valseth, Mark Loveland, Peter Rivera-Casillas, Shukai Cai, Sourav Dutta

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cs.LGphysics.flu-dyn
keywords deep operator networkssurrogate modelingradiation stressSWAN wave modelwave-induced forcescoastal modelingstorm surgeoperator learning
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The pith

Deep operator networks can accurately surrogate the SWAN wave model for predicting radiation stress and wave heights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether Deep Operator Networks can act as a fast replacement for the full SWAN numerical wave model when calculating wave-induced forces. Inputs are boundary wave conditions and wind fields; outputs are the components of the radiation stress gradient together with significant wave height. The surrogate is trained and evaluated on several 1-D and 2-D steady-state problems, including a realistic coastal domain at Duck, North Carolina, where it reproduces the target fields with consistently high accuracy. A reader would care because this surrogate could let wave information enter coupled circulation models at finer time steps than is currently feasible, improving storm-surge forecasts without the usual computational penalty.

Core claim

The authors show that a DeepONet, trained exclusively on steady-state wave simulations, reproduces the radiation stress gradient components and significant wave height produced by the full SWAN model across multiple representative scenarios with variable boundary conditions and wind forcing, including a realistic application to the Duck, NC coastline.

What carries the argument

Deep Operator Networks that learn the mapping from input functions (boundary wave spectra and wind fields) to output functions (radiation stress gradients and significant wave height) as a direct surrogate for the SWAN solver.

Load-bearing premise

Performance measured on steady-state examples will carry over when the surrogate is inserted into the unsteady, time-evolving coupled wave-circulation models used in practice.

What would settle it

A side-by-side run of the surrogate and the full SWAN model on an unsteady simulation with time-varying winds and boundaries, checking whether errors in the radiation stress gradient components remain small over many time steps.

Figures

Figures reproduced from arXiv: 2604.06433 by Clint Dawson, Corey Trahan, Eirik Valseth, Mark Loveland, Peter Rivera-Casillas, Shukai Cai, Sourav Dutta.

Figure 1
Figure 1. Figure 1: The branch network encodes the input function u, which can correspond to initial conditions, boundary conditions, or other forcing inputs, through its evaluations at a set of representative sampling points {xi ∈ X }m i=1, given by {u1 = u(x1), u2 = u(x2), . . . , um = u(xm)}. The output of the branch network is a feature vector b ∈ R p . The trunk network, on the other hand, encodes the coordinates y ∈ Y a… view at source ↗
Figure 1
Figure 1. Figure 1: Conceptual overview of the DeepONet architecture for wave modeling. For each [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of the 1-D example. The initial wave starts at [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Analysis of Hsig prediction errors for the 1-D example. (a) Histogram of scenario [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Analysis of Forces prediction errors for the 1-D example. (a) Histogram of [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between SWAN and DeepONet for the worst-case scenarios in the [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bathymetry field of the 2-D example. The initial wave starts at the left boundary [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Analysis of Hsig prediction errors for the 2-D example. (a) Histogram of scenario [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Analysis of x-forces prediction errors for the 2-D example. (a) Histogram of [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Analysis of y-forces prediction errors for the 2-D example. (a) Histogram of [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the worst-case scenarios for each variable between SWAN and [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The left figure shows the overlap of the computational grid centers (grey) with [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Analysis of Hsig prediction errors for the DUCK test example. (a) Histogram [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Analysis of x-forces prediction errors for the DUCK test example. (a) Histogram [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Analysis of y-forces prediction errors for the DUCK test example. (a) Histogram [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of the worst-case scenarios for each variable between SWAN and [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Location of the maximum absolute error of each scenario across all test scenarios [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
read the original abstract

Wave setup plays a significant role in transferring wave-induced energy to currents and causing an increase in water elevation. This excess momentum flux, known as radiation stress, motivates the coupling of circulation models with wave models to improve the accuracy of storm surge prediction, however, traditional numerical wave models are complex and computationally expensive. As a result, in practical coupled simulations, wave models are often executed at much coarser temporal resolution than circulation models. In this work, we explore the use of Deep Operator Networks (DeepONets) as a surrogate for the Simulating WAves Nearshore (SWAN) numerical wave model. The proposed surrogate model was tested on three distinct 1-D and 2-D steady-state numerical examples with variable boundary wave conditions and wind fields. When applied to a realistic numerical example of steady state wave simulation in Duck, NC, the model achieved consistently high accuracy in predicting the components of the radiation stress gradient and the significant wave height across representative scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Deep Operator Networks (DeepONets) as a surrogate for the SWAN numerical wave model to predict wave-induced forces, specifically the components of the radiation stress gradient and significant wave height, from boundary wave conditions and wind fields. The surrogate is trained and tested exclusively on steady-state solutions from three 1-D and 2-D numerical examples, including a realistic case in Duck, NC, where it reports consistently high accuracy.

