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

This paper proposes using Deep Operator Networks (DeepONets) as a computationally efficient surrogate for the SWAN numerical wave model to accurately predict wave-induced forces and significant wave heights, thereby enabling more precise and efficient coupling with circulation models for storm surge prediction.

The Gist

Imagine you are trying to predict how a massive ocean wave will crash against a coastline, how high the water will rise, and how hard it will push against a seawall or an oil rig.

To do this accurately, scientists usually use super-complex computer programs (like SWAN) that act like a digital ocean. They simulate every tiny ripple, the wind blowing, the shape of the ocean floor, and how waves crash into each other. The problem? These simulations are like trying to solve a million-piece puzzle while running a marathon. They are incredibly accurate, but they take a long time and a lot of computer power. If you need to run thousands of simulations to predict a hurricane's path, waiting hours for each one is impossible.

This paper introduces a "Magic Shortcut" called DeepONet.

Here is the simple breakdown of what they did, using some everyday analogies:

1. The Problem: The Slow Chef vs. The Fast Sous-Chef

Think of the traditional wave model (SWAN) as a Master Chef who cooks a perfect, gourmet meal from scratch. They chop every vegetable, measure every spice, and simmer the sauce for hours. The result is delicious (accurate), but it takes forever.

The researchers wanted a Sous-Chef (the DeepONet) who could taste the ingredients (wind speed, wave height, ocean depth) and instantly guess what the final dish would taste like, without actually cooking it from scratch.

2. The Solution: Learning the "Recipe," Not the "Ingredients"

Most AI models are like students who memorize specific answers. If you ask them about a wave in a specific spot, they know the answer. But if you move the wave slightly, they get confused.

DeepONet is different. Instead of memorizing specific answers, it learns the underlying physics (the recipe).

• The Branch Network: This part of the AI looks at the "ingredients" (the wind, the starting wave size, the direction).
• The Trunk Network: This part looks at the "location" (where on the map are we looking?).
• The Magic: By combining these two, the AI learns the relationship between the ingredients and the result. It learns that "If the wind blows this hard from this direction, the wave will grow this much at that specific spot."

Once it learns this "recipe," it can predict the outcome for any location, even ones it has never seen before, in a split second.

3. The Test Drive: From a Straight Hallway to a Real City

The researchers tested this "Sous-Chef" in three different scenarios to see if it could handle the heat:

• Level 1 (The Straight Hallway): They started with a simple, straight ocean with a flat slope. The AI was perfect, almost indistinguishable from the slow Master Chef.
• Level 2 (The Wide Room): They made it a 2D room with waves coming from different angles. The AI still did a great job, though it got slightly confused by the complex interactions (like waves hitting each other from the side).
• Level 3 (The Real City - Duck, NC): This was the big test. They used a real map of the coast in North Carolina, with weird bumps, sandbars, and real-world wind patterns.
  • The Result: The AI was incredibly accurate at predicting wave heights.
  • The Quirk: When it came to predicting the exact "push" (force) of the water, the AI was a bit too smooth. The real computer model showed jagged, sharp spikes in force where waves crashed over sandbars. The AI smoothed those out, like a photo filter blurring a sharp edge.
  • Why this is actually good: Those sharp spikes in the real model are often just computer "noise" (glitches). The AI's smooth version might actually be more useful for engineers because it gives a stable, reliable force without the computer glitches.

4. Why Should You Care?

This isn't just about math; it's about speed and safety.

• Speed: The traditional model takes about 30 seconds to run one simulation. The new AI model takes about 0.04 seconds. That is a 750x speedup.
• Real-World Impact: Imagine a hurricane is approaching. Meteorologists need to run thousands of simulations to see where the storm surge will hit.
  • Before: They might run 10 simulations and hope for the best because they don't have time for more.
  • With DeepONet: They can run 10,000 simulations in the time it used to take to run 10. This means they can give much more accurate warnings, saving lives and property.

The Bottom Line

The researchers built a "super-fast wave predictor" that learns the rules of the ocean rather than just memorizing data. It's not perfect at capturing every tiny, jagged detail, but it is fast, accurate enough for real-world safety, and it can run on a laptop instead of a supercomputer. It's like upgrading from a hand-drawn map to a GPS that updates in real-time.

Technical Summary

1. Problem Statement

Accurate prediction of storm surges and coastal hazards requires coupling circulation models (e.g., ADCIRC) with spectral wave models (e.g., SWAN). The interaction is mediated by radiation stress gradients, which represent the transfer of momentum from surface waves to currents.

• The Bottleneck: Traditional spectral wave models like SWAN solve the high-dimensional Wave Action Balance Equation (WABE). This process is computationally expensive, often requiring significantly coarser temporal resolution in coupled simulations than the circulation model, thereby limiting the accuracy and efficiency of real-time forecasting and ensemble modeling.
• The Gap: Existing data-driven surrogates (e.g., CNNs, LSTMs) often struggle with complex, unstructured coastal geometries, are tied to specific grid discretizations, or fail to generalize across varying physical conditions.

