Enhancing the Boundary Element Method with Scientific Machine Learning

A Hybrid Approach for Solving Diffraction-Radiation Problems

research
machine learning
hydrodynamics
Julia
Author

Kapil Khanal

Published

July 30, 2024

Introduction

The Boundary Element Method (BEM) is a cornerstone of computational hydrodynamics, widely used to solve the three-dimensional diffraction-radiation problem for floating bodies. BEM involves transforming a partial differential equation (PDE) into a boundary integral equation using Green’s theorem, which is then solved numerically. A key challenge in BEM is the numerical evaluation of the Green’s function, particularly its wave-related terms. These computations typically scale poorly, often at \(O(N^3)\) with the number of panels (\(N\)), creating a significant bottleneck. This is especially problematic in large-scale optimization tasks for offshore structures, which involve many panels and design parameters.

This creates a classic trade-off: accuracy vs. efficiency. We propose adding a third dimension to this trade-off: the ease of numerical differentiation. While surrogate models and semi-analytical methods exist, we introduce a hybrid approach that leverages a physics-informed neural network (PINN) to enhance the traditional BEM program.

Neural Networks in Scientific Machine Learning

The classical expression of the free surface Greens function.\(G(P,Q) = \text{fundamental solution} + \textit{harmonic function}\) : \[\begin{equation} G(P,Q) = \left(\frac{1}{X^2+ (Z_1-Z_2)^2}\right) + \left( \frac{1}{X^2+ Y^2} + F(X,Y) + 2 \pi ik_0 e^{-Y} J_0(X)\right) \end{equation}\] where horizontal (X) and vertical distances (Y), \ \(X:k_0 *\sqrt{x1-x2)^2 + (y1-y2)^2}\) and \(Y:-k_o(z1+z2)\) Newman simplified \(F(X,Y)\) from the original expression (not included here) which has many computational challenges to: \[\begin{equation} F(X, Y) = -2e^{-y} \int_{0}^{Y} \frac{e^t}{\sqrt{X^2 + t^2}} \, dt - \pi e^{-y} \left[B(X) + H(X)\right] \end{equation}\]

Q: Can we instead use a neural network to approximate the Green’s function?

Neural networks have a growing history in scientific machine learning, with recent publications demonstrating their power in hydrodynamics . However, a well-known challenge is that standard neural networks often struggle to learn periodic and oscillatory functions, which are fundamental to wave physics.

Several research avenues have explored solutions to this, including: - Transforming data into the Fourier space. - Developing novel activation functions. - Using deeper network architectures.

We’re exploring the use of neural networks to approximate the Green’s function and integrate it into our in-house BEM solver - MarineHydro.jl.

This is a work in progress.

References

  • John, F. (1950). On the motion of floating bodies II. Communications on Pure and Applied Mathematics.
  • Xie, M., et al. (2018). A review of Green’s functions for marine hydrodynamics. Journal of Hydrodynamics.
  • Huang, G., et al. (2022). Deep-learning-based fast-prediction method for hydrodynamic performance of a ship sailing in waves. Ocean Engineering.
  • Zhan, S., et al. (2023). A review of deep learning in naval architecture and ocean engineering. Ocean Engineering.
  • Boulle, N., et al. (2020). Rational neural networks. arXiv preprint.