Reduced-Order Surrogates for Forced Flexible Mesh Coastal-Ocean Models

This paper introduces a flexible Koopman autoencoder surrogate model that incorporates meteorological forcings and boundary conditions to achieve stable, accurate, and computationally efficient long-term predictions for forced flexible mesh coastal-ocean models, outperforming traditional POD-based approaches in several cases while enabling practical applications like ensemble forecasting.

The Gist

Imagine you are trying to predict the weather for a coastal city next year. You have a super-accurate, high-tech weather simulator (let's call it the "Master Simulator"). It's incredibly detailed, calculating every tiny ripple in the water and every gust of wind. But there's a catch: it's so heavy and complex that running it for just one day takes hours, and running it for a whole year would take months. It's like trying to drive a Formula 1 car to the grocery store; it works, but it's wildly inefficient for the job.

This paper is about building a lightweight, super-fast "smart assistant" that can mimic the Master Simulator's predictions almost perfectly, but in a fraction of the time.

Here is the breakdown of how they did it, using simple analogies:

1. The Problem: The "Heavy" Simulator

The existing models (like MIKE 21) are like a high-resolution 4K movie. They show every single detail of the ocean. But to play that movie, you need a massive supercomputer. If you want to run 100 different scenarios to see what happens if a storm hits (an "ensemble forecast"), or simulate 100 years of climate change, the supercomputer gets tired and takes forever.

2. The Solution: The "Smart Assistant" (Surrogates)

The authors built two types of "smart assistants" (called Surrogates) to replace the heavy simulator. Think of these assistants as students who have studied the Master Simulator's movies so well that they can predict the next scene without needing the supercomputer.

They tested two different ways of teaching these students:

• Method A: The "POD" Student (The Pattern Spotter)

  • How it works: This student looks at thousands of hours of ocean data and says, "I see a pattern! The water usually moves in these 50 main ways." It compresses the complex ocean into a simple list of 50 "moves" (like a dance routine).
  • Pros: It learns very quickly.
  • Cons: It's a bit rigid. If the ocean does something weird that doesn't fit the "dance routine," it might get confused.

• Method B: The "Koopman" Student (The Linearizer)

  • How it works: This student is a bit more magical. It tries to find a special "secret language" (a latent space) where the chaotic, messy ocean movements look like simple, straight lines. Imagine taking a tangled ball of yarn and magically straightening it out so you can predict exactly where the end will go.
  • Pros: It's better at handling long-term predictions and strange weather patterns.
  • Cons: It takes longer to study (train).

3. The Secret Sauce: "Temporal Unrolling"

Both students had a problem: if you ask them to predict 100 days in the future, they would start making small mistakes, and those mistakes would pile up like a snowball rolling down a hill, eventually becoming a disaster.

To fix this, the authors used a technique called Temporal Unrolling.

• The Analogy: Imagine teaching a student to walk. Instead of just saying, "Take one step," you say, "Take 10 steps in a row, and if you stumble at step 3, we fix your whole balance for steps 1 through 10."
• The Result: By forcing the students to practice long chains of predictions during training, they learned to keep their balance for the long haul. This made them incredibly stable, even when predicting a whole year ahead.

4. The Results: Speed vs. Accuracy

The team tested these assistants on three real-world ocean locations:

1. Øresund (Denmark/Sweden): A narrow strait.
2. Southern North Sea: A large, open shelf with big tides.
3. Adriatic Sea: A complex basin known for sudden flooding (like Venice).

The Verdict:

• Speed: The assistants were 300 to 1,400 times faster than the Master Simulator.
  • Analogy: If the Master Simulator takes 10 hours to simulate a year, the assistant does it in 30 seconds.
• Accuracy: They were surprisingly accurate.
  • For water levels, the error was often just a few centimeters (less than an inch).
  • In some cases, the "Koopman" student was slightly better than the "POD" student, especially for long-term predictions.
  • Even the "worst" case (Adriatic Sea) was only off by about 12% compared to the heavy simulator, which is acceptable for many real-world planning tasks.

