Reduced-Order Hydrodynamic Modelling of a Sphere Near a Wall Using Sparse Regression and Neural Operators

This paper presents a real-time, interpretable surrogate model for predicting the heave dynamics of a sphere near a wall by combining Sparse Identification of Nonlinear Dynamics (SINDy) to extract governing ODEs from CFD data with a neural operator network that maps geometric parameters to these dynamics.

The Gist

Imagine you are trying to predict how a small toy boat (a drone boat) bounces up and down in the water right next to a giant ship. This is a critical moment called "launch and recovery." If the boat bounces too wildly, it might crash into the big ship or flip over.

To predict this safely, scientists usually use CFD (Computational Fluid Dynamics). Think of CFD as a super-detailed, high-definition video game simulation of water. It's incredibly accurate, but it's also slow and expensive. Running one simulation can take hours of computer time. If you need to know what happens right now to save the boat, waiting hours for a computer to finish its math is too late.

This paper presents a clever shortcut: a "Smart Cheat Sheet" that is as fast as a video game but as smart as the slow simulation.

Here is how they built it, using simple analogies:

1. The Problem: The "Slow Motion" vs. "Real Time" Dilemma

• The Old Way (CFD): Imagine trying to predict the weather by simulating every single raindrop and wind gust in a computer. It's perfect, but it takes days to calculate tomorrow's forecast.
• The Goal: We need a weather app that tells us the forecast instantly, even if it's an approximation.

2. Step One: The "Detective" (SINDy)

First, the researchers ran the slow, expensive simulations for many different scenarios (dropping the boat from different heights, at different distances from the wall).

Then, they used a tool called SINDy (Sparse Identification of Nonlinear Dynamics).

• The Analogy: Imagine you have a messy room full of toys, clothes, and books (the complex water data). You want to know the simple rules that explain how the room got messy.
• What SINDy does: It acts like a detective who looks at the mess and says, "Okay, 90% of this mess is caused by the cat knocking things over (gravity), 9% is the dog (damping), and 1% is the wind (waves)." It ignores the tiny, unimportant details and finds the simplest mathematical equation that explains the movement.
• The Result: They found that the boat's bounce could be described by a simple formula involving a "spring" (buoyancy), a "shock absorber" (water resistance), and a "push" (waves).

3. The Twist: The Rules Change

Here is the tricky part. The "spring" and "shock absorber" aren't constant.

• If you drop the boat from high up, the water pushes back harder (non-linear).
• If the boat is very close to the wall, the water bounces off the wall and hits the boat differently.

So, the "detective" (SINDy) gave them a different equation for every single scenario. But we can't memorize a million different equations. We need one master rulebook.

4. Step Two: The "Translator" (Neural Operator)

This is where the Neural Operator (ONet) comes in.

• The Analogy: Imagine you have a dictionary that translates "Distance from Wall" and "Drop Height" into "The Right Equation."
• How it works: The researchers taught a computer (a Neural Network) to look at the input (e.g., "Drop from 5 meters, 10 meters from wall") and instantly output the correct coefficients (the numbers for the spring and shock absorber) that the "Detective" found earlier.
• The Magic: The computer didn't just memorize the answers; it learned the shape of the relationship. It created a smooth, continuous map. Now, if you ask for a scenario the computer has never seen before (e.g., "Drop from 4.2 meters"), it can guess the right equation instantly.

5. The Final Product: The "Real-Time Pilot"

The result is a Surrogate Model.

• Speed: It runs in milliseconds (real-time).
• Accuracy: It is almost as accurate as the slow, expensive simulation.
• Interpretability: Unlike a "black box" AI that just gives a number, this model tells you why. It says, "The boat will bounce because the water is acting like a stiff spring right now."

Why Does This Matter?

In the real world, when a Navy drone boat is being lowered into rough seas next to a massive ship, the operators need to know immediately if the boat will crash.

• Before: They might have to guess or wait for a slow computer to run a simulation.
• Now: They can use this "Smart Cheat Sheet" to predict the boat's path instantly, allowing them to adjust the crane or the boat's thrusters in real-time to keep it safe.

In a nutshell: They took a slow, perfect physics simulation, used a detective to find the simple rules inside it, and then taught a computer to instantly write those rules for any situation. It's like turning a 10-hour movie into a 1-second summary that still tells the whole story.

Technical Summary

1. Problem Statement

The paper addresses the challenge of predicting the hydrodynamic trajectory of small uncrewed surface vessels (USVs) during launch-and-recovery operations (LARS) from larger parent vessels.

• The Challenge: In close proximity to a large vessel (modeled as a vertical wall) and in irregular/steep seas, linear potential-flow theories fail to capture critical non-linear effects (e.g., non-linear hydrostatic restoring, radiation damping, and wave diffraction).
• The Bottleneck: While high-fidelity Computational Fluid Dynamics (CFD) can model these non-linearities, it is computationally prohibitive for real-time applications (requiring $O(10^2 - 10^3)$ CPU core-hours per simulation).
• The Goal: Develop a physics-informed, interpretable, reduced-order surrogate model capable of real-time prediction of heave motion for arbitrary wall distances ($WD$) and drop heights ($DH$) without re-running CFD.

