Estimating bottom topography in shallow water flows

This paper introduces and validates two robust methods, based on Physics-Informed Neural Networks and the Adjoint State Method, for reconstructing bottom topography and surface velocity in shallow water flows using only surface deformation measurements, demonstrating their effectiveness in both 1D and 2D synthetic scenarios even with noisy or sparse data.

The Gist

Imagine you are standing on the deck of a ship in the middle of the ocean. You can see the waves rolling over the surface, but you have no idea what the ocean floor looks like beneath you. Is it flat? Are there hidden mountains? Deep trenches?

For centuries, figuring out the shape of the ocean floor (called bathymetry) has been like trying to guess the shape of a cake by only looking at the frosting. You have to send ships down with sonar (sound waves) to map it, which is slow, expensive, and dangerous.

This paper introduces two new, clever "superpowers" that let us figure out the shape of the ocean floor just by watching the waves on the surface. The authors tested two different mathematical "detectives" to see which one is better at solving this puzzle.

The Two Detectives

The paper compares two methods to solve this mystery:

1. The "Physics-Informed" Neural Network (PINN)
Think of this as a super-smart student who has memorized the laws of physics (how water moves) but hasn't seen the specific ocean floor yet.

• How it works: You show the student a few snapshots of the waves on the surface. The student then tries to guess the shape of the bottom. Every time they guess, they check their work against the laws of physics. "If the bottom looked this way, would the waves look that way?" If the answer is no, they adjust their guess. They keep doing this until the waves they predict match the real waves you showed them.
• The Analogy: It's like trying to guess the shape of a hidden rock in a river by watching how the water flows around it. The student learns by trial and error, guided by the rules of nature.

2. The Adjoint State Method (ASM)
Think of this as a highly precise engineer using a specialized reverse-engineering tool.

• How it works: This method starts with a guess of the ocean floor and runs a simulation forward in time to see what the waves should look like. Then, it runs the simulation backward in time, using the difference between the real waves and the predicted waves to calculate exactly how to tweak the ocean floor guess to make them match.
• The Analogy: Imagine you hear a sound echo in a cave. Instead of guessing the cave shape, you mathematically "rewind" the sound to its source to pinpoint exactly where the walls must be to create that specific echo.

The Experiment: The "Ocean" in a Computer

The authors didn't go to the ocean; they built a virtual ocean inside a computer.

• They created a fake seafloor with bumps and valleys (some big, some tiny).
• They sent a giant wave (like a tsunami) rolling over it.
• They recorded how the water surface moved.
• The Challenge: They then hid the map of the seafloor and gave the two detectives only a few scattered measurements of the water surface (sometimes very few, sometimes with "noise" or static added to the data) and asked them to redraw the map.

What Did They Find?

Both detectives were surprisingly good at the job, but they had different personalities:

• The Student (PINN) is smooth and robust: When the data was messy or very sparse (few measurements), the student tended to draw a smooth, slightly blurry map. It missed some tiny, jagged details, but it didn't get confused or make wild mistakes. It was very good at handling "noisy" data.
• The Engineer (ASM) is sharp and precise: When the data was clear and plentiful, the engineer could see the tiny details of the seafloor that the student missed. However, if the data was too sparse or too noisy, the engineer's map started to get "jittery" and erratic.

The Verdict:

• If you have lots of data, the Engineer (ASM) gives you the sharpest picture.
• If you have very little data or the data is noisy, the Student (PINN) is more reliable and produces a cleaner, more realistic map.

Why Does This Matter?

This is a big deal because:

1. Safety: Knowing the shape of the ocean floor helps predict tsunamis and landslides.
2. Climate: The shape of the bottom affects how ocean currents move, which controls our global climate.
3. Efficiency: If we can use satellites to measure surface waves and then use these math tricks to guess the bottom, we might not need to send as many expensive ships to map the entire ocean.

The Bottom Line

The paper shows that we don't need to touch the ocean floor to know what it looks like. By using the laws of physics and smart computer algorithms, we can "see" the hidden world beneath the waves just by watching the ripples on top. It's like being able to guess the shape of a hidden object just by watching how a shadow moves across a wall.

Technical Summary

1. Problem Statement

The paper addresses the inverse problem of estimating bottom topography ($h_b$) and the surface velocity field ($u$) in shallow water flows using only surface deformation measurements (fluid height $h$ or surface velocity $u$).

• Context: Bathymetry (mapping the seafloor) is crucial for climate modeling, tsunami prediction, and understanding ocean circulation, yet only a small fraction of the ocean floor is charted. Traditional methods (satellite altimetry, acoustic echo-sounders) have limitations regarding resolution, cost, or water depth.
• Challenge: The authors aim to recover the hidden bottom boundary conditions and flow dynamics from sparse and potentially noisy surface data, governed by the Shallow Water (SW) equations.
• Goal: To compare two distinct data assimilation methodologies: Physics-Informed Neural Networks (PINNs) and the Adjoint State Method (ASM).

2. Methodology

The study utilizes synthetic data generated from numerical simulations of the 2D Shallow Water equations (reduced to 1D via symmetry for some tests). The governing equations are:
$$\frac{\partial u}{\partial t} + (u \cdot \nabla)u + g \nabla h = 0$$
$$\frac{\partial h}{\partial t} + \nabla \cdot (u(h - h_b)) = 0$$

Two distinct approaches are implemented to solve the optimization problem: minimizing the difference between observed and predicted surface data while satisfying the physical constraints.

