physics.comp-phApr 7, 2026

Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

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

Organizations: Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Information Technology Laboratory, U.S. Army Engineer Research and Development Center, MS, USA · Department of Mechanical Engineering and Technology Management, Norwegian University of Life Sciences, Ås, Norway · Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory, Oslo, Norway

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Abstract

The impact of wave-induced forcing on the mean water level and nearshore currents is typically modeled through excess momentum fluxes, also known as radiation stresses, and their spatial gradients. Accurate storm surge prediction requires coupled circulation and wave models, but the high computational cost of numerical wave models limits their temporal resolution. In this work, we explore a proof-of-concept application of Deep Operator Networks (DeepONets) as a surrogate for the Simulating WAves Nearshore (SWAN) numerical wave model. Unlike grid-dependent surrogate models, DeepONets learn the underlying continuous operator, and thus, can provide highly efficient prediction while enabling discretization-invariant inference. The proposed surrogate model is evaluated using two 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 DeepONet surrogate improves computational efficiency by four orders of magnitude. Furthermore, the model demonstrates consistently high accuracy in predicting the significant wave height and the x- and y- components of the radiation stress gradient, by achieving relative L_2 errors bounded by 1.91%, 10.98%, and 6.88%, respectively, across all unseen test scenarios.

Figures & tables

Figure 1: Conceptual overview of the DeepONet architecture for modeling bulk wave parameters. For each wave–wind scenario, boundary wave information and wind forcing are encoded by the branch network, while spatial coordinates for evaluating the output function are encoded by the trunk network to their corresponding pp -dimensional vector fields. The DeepONet approximation of the solution function at the queried coordinates is obtained by taking a dot product of the branch and trunk outputs, thus implicitly approximating the SWAN solution operator. The high-fidelity SWAN solutions are used as the ground truth to supervise training through a point-wise loss function.
 Parameter1-D ExampleDUCK Example
 Reference FramePositive x-axis = 0∘0^{\circ}True North (+y) = 0∘0^{\circ}
 Direction ConventionPropagation Direction (toward)Nautical Direction (from)
WindWind Speed11−17 m/s11-17~m/s1−8 m/s1-8~m/s
Increment1 m/s1~m/s1 m/s1~m/s
Wind Direction−7∘-7^{\circ} to 7∘7^{\circ}0∘0^{\circ} to 315∘315^{\circ}
Increment1∘1^{\circ}45∘45^{\circ}
Table 1: Summary of input parameter spaces, coordinate reference frames, and directional conventions utilized for generating the training and testing datasets in the 1-D uniform slope and 2-D DUCK field experiment examples.
Figure 2: Schematic diagram of the 1-D example. The wave condition at x=0x=0 and propagates along the positive x-axis.
Figure 3: The left figure shows the overlap of the computational grid centers (gray) with the bathymetry (color). The computational grid extends beyond the surveyed bathymetry to minimize boundary effects. After SWAN is run, it generates the x, y coordinates along with their corresponding water depths, Hsig, and other target wave properties. The figure on the right illustrates the spatial distribution of water depth.
Cross-shore Stations (Offshore →\rightarrow Shore)Hsig Testbed (m)Hsig JONSWAP (m)Relative Difference(%)
SAMSON1.646301.640060.38
Nearshore #1\#11.805011.750833.00
Nearshore #2\#21.794151.757272.06
Nearshore #3\#31.711971.701490.61
Nearshore #4\#41.621951.627010.31
Nearshore #5\#51.429231.438560.65
Table 2: Comparison of significant wave height ( HsigH_{\mathrm{sig}} ) of the F71 test case obtained from the SWAN Test Bed observation-based simulation and the parametric JONSWAP-based simulation, at the SAMSON station and 9 different cross-shore stations, listed from farthest to nearest to shore.
 HyperparameterWave Prediction ModelForces Prediction Model
GeneralInitial Learning Rate10−310^{-3}10−410^{-4}
Batch Size25664 ∗
Branch/Trunk Output Shape 12030
BranchNumber of Layers44
Neurons Per Layer16128
Activation FunctionELULeakyReLU
Table 3: The best-performing set of hyperparameters obtained from Optuna. ∗ A larger batch size (256) was used for the DUCK case to reduce computational overhead, as smaller batch sizes resulted in excessive training cost. 1 This is the dimension denoted by pp in Section 2.2 .