Significance. If the operator generalizes beyond the steady-state regime, the work could enable computationally efficient surrogates for wave modeling in coupled circulation-wave systems, improving storm surge predictions. The choice of DeepONet for field-to-field mapping is appropriate for this physics application, and the inclusion of a realistic coastal geometry (Duck, NC) provides a concrete test of practical relevance.

major comments (2)
  1. [Abstract and Introduction] Abstract and Introduction: The motivation explicitly states that practical coupled simulations run wave models at coarser temporal resolution under unsteady conditions, yet all training, validation, and testing data consist of steady-state snapshots with fixed boundary conditions and wind fields. No time-dependent marching, circulation feedback, or temporal subsampling experiments are reported, so the learned operator has no demonstrated applicability to the stated target regime.
  2. [Duck, NC example] Duck, NC results: The claim of 'consistently high accuracy' in predicting radiation stress gradient components and significant wave height is presented without quantitative metrics (e.g., RMSE or relative L2 error with error bars), training/validation split details, baseline comparisons, or hyperparameter sensitivity analysis, weakening the evidential support for the surrogate's performance.
minor comments (1)
  1. [Methods] Clarify the precise input/output function spaces for the DeepONet (branch and trunk networks) and how they align with the SWAN discretization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which have helped us improve the clarity and rigor of our manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract and Introduction] Abstract and Introduction: The motivation explicitly states that practical coupled simulations run wave models at coarser temporal resolution under unsteady conditions, yet all training, validation, and testing data consist of steady-state snapshots with fixed boundary conditions and wind fields. No time-dependent marching, circulation feedback, or temporal subsampling experiments are reported, so the learned operator has no demonstrated applicability to the stated target regime.

    Authors: We agree that the current work focuses exclusively on steady-state conditions. This choice is motivated by the practical use of wave models at coarser temporal resolutions in coupled simulations, where the wave field can be approximated as steady-state for each time step. The DeepONet is trained to learn the mapping operator under these conditions. We will update the abstract and introduction to clarify that the surrogate is for steady-state wave modeling and include a discussion on the limitations regarding unsteady dynamics, along with plans for future extensions to time-dependent cases. revision: partial

  2. Referee: [Duck, NC example] Duck, NC results: The claim of 'consistently high accuracy' in predicting radiation stress gradient components and significant wave height is presented without quantitative metrics (e.g., RMSE or relative L2 error with error bars), training/validation split details, baseline comparisons, or hyperparameter sensitivity analysis, weakening the evidential support for the surrogate's performance.

    Authors: We thank the referee for highlighting this. We will revise the manuscript to include explicit quantitative metrics, including RMSE and relative L2 errors with error bars computed over multiple independent training runs, detailed information on the training, validation, and test data splits, comparisons against baseline methods such as linear interpolation or simpler neural network architectures, and a hyperparameter sensitivity study. These additions will provide stronger evidential support for the reported accuracy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; data-driven surrogate trained on independent SWAN simulations.

full rationale

The paper trains DeepONets on data generated by running the SWAN numerical wave model on steady-state 1-D and 2-D examples with prescribed boundary conditions and wind fields, then evaluates the learned operator on held-out test cases from the same steady-state regime (including the Duck, NC example). The central results are learned mappings from input fields to radiation-stress gradients and significant wave height; these are not equivalent to the inputs by construction, nor are any fitted parameters renamed as predictions. No load-bearing self-citations, uniqueness theorems, or ansatzes reduce the claim to prior author work. The derivation is self-contained against the numerical benchmarks used for training and testing.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical validation of the trained surrogate against SWAN outputs; the main added value is the trained model rather than new physical axioms.

free parameters (1)
  • DeepONet hyperparameters (branch and trunk net sizes, etc.)
    The network architecture parameters are chosen and likely tuned to fit the training data from SWAN simulations.
axioms (1)
  • domain assumption DeepONet can approximate the operator mapping from wave boundary conditions to radiation stress fields
    Relies on the universal approximation property of DeepONets for operators, assumed to hold for this wave physics operator.

pith-pipeline@v0.9.0 · 5490 in / 1227 out tokens · 46454 ms · 2026-05-10T17:56:34.308522+00:00 · methodology

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