5. Why This Matters

This isn't just about saving time; it's about saving lives and money.

• Emergency Planning: If a hurricane is coming, you can run 500 different scenarios in minutes to see which one is most likely, rather than waiting days.
• Climate Change: You can simulate 100 years of rising sea levels on a standard laptop instead of needing a supercomputer.
• Infrastructure: Engineers can design sea walls and bridges with much better data on how the ocean will behave over decades.

Summary

The authors took a heavy, slow, super-accurate ocean model and built a "lightweight" version that learns the ocean's patterns. By teaching this lightweight model to practice long-term predictions, they created a tool that is fast enough to run on a laptop but accurate enough to trust for coastal safety. It's like turning a massive, slow-moving cruise ship into a nimble speedboat that still knows exactly where the reefs are.

Technical Summary

1. Problem Statement

Physics-based hydrodynamic models (e.g., MIKE 21, ADCIRC) are essential for coastal management, flood forecasting, and climate impact studies. However, these models solve the shallow-water equations on high-resolution unstructured meshes with small time steps, making them computationally expensive. This limits their applicability for:

• Long-term climate simulations (years to decades).
• Ensemble forecasting (running hundreds of scenarios for uncertainty quantification).
• Real-time emergency response requiring rapid iteration.

While Machine Learning (ML) surrogates have shown promise, existing methods often struggle with:

• Long-horizon stability: Errors compound over time, leading to drift.
• External forcing: Most Koopman-based approaches assume autonomous systems, whereas coastal dynamics are driven by time-varying meteorological forcings (wind, pressure) and boundary conditions.
• Generalizability: Performance often degrades when applied to complex, real-world domains with distinct dynamical regimes.

2. Methodology

The authors propose and evaluate Reduced-Order Surrogates (ROS) that approximate the flowmap of the physics-based model. The core methodology involves two main approaches compared systematically:

A. Model Architectures

1. Koopman Autoencoder (KAE):

   • Concept: Learns a transformation (encoder) from the physical state space to a latent space where the dynamics are linear (governed by a Koopman operator).
   • Forced Systems: Explicitly incorporates external forcings (wind, pressure, boundary conditions) into the latent dynamics. The encoder processes both state ($x$) and forcing ($u$) at time $k$ and $k+1$.
   • Linear Propagator: Uses a learned linear matrix $K_f$ in the latent space to propagate the state forward.
   • Stability: Allows for eigenvalue regularization to ensure the linear operator is asymptotically stable (eigenvalues within the unit circle).

2. Proper Orthogonal Decomposition (POD) Surrogates:

   • Concept: Uses POD (PCA) to extract dominant spatial modes, reducing the dimensionality of the state.
   • Temporal Propagation: The reduced latent states are propagated using various regressors:
     • Linear Regression (LR)
     • Multi-Layer Perceptron (MLP)
     • Gated Recurrent Unit (GRU)
   • Training: Typically involves a two-step process (reconstruct basis, then train regressor), though the authors also explore end-to-end training for linear regressors.

B. Training Strategies for Stability

To address error accumulation in long-term predictions, the authors implement two specific strategies:

1. Eigenvalue Regularization: A loss term is added to penalize any eigenvalues of the system matrix that exceed magnitude 1. This enforces stability in the latent dynamics, preventing exponential growth of errors.
2. Temporal Unrolling: Instead of training solely on one-step predictions, the model is trained by unrolling the temporal propagator over multiple steps ($N_{TU}$). The loss is computed over the entire unrolled trajectory, forcing the model to learn to minimize error accumulation over time.

C. Variable-Separated Architecture

Recognizing that surface elevation ($S$) and current velocities ($U, V$) have different smoothness properties, the authors experiment with separated autoencoders. This allows distinct encoding strategies for different variables (e.g., using fewer modes for smooth $S$ and more for turbulent $U/V$), improving reconstruction efficiency.

