Machine Learning Based Mesh Movement for Non-Hydrostatic Tsunami Simulation

Matthew D. Piggott; Stephan C. Kramer; Yezhang Li

Machine Learning Based Mesh Movement for Non-Hydrostatic Tsunami Simulation

T0 review · 2 major / 0 minor · reviewed 2026-07-15 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A machine-learning mesh mover tracks tsunami waves more accurately and robustly than classical Monge–Ampère adaptation, cutting nearshore peak error by ~91% on a laboratory inundation benchmark while speeding the movement step.

desk verdict Solid first coupling of UM2N into Thetis non-hydrostatic tsunami runs; useful robustness and speed story, but the headline 91%/74% Monai peak-error claim is not backed by a transparent metric or table. read the letter →

arxiv 2603.06152 v2 pith:PJWXNNP3 submitted 2026-03-06 physics.ao-ph physics.comp-phphysics.flu-dyn

Yezhang Li , Stephan C. Kramer , Matthew D. Piggott This is my paper · ORCID

classification physics.ao-phphysics.comp-phphysics.flu-dyn

keywords non-hydrostaticshallowwatermeshmovementUM2NMonge–Ampèretsunamirun-upwetting-dryingdiscontinuousGalerkincoastalhazard

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Coastal tsunami models that keep non-hydrostatic pressure need fine resolution near waves and wet–dry fronts, but classical mesh-movement methods that solve a Monge–Ampère equation at every step are expensive and can tangle under strong nonlinearity. This paper shows that a pre-trained Universal Mesh Movement Network can replace that solve: it takes only a monitor built from free-surface curvature (and a wet–dry tracker) and relocates the same number of nodes so that resolution follows the wave. On solitary-wave refraction over a shoal, run-up on a conical island, and the Monai Valley laboratory inundation test, the adapted solutions stay close to fine-mesh and measured free-surface records. On Monai Valley the network reduces wave-peak error at the critical nearshore gauge by about 91 percent relative to the coarse fixed mesh (versus about 74 percent for Monge–Ampère), while roughly halving the movement-step time on GPU and remaining stable over long, nonlinear integrations where the classical solver often fails. The practical claim is that an off-the-shelf mesh-movement surrogate can make adaptive non-hydrostatic coastal modelling cheap enough for the many runs required by probabilistic hazard assessment.

What carries the argument

Universal Mesh Movement Network (UM2N): a graph-transformer encoder plus graph-attention decoder that maps mesh coordinates and a Hessian-based monitor (optionally augmented by a wet–dry interface tracker) into new node positions, trained by element-volume and Chamfer losses against Monge–Ampère reference meshes.

What would settle it

On the Monai Valley or conical-island benchmarks, re-run the identical non-hydrostatic solver with a freshly trained or fine-tuned UM2N and show that the nearshore wave-peak error reduction falls below the classical Monge–Ampère result, or that the mesh still tangles within the first hundred time steps.

Watch

Extended reading notes

Core claim

A Universal Mesh Movement Network trained only on synthetic random fields can be coupled, without retraining, to a discontinuous-Galerkin non-hydrostatic shallow-water solver and produce adapted meshes that resolve refraction, run-up and inundation at least as well as classical Monge–Ampère movement, cut wave-peak error by ~91% versus a coarse fixed mesh on the Monai Valley gauge, and remain robust where the Monge–Ampère solver diverges, while accelerating the mesh-movement step.

Load-bearing premise

A network trained solely on synthetic Gaussian-sum fields and generic Hessian monitors will transfer, without fine-tuning, to real non-hydrostatic free-surface flows that include wetting–drying and strong refraction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news is the first tight integration of Zhang et al.’s UM2N into a DG non-hydrostatic Thetis solver for coastal tsunami work, plus a wet–dry-aware monitor (Eq. 32) and head-to-head runs on N-wave, conical shoal, conical island, and Monai Valley. That is new and useful for people who actually run adaptive coastal models.

What they do well is clear. UM2N tracks refraction, run-up, and inundation on the standard lab benchmarks and stays close to fine-mesh and gauge data where the coarse fixed mesh is visibly worse. On the shoal and island cases MA crashes early from tangling or MA-solver divergence; UM2N keeps going. That robustness point is real and matters more for long nonlinear runs than another 10 % on a single peak. Training on synthetic Gaussian-sum monitors is PDE-independent, so the transfer claim is at least honest about its basis. Self-citations are mostly the tools (Thetis, Pan non-hydrostatic, Clare MA, Zhang UM2N), not the measured gains.

Soft spots are real but limited. The abstract’s ~91 % vs ~74 % wave-peak error reduction at the Monai nearshore gauge is the strongest quantitative claim, yet Table 1 only gives inference times; there is no definition of the peak-error metric, no raw peak values for FIX-COARSE / MA / UM2N / experiment, and no uncertainty. Fig. 15 looks qualitatively similar, so the percentages cannot be recovered. The ~32 % total-runtime cut is also only loosely supported by the published numbers (mesh-step speed-ups are clearer). Monitor parameters (α, diffusion, wet–dry μ/λ/p_cap) are hand-tuned; no code or data release. These are presentation and reproducibility gaps, not evidence that the method fails.