Figure 4: Analysis of Hsig prediction errors for the 1-D example. (a) Histogram of scenario-based relative L2L_{2} errors across all test scenarios. (b-g) Parity plots comparing predicted and ground-truth values for the three best-performing scenarios (middle row) and the three worst-performing scenarios (bottom row), where the dashed diagonal lines indicate ideal agreement.
Figure 5: Analysis of radiation stress gradients prediction errors for the 1-D example. (a) Histogram of scenario-based relative L2L_{2} errors across all test scenarios. (b-g) Parity plots comparing predicted and ground-truth values for the three best-performing scenarios (middle row) and the three worst-performing scenarios (bottom row), where the dashed diagonal lines indicate ideal agreement.
VariableScenario 1 1 1 Scenarios are labeled using the format N1VN2N_{1}VN_{2} , where: N1N_{1} represents the example (1: 1-D, 2: DUCK, C: Cross-Grid), VV represents the variable (H: Hsig, X: X-forces, Y: Y-forces), and N2N_{2} represents the severity rank (1: worst, 2: second worst, 3: third worst). For example, “1H3” represents the scenario with the third highest relative L2L_{2} error in predicting Hsig for the 1-D example.RLE(s)(%)\mathbf{RLE^{(s)}(\%)}AEM(s)\mathbf{AE_{M}^{(s)}}MAE(s)\mathbf{MAE^{(s)}}RMSE(s)\mathbf{RMSE^{(s)}}
Hsig1H10.721.61×10−21.61\times 10^{-2}7.00×10−37.00\times 10^{-3}8.31×10−38.31\times 10^{-3}
 1H20.541.54×10−21.54\times 10^{-2}5.50×10−35.50\times 10^{-3}7.13×10−37.13\times 10^{-3}
 1H30.471.11×10−21.11\times 10^{-2}4.40×10−34.40\times 10^{-3}5.71×10−35.71\times 10^{-3}
Forces1X16.412.98×10−22.98\times 10^{-2}4.85×10−34.85\times 10^{-3}8.03×10−38.03\times 10^{-3}
 1X25.672.47×10−22.47\times 10^{-2}2.58×10−32.58\times 10^{-3}5.07×10−35.07\times 10^{-3}
 1X35.653.66×10−23.66\times 10^{-2}3.47×10−33.47\times 10^{-3}7.13×10−37.13\times 10^{-3}
Table 4: Error metrics of the three worst-performing scenarios for each variable in the 1-D example, identified by the three highest RLE(s)RLE^{(s)} magnitudes across all test cases.
VariableScenarioWindWind DirWave H.Wave Dir
Hsig1H111-7.00.96.0
 1H213-4.00.60.0
 1H311-7.01.26.0
Forces1X115-7.00.67.0
 1X2120.01.20.0
 1X315-6.01.07.0
Table 5: Configurations of the three worst-performing scenarios reported in Table 4 . Wind speed is given in m/s, wave height in m, and directional quantities in degrees.
Figure 6: Comparison between SWAN and DeepONet for the worst-case scenarios in the 1-D example. The top panel shows the Hsig prediction and the bottom panel shows the radiation stress gradient predictions.
Figure 7: Sensitivity of model prediction (best-case, mean, and worst-case RLE(s)RLE^{(s)} ) and computational training overhead to varying spatial coordinate sampling percentages ( 1%1\% , 5%5\% , 10%10\% , 20%20\% , and 100%100\% ) over the idealized 1-D uniform slope domain: (a) evaluation for the HsigH_{sig} , and (b) evaluation for the wave-induced radiation-stress force.
Figure 8: Analysis of Hsig prediction errors for the DUCK test example. (a) Histogram of scenario-based relative L2L_{2} errors across all test scenarios. (b-g) Parity plots comparing predicted and ground-truth values for the three best-performing scenarios (middle row) and the three worst-performing scenarios (bottom row), where the dashed diagonal lines indicate ideal agreement.
Figure 9: Analysis of x-forces prediction errors for the DUCK test example. (a) Histogram of scenario-based relative L2L_{2} errors across all test scenarios. (b-g) Parity plots comparing predicted and ground-truth values for the three best-performing scenarios (middle row) and the three worst-performing scenarios (bottom row), where the dashed diagonal lines indicate ideal agreement.
Figure 10: Analysis of y-forces prediction errors for the DUCK test example. (a) Histogram of scenario-based relative L2L_{2} errors across all test scenarios. (b-g) Parity plots comparing predicted and ground-truth values for the three best-performing scenarios (middle row) and the three worst-performing scenarios (bottom row), where the dashed diagonal lines indicate ideal agreement.
VariableScenarioRLE(s)(%)\mathbf{RLE^{(s)}(\%)}AEM(s)\mathbf{AE_{M}^{(s)}}MAE(s)\mathbf{MAE^{(s)}}RMSE(s)\mathbf{RMSE^{(s)}}
Hsig2H11.919.87×10−29.87\times 10^{-2}1.44×10−21.44\times 10^{-2}1.86×10−21.86\times 10^{-2}
 2H21.819.54×10−29.54\times 10^{-2}1.29×10−21.29\times 10^{-2}1.75×10−21.75\times 10^{-2}
 2H31.581.00×10−11.00\times 10^{-1}1.15×10−21.15\times 10^{-2}1.53×10−21.53\times 10^{-2}
x-forces2X110.982.552.551.30×10−11.30\times 10^{-1}2.54×10−12.54\times 10^{-1}
 2X28.82.752.759.81×10−29.81\times 10^{-2}2.12×10−12.12\times 10^{-1}