2. Methodology

The authors propose a hybrid framework combining Sparse Identification of Nonlinear Dynamics (SINDy) with Neural Operators (ONet). The workflow consists of four main stages:

A. High-Fidelity Data Generation

• Setup: A 1-DoF heaving sphere near a vertical wall is simulated using Ansys Fluent 2025R1.
• Physics: The solver uses the Navier-Stokes equations with a Volume of Fluid (VOF) method for the air-water interface and the $k-\omega$ SST turbulence model.
• Dataset: A parametric dataset of heave-decay responses was generated across varying wall distances ($WD \in [9, 25]$ m) and drop heights ($DH \in [0.5, 5]$ m).

B. Local Sparse Regression (SINDy)

Instead of using a black-box neural network to predict trajectories directly, the authors first identify the governing Ordinary Differential Equations (ODEs) for specific $(WD, DH)$ points.

• Candidate Library: A library of basis functions is constructed, including polynomial terms ($x, x^2, x^3, \dot{x}, \dot{x}^2, \dot{x}^3$) for hydrostatic and radiation forces, and harmonic terms ($\sin(\omega t), \cos(\omega t)$) for wave excitation.
• Sparse Identification: SINDy (using Sequential Thresholded Least Squares) is applied to CFD time-series data to select the sparsest set of coefficients ($\Xi$) that accurately reconstruct the dynamics.
• Key Finding: The optimal local model requires a 3rd-order polynomial (capturing non-linear hydrostatics) and 1st-order harmonics (capturing wave excitation).

C. Global Neural Operator (ONet)

Since SINDy only provides coefficients at discrete parameter points, a Neural Operator is trained to map the input space $(WD, DH)$ to the ODE coefficient vector $\Xi$.

• Architecture: A fully connected feed-forward network (2 inputs $\to$ 32 hidden units $\to$ 8 outputs) with $\tanh$ activation.
• Physics-Informed Constraints:
  • The network outputs are normalized and rescaled using the min/max bounds derived from the SINDy results. This acts as a prior, ensuring the learned coefficients remain within physically plausible ranges.
  • Loss Function: The network is not trained to minimize the error between predicted and SINDy coefficients directly. Instead, the loss is the Mean Squared Error (MSE) between the numerically integrated ODE trajectory and the raw CFD trajectory. This acknowledges that multiple coefficient sets can produce similar dynamics in non-linear systems.

D. Training and Integration

• The ODE is integrated using a 4th-order Runge-Kutta (RK4) scheme within the training loop.
• The model is trained to minimize the trajectory error over the entire time history, effectively learning the smooth manifold of hydrodynamic coefficients across the parameter space.

3. Key Contributions

1. Hybrid Physics-Data Framework: The paper introduces a novel pipeline where sparse regression extracts interpretable physics (ODE structure) from high-fidelity data, and a neural operator generalizes these physics across the parameter space.
2. Interpretability: Unlike standard "black-box" deep learning models, this approach yields a surrogate model where coefficients explicitly correspond to physical quantities (added mass, damping, stiffness, excitation).
3. Real-Time Capability: The resulting surrogate can predict complex non-linear heave dynamics in real-time, bypassing the need for expensive CFD runs.
4. Insight into Excitation Forces: The study reveals that wave excitation forces in close proximity to a wall are non-linearly coupled to body motion and radiation fields, failing simple linear superposition assumptions.

4. Results

• Accuracy:
  • The SINDy-identified local models achieve an MSE of $\approx 5.1 \times 10^{-4}$.
  • The global Neural Operator achieves a test MSE of $\approx 5.76 \times 10^{-4}$, which is remarkably close to the theoretical lower bound set by the SINDy coefficients.
  • The surrogate reproduces CFD heave-decay envelopes with near-optimal accuracy.
• Coefficient Manifolds:
  • The Neural Operator successfully learns a smooth mapping from $(WD, DH)$ to ODE coefficients.
  • Interestingly, the coefficients learned by the NN deviate from the direct SINDy linear-algebra results but produce identical dynamic trajectories. This suggests that for non-linear systems, there is no single "true" coefficient set; the NN finds an alternative representation that minimizes trajectory error.
• Physical Validation:
  • The model correctly captures the cubic hardening behavior of the hydrostatic restoring force for large drop heights.
  • The phase of the excitation force aligns with analytical wave travel-time predictions ($t_{delay} = 2WD/c$), though the magnitude remains complex and non-linear.

5. Significance and Future Work

• Operational Impact: This methodology provides a practical pathway for real-time decision support in naval launch-and-recovery operations, allowing operators to predict USV trajectories under varying sea states and proximity conditions instantly.
• Scientific Contribution: It demonstrates that combining sparse regression (for structure discovery) with neural operators (for generalization) creates a robust, physics-constrained surrogate that outperforms purely data-driven models in terms of physical interpretability and extrapolation stability.
• Limitations & Future Directions:
  • Current work is limited to a 1-DoF heaving sphere and regular wave decay.
  • Future work aims to extend this to 6-DoF dynamics, complex hull forms, and irregular sea states.
  • The excitation force model requires refinement, as current harmonic terms are a crude approximation of complex diffraction effects.
  • The framework is designed to be compatible with experimental data, not just CFD, for future validation.