Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

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

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

p

-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.

	Parameter	1-D Example	DUCK Example
	Reference Frame	Positive x-axis = $0^{\circ}$	True North (+y) = $0^{\circ}$
	Direction Convention	Propagation Direction (toward)	Nautical Direction (from)
Wind	Wind Speed	$11-17~m/s$	$1-8~m/s$
	Increment	$1~m/s$	$1~m/s$
	Wind Direction	$-7^{\circ}$ to $7^{\circ}$	$0^{\circ}$ to $315^{\circ}$
	Increment	$1^{\circ}$	$45^{\circ}$

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=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(%)
SAMSON	1.64630	1.64006	0.38
Nearshore $\#1$	1.80501	1.75083	3.00
Nearshore $\#2$	1.79415	1.75727	2.06
Nearshore $\#3$	1.71197	1.70149	0.61
Nearshore $\#4$	1.62195	1.62701	0.31
Nearshore $\#5$	1.42923	1.43856	0.65

Table 2: Comparison of significant wave height (

H_{\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.

	Hyperparameter	Wave Prediction Model	Forces Prediction Model
General	Initial Learning Rate	$10^{-3}$	$10^{-4}$
	Batch Size	256	64 ∗
	Branch/Trunk Output Shape 1	20	30
Branch	Number of Layers	4	4
	Neurons Per Layer	16	128
	Activation Function	ELU	LeakyReLU

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

p

in Section 2.2 .

Figure 4: Analysis of Hsig prediction errors for the 1-D example. (a) Histogram of scenario-based relative

L_{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

L_{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.

Variable	Scenario 1 1 1 Scenarios are labeled using the format $N_{1}VN_{2}$ , where: $N_{1}$ represents the example (1: 1-D, 2: DUCK, C: Cross-Grid), $V$ represents the variable (H: Hsig, X: X-forces, Y: Y-forces), and $N_{2}$ represents the severity rank (1: worst, 2: second worst, 3: third worst). For example, “1H3” represents the scenario with the third highest relative $L_{2}$ error in predicting Hsig for the 1-D example.	$\mathbf{RLE^{(s)}(\%)}$	$\mathbf{AE_{M}^{(s)}}$	$\mathbf{MAE^{(s)}}$	$\mathbf{RMSE^{(s)}}$
Hsig	1H1	0.72	$1.61\times 10^{-2}$	$7.00\times 10^{-3}$	$8.31\times 10^{-3}$
	1H2	0.54	$1.54\times 10^{-2}$	$5.50\times 10^{-3}$	$7.13\times 10^{-3}$
	1H3	0.47	$1.11\times 10^{-2}$	$4.40\times 10^{-3}$	$5.71\times 10^{-3}$
Forces	1X1	6.41	$2.98\times 10^{-2}$	$4.85\times 10^{-3}$	$8.03\times 10^{-3}$
	1X2	5.67	$2.47\times 10^{-2}$	$2.58\times 10^{-3}$	$5.07\times 10^{-3}$
	1X3	5.65	$3.66\times 10^{-2}$	$3.47\times 10^{-3}$	$7.13\times 10^{-3}$

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)}

magnitudes across all test cases.

Variable	Scenario	Wind	Wind Dir	Wave H.	Wave Dir
Hsig	1H1	11	-7.0	0.9	6.0
	1H2	13	-4.0	0.6	0.0
	1H3	11	-7.0	1.2	6.0
Forces	1X1	15	-7.0	0.6	7.0
	1X2	12	0.0	1.2	0.0
	1X3	15	-6.0	1.0	7.0

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)}

) and computational training overhead to varying spatial coordinate sampling percentages (

1\%

,

5\%

,

10\%

,

20\%

, and

100\%

) over the idealized 1-D uniform slope domain: (a) evaluation for the

H_{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

L_{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

L_{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

L_{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.

Variable	Scenario	$\mathbf{RLE^{(s)}(\%)}$	$\mathbf{AE_{M}^{(s)}}$	$\mathbf{MAE^{(s)}}$	$\mathbf{RMSE^{(s)}}$
Hsig	2H1	1.91	$9.87\times 10^{-2}$	$1.44\times 10^{-2}$	$1.86\times 10^{-2}$
	2H2	1.81	$9.54\times 10^{-2}$	$1.29\times 10^{-2}$	$1.75\times 10^{-2}$
	2H3	1.58	$1.00\times 10^{-1}$	$1.15\times 10^{-2}$	$1.53\times 10^{-2}$
x-forces	2X1	10.98	$2.55$	$1.30\times 10^{-1}$	$2.54\times 10^{-1}$
	2X2	8.8	$2.75$	$9.81\times 10^{-2}$	$2.12\times 10^{-1}$
	2X3	7.8	$2.51$	$8.86\times 10^{-2}$	$1.85\times 10^{-1}$

Table 6: Error metrics of the three worst-performing scenarios for each variable in the Duck example, identified by the three highest

RLE^{(s)}

magnitudes across all test cases.

Variable	Scenario	Wind	Wind Dir	Wave H.	Wave Dir
Hsig	2H1	7	315	1.6	-20
	2H2	7	90	1.6	-20
	2H3	6	135	1.6	-20
x-forces	2X1	8	0	1.6	160
	2X2	7	90	1.6	-20
	2X3	6	135	1.6	-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

L_{2}

error for the worst Hsig scenario. Panels (d–f) and Panels (g–i) present the ground truth, DeepONet prediction, and relative

L_{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.

Variable	DeepONet	MLP-2L	MLP-3L
Hsig	40,729	60,397	225,583
x-forces	94,621	60,397	225,583
y-forces	94,621	60,397	225,583

Table 8: Total trainable parameters of the DeepONet and MLP benchmark models for each variable in the DUCK example.

Variable	Metrics	DeepONet	MLP-2L	MLP-3L
Hsig	Mean $RLE^{(s)}$	0.56%	1.17%	0.20%
	Median $RLE^{(s)}$	0.51%	1.03%	0.10%
	Worst $RLE^{(s)}$	1.91%	4.27%	1.31%
x-forces	Mean $RLE^{(s)}$	3.46%	10.18%	0.98%
	Median $RLE^{(s)}$	3.27%	9.71%	0.82%
	Worst $RLE^{(s)}$	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

RLE^{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.

Variable	Metrics	Full Mesh	Truncated Mesh	DeepONet
Hsig	Mean $RLE^{(s)}$	0.93%	0.61%	1.25%
	Median $RLE^{(s)}$	0.79%	0.48%	1.09%
	Worst $RLE^{(s)}$	2.27%	1.40%	3.17%
x-forces	Mean $RLE^{(s)}$	6.50%	4.25%	9.13%
	Median $RLE^{(s)}$	6.00%	4.11%	8.67%
	Worst $RLE^{(s)}$	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.

Variable	Scenario	Wind	Wind Dir	Wave H.	Wave Dir
Hsig	CH1	7	315	1.6	-20
x-forces	CX1	8	0	1.6	160
y-forces	CY1	7	315	2.6	160

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

RLE

is clipped at the 99th percentile to reduce the influence of extreme values and better highlight the overall spatial error patterns.

Model	1-D Example	DUCK Example
Hsig	$26~m~41~s$	$12~h~15~m~33~s$
x-forces	$27~m~06~s$	$14~h~04~m~34~s$
y-forces	-	$14~h~02~m~58~s$

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^{\circ}

and boundary wave height 1.6 m at

20^{\circ}

. The color bar is clipped between -1 and 1

N/m^{2}

to highlight spatial distribution patterns, illustrating DeepONet’s spatial smoothing relative to the localized fluctuations in SWAN.

Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

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