This is for coastal-ocean and adaptive-mesh CFD people who already care about Thetis/Firedrake or r-adaptivity cost. It is not a field-reorganizing result, but it is a clean systems paper with multi-benchmark evidence. I would send it to peer review; the coupling and robustness story deserve a referee even if the error percentages need a proper table.

Referee Report

2 major / 0 minor

Summary. The manuscript couples the Universal Mesh Movement Network (UM2N) of Zhang et al. (2024) to the depth-integrated non-hydrostatic shallow-water solver in Thetis/Firedrake, and evaluates the resulting adaptive scheme on four benchmarks: an analytical N-wave strip source, solitary-wave refraction over a truncated conical shoal, solitary-wave run-up on a conical island with wetting–drying, and the Monai Valley laboratory inundation experiment. A Hessian-based monitor (with an additional wet–dry interface term, Eq. 32) drives mesh movement; UM2N is used as a PDE-independent surrogate for the conventional Monge–Ampère (MA) solver. The authors report that UM2N-adapted solutions remain close to fine-mesh and laboratory references, that MA frequently diverges under strong refraction or wetting–drying while UM2N remains robust, and that, on Monai Valley, UM2N yields a ~91 % reduction in nearshore wave-peak error versus ~74 % for MA (both relative to the coarse fixed mesh), together with a ~32 % total-runtime reduction and roughly 2 imes speed-up of the mesh-movement step on GPU.

Significance. If the quantitative claims hold, the work supplies a practical route to r-adaptive non-hydrostatic coastal modelling that is both more robust and substantially cheaper than classical MA mesh movement, while remaining compatible with an existing high-order DG code base. That combination is directly relevant to probabilistic tsunami hazard assessment, where many realisations must be run at coastal resolution. Strengths that are already visible include the transparent coupling pipeline (Figs. 1–2), the use of external laboratory gauges and an analytical N-wave solution, and the explicit demonstration that a network trained only on synthetic Gaussian-sum monitors can be deployed without retraining on free-surface flows. These are genuine engineering contributions even if the headline error percentages require clearer documentation.

major comments (2)

Abstract and §5.4 assert that UM2N achieves a ~91 % reduction in wave-peak error at the nearshore gauge versus ~74 % for MA (both relative to the coarse fixed mesh). Table 1 reports only average inference times; no equation, table or caption defines the peak-error metric (absolute difference at a single time? L2 over a window? which gauge and which peak?), nor are the raw peak elevations for FIX-COARSE / FIX-FINE / MA / UM2N / experiment supplied. Fig. 15 shows qualitatively similar time series from which the stated percentages cannot be recovered. Because this comparison is the central quantitative claim that distinguishes UM2N from MA on a laboratory benchmark, the definition and intermediate numbers must be added (or the percentages removed) before the claim can be verified.
Abstract states a ~32 % reduction in total runtime and ~2 imes speed-up of the mesh-movement step over the MA solver on GPU. Table 1, however, lists only average inference times for the movement step itself (MA-CPU 12 563 s, UM2N-CPU 2 229 s, UM2N-GPU 43 s) and does not report wall-clock times for the full PDE solve, the number of adaptation calls, or a GPU-based MA baseline. The ~32 % total-runtime figure and the ~2 imes GPU comparison therefore cannot be reconstructed from the published material and should be supported by an explicit timing breakdown (or the abstract wording adjusted to match Table 1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: empirical performance claims rest on external analytical/lab/fine-mesh benchmarks and a PDE-independent synthetic training set, not on self-referential definitions or fitted targets.

full rationale

The paper applies a pre-trained UM2N surrogate (Zhang et al. 2024) as a drop-in mesh-movement operator inside an existing DG non-hydrostatic solver (Thetis/Pan). Training uses only synthetic random Gaussian-sum fields and Hessian monitors that are deliberately PDE-independent (§4.3–4.4); the network never sees the tsunami equations or the laboratory gauges. Evaluation metrics (RMS free-surface error on the N-wave analytic solution, time-series peaks at conical-island and Monai Valley gauges, wall-clock timings) are computed against external references (analytic solution of Kanoğlu et al., laboratory data of Briggs et al./CRIEPI, and independent fine-mesh runs). Self-citations supply the underlying solver, the MA baseline, and the UM2N architecture, but none of those citations is used to justify the numerical superiority percentages or the robustness claims; those percentages are measured outputs of new simulations. No parameter is fitted to a target quantity and then re-presented as a prediction, no uniqueness theorem is imported to force the method, and no ansatz is smuggled in as a derivation. The derivation chain is therefore self-contained and non-circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central accuracy/robustness/speed claims rest on standard non-hydrostatic depth-integrated SWEs and wet–dry regularization, classical MA equidistribution as the teacher, a pretrained UM2N surrogate assumed to generalize from synthetic Gaussians, and several hand-chosen monitor and friction parameters that control where nodes move. No new physical entity is postulated; the main ad-hoc pieces are the monitor design and transfer assumption.