3. Experimental Setup

The models were evaluated on three distinct real-world coastal domains using data generated by the MIKE 21 Flow Model FM:

1. Øresund (Denmark/Sweden): Small domain, low tidal range, dominated by wind/pressure surges.
2. Southern North Sea: Large shelf sea, strong semi-diurnal tides, complex bathymetry.
3. Adriatic Sea: Semi-enclosed basin, strong wind-driven circulation (Bora/Sirocco), resonant seiching (flooding in Venice). Note: Tidal forcing was removed here to isolate meteorological dynamics.

Metrics:

• Accuracy: $R^2$, Root Mean Squared Error (RMSE), and Relative RMSE (normalized by local variability).
• Validation: Comparison against both the physics-based model (MIKE 21) and in-situ observational data.
• Speed: Inference time comparison.

4. Key Results

Accuracy and Performance

• Overall Skill: All reduced-order surrogates with temporal unrolling achieved high accuracy. Relative RMSEs ranged from 0.0068 to 0.14, with $R^2$ values between 0.61 and 0.995.
• Variable Performance: Surface elevation predictions were highly accurate ($R^2 > 0.95$), while current velocities were slightly less accurate ($R^2$ as low as 0.61 in the complex Adriatic case).
• KAE vs. POD:
  • The Koopman Autoencoder (KAE) generally outperformed POD-based surrogates in the Øresund and Southern North Sea cases, particularly in the upper percentiles of error distribution.
  • In the complex Adriatic case, a POD-based model with temporal unrolling (PODLRt) performed slightly better than KAE, likely due to the difficulty of learning resonant dynamics with limited training data.
• Impact of Strategies:
  • Temporal Unrolling: Crucial for stability. It improved accuracy by an order of magnitude in some cases (e.g., reducing relative RMSE from 0.13 to 0.016 for Southern North Sea surface elevation).
  • Eigenvalue Regularization: Significantly improved KAE stability, reducing surface elevation RMSE from 0.307 to 0.07 in the Adriatic case.

Comparison with Observations

When validated against real-world measurements:

• The surrogates increased the prediction error (RMSE) by -0.64% to 12% compared to the physics-based MIKE 21 model.
• In the Øresund case, the surrogate actually improved accuracy (-0.64% error increase), suggesting it may filter out numerical noise present in the physics model.
• The maximum error increase (12%) corresponded to a few centimeters, which is acceptable for many operational applications.

Computational Efficiency

• Speed-up: The surrogates achieved a 300x to 1400x speed-up compared to the physics-based model.
  • Example: A 1-year simulation took ~12 seconds on a laptop CPU for the surrogate, compared to ~16,800 seconds (4.6 hours) on a GPU for MIKE 21 in the Adriatic case.
• Training vs. Inference: POD models have very low offline training costs. KAEs require more expensive iterative training but offer superior long-term stability and accuracy.

5. Key Contributions

1. Forced System Formulation: Successfully extended Koopman autoencoders to handle time-varying external forcings (wind, pressure, boundaries), a gap in previous hydrodynamic surrogate literature.
2. Systematic Comparison: Provided a rigorous benchmark between KAE and POD-based surrogates across three distinct dynamical regimes.
3. Stability Mechanisms: Demonstrated that combining eigenvalue regularization and temporal unrolling is essential for achieving stable, long-horizon predictions (up to one year) in coastal ocean models.
4. Operational Viability: Proved that reduced-order surrogates can maintain near-physics-model accuracy while enabling workflows (ensemble forecasting, climate simulations) that are currently computationally infeasible.

6. Significance and Conclusion

This work bridges the gap between theoretical Koopman operator learning and practical coastal engineering. The findings indicate that Koopman Autoencoders with temporal unrolling are a robust alternative to traditional POD methods, offering better stability and slightly higher accuracy in complex scenarios.

The ability to run 1-year simulations in seconds with centimeter-level accuracy relative to observations suggests these surrogates are ready for deployment in:

• Ensemble forecasting for storm surge prediction.
• Long-term climate change impact assessments on coastal infrastructure.
• Real-time decision support systems for flood management.

The study concludes that while some accuracy is traded for speed, the error margins are negligible for most practical applications, making these reduced-order models a powerful tool for the future of operational oceanography.