 2X37.82.512.518.86×10−28.86\times 10^{-2}1.85×10−11.85\times 10^{-1}
Table 6: Error metrics of the three worst-performing scenarios for each variable in the Duck example, identified by the three highest RLE(s)RLE^{(s)} magnitudes across all test cases.
VariableScenarioWindWind DirWave H.Wave Dir
Hsig2H173151.6-20
 2H27901.6-20
 2H361351.6-20
x-forces2X1801.6160
 2X27901.6-20
 2X361351.6-20
Table 7: Configurations of the three worst-performing scenarios for each variable in DUCK example, as reported in Table 6 . Wind speed is given in m/s, wave height in m, and directional quantities in degrees.
Figure 11: Comparison of the worst-case scenarios for each variable between SWAN and DeepONet in the DUCK example. Wind and wave settings are specified in Table 7 . Panels (a–c) show the ground truth, DeepONet prediction, and relative L2L_{2} error for the worst Hsig scenario. Panels (d–f) and Panels (g–i) present the ground truth, DeepONet prediction, and relative L2L_{2} error for the worst x-force and y-force predictions, respectively. For enhanced spatial visualization, the color bar in (c), (f), and (i) is clipped at the 99th percentile to reduce the influence of extreme values and better highlight the overall spatial error patterns. Finally, panels (j-k), (l-m) and (n-o) show the cross-shore profiles and RLE of Hsig, x-forces, and y-forces, respectively, along the transects shown in (c), (f) and (i), respectively.
Figure 12: Location of the maximum absolute error for every test scenario of the DUCK example.
VariableDeepONetMLP-2LMLP-3L
Hsig40,72960,397225,583
x-forces94,62160,397225,583
y-forces94,62160,397225,583
Table 8: Total trainable parameters of the DeepONet and MLP benchmark models for each variable in the DUCK example.
VariableMetricsDeepONetMLP-2LMLP-3L
HsigMean RLE(s)RLE^{(s)}0.56%1.17%0.20%
 Median RLE(s)RLE^{(s)}0.51%1.03%0.10%
 Worst RLE(s)RLE^{(s)}1.91%4.27%1.31%
x-forcesMean RLE(s)RLE^{(s)}3.46%10.18%0.98%
 Median RLE(s)RLE^{(s)}3.27%9.71%0.82%
 Worst RLE(s)RLE^{(s)}10.38%31.42%5.13%
Table 9: Error metrics for the DeepONet and MLP models across all test scenarios of the DUCK example
Figure 13: Distribution of test scenario RLEsRLE^{s} values of the DeepONet and MLP models for each variable
Figure 14: Coarse and fine meshes used in the discretization invariance study. The region used in the evaluation of the error metrics, after applying boundary truncation, is shaded in light blue color.
VariableMetricsFull MeshTruncated MeshDeepONet
HsigMean RLE(s)RLE^{(s)}0.93%0.61%1.25%
 Median RLE(s)RLE^{(s)}0.79%0.48%1.09%
 Worst RLE(s)RLE^{(s)}2.27%1.40%3.17%
x-forcesMean RLE(s)RLE^{(s)}6.50%4.25%9.13%
 Median RLE(s)RLE^{(s)}6.00%4.11%8.67%
 Worst RLE(s)RLE^{(s)}10.47%7.01%14.85%
Table 10: Cross-grid verification error metrics across 576 DUCK Test scenarios. The coarse-mesh SWAN results are compared with the fine-mesh SWAN results on the common grid points across (a) the entire fine mesh (column 3) and (b) the boundary-truncated fine-mesh (column 4). The last column shows the error metrics between the fine-mesh SWAN results and the DeepONet inference on the common grid points in the boundary-truncated fine-mesh.
VariableScenarioWindWind DirWave H.Wave Dir
HsigCH173151.6-20
x-forcesCX1801.6160
y-forcesCY173152.6160
Table 11: Scenario conditions corresponding to the worst-performing test scenarios for each variable in the DUCK cross-grid verification. Wind speed is given in m/s, wave height in m, and directional quantities in degrees.
Figure 15: Comparison between SWAN and DeepONet for the worst-case scenarios in the DUCK example with different mesh resolutions. For visualization purposes, the color bar of RLERLE is clipped at the 99th percentile to reduce the influence of extreme values and better highlight the overall spatial error patterns.
Model1-D ExampleDUCK Example
Hsig26 m 41 s26~m~41~s12 h 15 m 33 s12~h~15~m~33~s
x-forces27 m 06 s27~m~06~s14 h 04 m 34 s14~h~04~m~34~s
y-forces-14 h 02 m 58 s14~h~02~m~58~s
Table 12: Training times for the Hsig, x-forces, and y-forces DeepONet models of the 2-D DUCK example on the TACC Vista cluster (NVIDIA GH200)
Figure 16: Scenarios-by-scenario computational runtime comparison between the SWAN model and DeepONet across all 576 test cases.
Figure 17: Spatial distribution of predicted x-forces compared with the benchmark SWAN under 7 m/s at 90∘90^{\circ} and boundary wave height 1.6 m at 20∘20^{\circ} . The color bar is clipped between -1 and 1 N/m2N/m^{2} to highlight spatial distribution patterns, illustrating DeepONet’s spatial smoothing relative to the localized fluctuations in SWAN.
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