Machine Learning Based Mesh Movement for Non-Hydrostatic Tsunami Simulation

TL;DR

Numerical results indicate that the UM2N surrogate based approach significantly accelerates conventional mesh movement techniques and has high robustness over long integration periods and under strongly nonlinear wave conditions.

Abstract

This study investigates the use of machine learning based mesh adaptivity, specifically mesh movement methods (UM2N), with depth integrated non-hydrostatic shallow water models. Motivation for this comes from the need for models which balance efficiency and accuracy for use in probabilistic coastal hazard assessment. Implementations are built on the discontinuous Galerkin finite-element (DG-FE) based software, Thetis, which leverages the partial differential equation (PDE) framework Firedrake for automated code generation. Verification on benchmark test cases and validation against laboratory measurements of coastal hazards, focusing on tsunami propagation, run-up, and inundation is performed. In these tests, the UM2N-driven meshes help resolve key non-hydrostatic dynamics and yield numerical solutions in close agreement with reference computations and measured data. Numerical results indicate that the UM2N surrogate based approach significantly accelerates conventional mesh movement techniques and has high robustness over long integration periods and under strongly nonlinear wave conditions.

Figures (15)

• Figure 1: Overview of Universal Mesh Movement Network (UM2N) shown in (a), cited from zhang2024, and UM2N based PDE solver pipeline shown in (b). The monitor interpolation in (b) includes a step of monitor smoothing.
• Figure 2: The solution procedure of one iteration in the UM2N based PDE solver framework shown in Fig. \ref{['fig:UM2N_PDEsolver']} (b). Blue and green background panels: the solution procedure of one iteration from the initial timestep $t^{(0)}$ to $t^{(1)}$; Orange and purple background panels: the solution procedure of one iteration from the timestep $t^{(i)}$ to $t^{(i+1)}$, where $i=1,2,3, \cdots,n$, $n\in\mathbb{N}$. Top right: Color legends used to represent the inputs and outputs in the solution procedure.
• Figure 3: Comparison of N-wave test case: Two-dimensional plan view of wave patterns after propagating at $t=200$ (upper left) and $t=600$ (bottom left), and their corresponding meshes via different mesh configurations: MA movement (middle column) and UM2N (right column). The scale of elevation range for 2D plan view (left column) is from $-0.5\times10^{-4}$ to $0.75\times10^{-4}$ shown by blue to red.
• Figure 4: Convergence of RMS error of free-surface elevations in the N-wave strip source test case for fixed meshes, MA moved meshes and UM2N moved meshes at times $t$=200 (left) and $t$=600 (middle), respectively. Comparisons of computational cost (average inference time of mesh adaptation on CPU/GPU) for fixed meshes, MA meshes and UM2N meshes (right). Mesh sizes correspond to $\Delta x = 8.0$, $5.0$, and $4.0\,\unit{m}$ in convergence plots at left and middle columns, and $\Delta x = 5.0$, $4.0$, $2.0$ and $1.0\,\unit{m}$ in plot of computational cost at right column. The numbers show the slope of the least-squares fitted linear lines (black: fixed meshes; blue: MA moved meshes; red: UM2N moved meshes).
• Figure 5: Three-dimensional view snapshots of wave patterns after propagating over a truncated conical shoal at $t = 15\,\unit{s}$, from different perspectives.