2. Methodology

The authors propose using Deep Operator Networks (DeepONets) as a surrogate for the SWAN model. Unlike standard neural networks that map finite-dimensional vectors, DeepONets approximate mappings between infinite-dimensional function spaces, making them inherently discretization-invariant.

A. Operator Formulation

The study frames the steady-state SWAN simulation as a nonlinear operator $\mathcal{G}_S$ that maps:

• Inputs (Branch Network): Boundary wave conditions (wave height, direction) and domain-wide wind forcing (speed, direction).
• Outputs (Trunk Network): Spatially distributed wave quantities, specifically Significant Wave Height ($H_{sig}$) and Radiation Stress Gradients (Forces).
• Goal: Learn the mapping $\hat{\mathcal{G}}: (\text{Wind}, \text{Boundary Waves}) \to (H_{sig}, \nabla \cdot \text{Radiation Stress})$.

B. Network Architecture

• Branch Network: Encodes the input functions (scenario parameters) evaluated at specific sampling points.
• Trunk Network: Encodes the spatial coordinates $(x, y)$ where the output is queried.
• Combination: The outputs are combined via an inner product to predict the field value at any coordinate, allowing the model to predict values at locations not present in the training mesh.

C. Experimental Design

The model was trained and tested on three distinct numerical examples using data generated by SWAN:

1. 1-D Uniform Slope: A simplified domain to test basic wave propagation and transformation.
2. 2-D Planar Slope: An extension to 2-D to test lateral refraction and multidirectional interactions.
3. DUCK (Realistic Coastal): Based on the ONR Test Bed F71 case at the Field Research Facility (Duck, NC). This features complex, real-world bathymetry, irregular boundaries, and realistic wind/wave conditions.

Data Generation:

• Thousands of parametric scenarios were generated by varying wind speed/direction and initial wave height/direction.
• Training/Validation/Testing splits were approximately 70/15/15.
• Hyperparameters were optimized using Optuna on the 1-D case and transferred to the 2-D and DUCK cases.

3. Key Contributions

1. First Application to Radiation Stress: This work extends DeepONet applications beyond predicting wave heights to predicting radiation stress gradients, a critical forcing term for coupled hydrodynamic models.
2. Discretization Invariance: Demonstrates that the surrogate can predict wave fields at arbitrary spatial coordinates, independent of the training mesh resolution, a crucial feature for integration into flexible coastal models.
3. Real-World Validation: Successfully validated the approach on a realistic, complex coastal domain (DUCK) with irregular bathymetry, moving beyond idealized academic examples.
4. Computational Efficiency: Achieved a massive speedup (approx. 3 orders of magnitude) compared to the native SWAN solver.

4. Results

The surrogate model demonstrated high accuracy across all three test cases:

• 1-D Example:
  • $H_{sig}$: Relative $L_2$ errors (RLE) clustered between 0.1% and 0.4%.
  • Forces: RLE generally below 5%, with the model capturing general trends despite slight smoothing of sharp gradients.
• 2-D Example:
  • $H_{sig}$: RLE between 0.2% and 0.6%.
  • Forces: RLE typically between 1.5% and 4%, with a maximum of ~8.8% in worst-case scenarios involving complex directional interactions.
• DUCK (Realistic) Example:
  • $H_{sig}$: RLE remained below 1.0% for the majority of scenarios.
  • Forces: RLE ranged between 2% and 5%, with a maximum of 10.98% for x-forces.
  • Error Localization: Errors were concentrated in the nearshore zone and offshore boundaries where bathymetry is steep and wave breaking occurs. The model tended to "smooth" high-frequency numerical noise present in the SWAN ground truth, which may actually be beneficial for stability in coupled circulation models.

Performance Metrics:

• Speed: The DeepONet surrogate predicts a scenario in ~0.04 seconds, compared to ~30 seconds for a full SWAN simulation.
• Robustness: The model maintained consistent accuracy across unseen wind and wave conditions, demonstrating strong generalization capabilities.

5. Significance and Implications

• Accelerated Coupled Modeling: By replacing the expensive SWAN component with a near-instantaneous DeepONet surrogate, the paper enables high-resolution, real-time coupled wave-current simulations. This is vital for operational storm surge forecasting and ensemble-based hazard assessment.
• Physical Consistency: Despite being a data-driven model, the surrogate successfully learned the underlying physics of shoaling, refraction, and breaking, as evidenced by its ability to reproduce spatial patterns in complex environments.
• Stabilization of Forcing Terms: The model's tendency to smooth high-frequency numerical artifacts found in traditional spectral models could provide more stable forcing terms for circulation models (like ADCIRC), potentially reducing numerical instability in coupled systems.
• Future Directions: The authors note that while the current model predicts bulk properties ($H_{sig}$ and forces), future work aims to extend this to predict the full wave action density spectrum and handle time-varying, non-uniform wind fields.

In conclusion, this study establishes DeepONets as a viable, highly efficient, and accurate surrogate for spectral wave models, offering a transformative approach to overcoming the computational bottlenecks in coastal engineering and ocean forecasting.