A. Physics-Informed Neural Networks (PINNs)

• Architecture: Two separate fully connected neural networks are used.
  • Network 1: Parametrizes velocity ($u$) and height ($h$) as functions of space ($x, y$) and time ($t$).
  • Network 2: Parametrizes bottom topography ($h_b$) as a function of space only.
• Loss Function: A composite loss function is minimized:
  $$L_{PINN} = L_d + \lambda_p (L_u + \lambda_h L_h)$$
  • $L_d$: Data mismatch term (difference between predicted and observed surface data).
  • $L_u, L_h$: Physics residuals (errors in satisfying the momentum and continuity equations) evaluated at collocation points.
• Training: Uses the Adam optimizer with automatic differentiation to compute derivatives. Hyperparameters ($\lambda_p, \lambda_h$) are dynamically adjusted during training to balance data and physics terms.
• Special Handling: For sparse data, an ensemble of PINNs with varying architectures is used to average out stochastic variations.

B. Adjoint State Method (ASM)

• Formulation: A variational approach where a Lagrangian is constructed combining the data cost function and the SW equations via adjoint variables ($u^\dagger, h^\dagger$).
• Algorithm:
  1. Forward Pass: Solve the SW equations forward in time using an initial guess for $h_b$ and initial conditions.
  2. Backward Pass: Solve the derived adjoint equations backward in time. These equations propagate the error from the surface measurements back to the bottom topography.
  3. Gradient Calculation: Compute gradients of the cost function with respect to $h_b$ and initial conditions using the adjoint fields.
  4. Update: Use an optimization algorithm (L-BFGS) to update the estimates and repeat until convergence.
• Implementation: Solved using a pseudo-spectral scheme with Fourier interpolation to handle forcing terms.

3. Key Contributions

1. Comparative Framework: Provides a rigorous side-by-side comparison of PINNs and ASM for the specific inverse problem of bathymetry estimation in shallow water flows.
2. Derivation of Adjoint System: Derives the specific adjoint system of equations required to solve the shallow water bathymetry inverse problem, including the treatment of unknown initial conditions.
3. Robustness Analysis: Systematically tests both methods against:
   • Data Sparsity: Varying the distance between measurement points ($\delta x$).
   • Data Corruption: Adding Gaussian noise to surface measurements.
   • Input Types: Testing reconstruction using surface height ($h$) vs. surface velocity ($u$) measurements.
4. 2D Extension: Successfully extends the PINN approach to 2D flows, demonstrating the ability to recover complex 2D topography and velocity fields.

4. Results

Performance in 1D Cases

• Accuracy: Both methods successfully reconstruct the bottom topography and velocity fields.
• Sparsity ($\delta x$):
  • Dense Data: PINNs slightly outperform ASM in topography reconstruction but produce smoother velocity fields.
  • Sparse Data: ASM outperforms PINNs in capturing fine-scale details (high-frequency structures) of the topography. PINNs exhibit "spectral bias," struggling with high-frequency components and producing smoother, less jagged results.
• Noise Robustness:
  • Both methods remain robust up to noise levels of $\approx 10\%$ of the wave amplitude.
  • PINNs show slightly better topography reconstruction under noise, while ASM maintains better velocity reconstruction accuracy.
• Velocity Artifacts: PINNs occasionally produce non-physical oscillations (artifacts) in the velocity field near steep gradients, likely due to insufficient training or sampling. ASM produces smoother velocity fields in sparse cases but can show larger fluctuations in noisy scenarios.
• Velocity Input: When using velocity measurements instead of height, both methods remain accurate, though ASM required fixing initial conditions to remain stable.

Performance in 2D Cases

• PINN Success: The PINN successfully reconstructed 2D bottom topography and velocity fields ($u, v$) from surface height data.
• Resolution: The method could resolve structures approximately half the size of the incoming wavelength (mapped to 4–6 km in physical ocean scales).
• Error Distribution: Errors were highest at the peaks of the topography structures but remained within acceptable limits.

Computational Cost

• ASM: Faster (2–4 hours) once implemented, as it relies on standard numerical solvers.
• PINN: Slower (10–24 hours for 1D, ~3 days for 2D) due to the iterative training process, but significantly easier to implement using modern deep learning libraries (e.g., TensorFlow/PyTorch) without custom adjoint code.

5. Significance and Conclusion

• Feasibility: The study demonstrates that surface measurements alone are sufficient to infer complex seafloor topography and flow dynamics in shallow water regimes, a finding relevant to tsunami modeling and coastal oceanography.
• Method Selection:
  • ASM is preferred when high-resolution recovery of fine-scale features is required and computational efficiency is a priority, provided the user can implement the complex adjoint derivation.
  • PINNs are preferred for their ease of implementation, flexibility with different data types, and robustness in producing smooth solutions when data is very sparse, despite the higher computational cost and spectral bias.
• Future Work: The authors plan to apply these methods to real-world experimental data and extend the models to include breaking waves, dispersive effects, and wind interactions, which are currently outside the scope of the standard Shallow Water approximation.

Code Availability: The authors have made all code, including ground truth generation, PINN/ASM implementations, and plotting scripts, available on Zenodo.