free parameters (5)

Hessian monitor intensity α = 5
User-specified scale in m := 1 + α ||H(η_h)|| / max||H||; set to α=5 (§4.2.1) and controls how aggressively nodes cluster on free-surface curvature.
Monitor diffusion coefficient ẽα = 0.3–0.5
Pseudo-time heat-equation smoothing strength for the monitor; set in 0.3–0.5 range (§4.2.2) and changes mesh regularity.
Wet–dry monitor parameters μ, α, λ, p_cap, b_λ
User-defined weights and caps in the conical-island/Monai monitor m_α (Eq. 32, §5.3) that blend Hessian refinement with a sech² wet–dry tracker; not uniquely determined by data.
Bottom friction / viscosity (case-dependent) = Cd=0.01; μ=0.019; ν=0.015
Cd=0.01 and ν_h=0.015 on conical island; quadratic drag μ=0.019 s·m^{-1/3} and velocity cap 5 m/s on Monai (§5.3–5.4). Affect run-up and gauge peaks used in the accuracy claim.
UM2N loss weights λ_vol, λ_cd
Hyperparameters balancing element-volume and Chamfer losses in training (Eq. 27); inherited from Zhang et al. and affect the surrogate mesh quality.

assumptions (5)

domain assumption Depth-integrated single-layer non-hydrostatic free-surface equations with Keller-box vertical structure and pressure split p = p_h + q are an adequate model for the chosen coastal benchmarks.
§2.1–2.2 and Pan et al. (2019); hydrostatic NSWE recovered when q=0. Limits wave breaking/overturning capture as the paper notes.
domain assumption Kärnä-style smooth wet–dry depth H̃ = H + f(H) with f(H)=(√(H²+α²)−H)/2 yields a well-posed positive-depth formulation.
§2.2; used for island and Monai run-up.
standard math Optimal-transport Monge–Ampère equidistribution m(x) det(I+H(Φ))=θ produces a unique injective mesh map for a given monitor.
§4.1 citing Brenier, McRae et al.; defines the teacher meshes for UM2N and the MA baseline.
ad hoc to paper A Graph-Transformer + GAT UM2N trained on synthetic Gaussian-sum fields is a valid PDE-independent surrogate for MA mesh movement when fed only monitor values.
§4.3–4.5 and Zhang et al. (2024); load-bearing for all UM2N results without per-case retraining.
domain assumption Hessian (Frobenius) of free-surface elevation, optionally plus wet–dry tracker, is a sufficient monitor for tsunami-relevant features.
§4.2 and Eq. 32; standard r-adaptivity practice but not unique or proven optimal for non-hydrostatic inundation.

invented entities (1)

Wet–dry-aware monitor m_α (Eq. 32)
purpose: Drive mesh refinement on free-surface curvature only in wet regions and track the wet–dry interface without refining dry land after the wave passes.
New composite monitor for this paper’s island/Monai runs; engineering construct rather than a physical entity. Independent evidence is only the numerical gauge improvement shown here.

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Cite this review

Pith. "Pith review of Machine Learning Based Mesh Movement for Non-Hydrostatic Tsunami Simulation." pith.science (2026). https://pith.science/paper/PJWXNNP3

@misc{pith2026260306152,
  author       = {Pith},
  title        = {Pith review of: Machine Learning Based Mesh Movement for Non-Hydrostatic Tsunami Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJWXNNP3}},
  note         = {Machine review of arXiv:2603.06152}
}

read the original abstract

This study investigates the use of machine learning based mesh movement method, specifically the Universal Mesh Movement Network (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 including wave refraction over a conical shoal, run-up with wetting-drying on a conical island, and tsunami inundation in the Monai Valley laboratory benchmark, and yield numerical solutions in close agreement with reference fine-mesh computations and measured data. Notably, in the Monai Valley case, UM2N achieves a ~91% reduction in wave-peak error at the nearshore gauge compared with ~74% for the conventional Monge--Amp\`ere (MA) mesh movement, both relative to the coarse fixed mesh. The UM2N surrogate based approach accelerates the conventional mesh movement step, achieving a ~32% reduction in total runtime and ~2 times speed-up in mesh movement step time over the MA solver on GPU, while offering a significant improvement in robustness over long integration periods and under strongly nonlinear wave conditions.

Figures

Figures reproduced from arXiv: 2603.06152 by the authors.

**Figure 2.** The solution procedure of one iteration in the UM2N based PDE solver frame [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Comparison of N-wave test case: Two-dimensional plan view of wave patterns [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Convergence of RMS error of free-surface elevations in the N-wave strip source [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Three-dimensional view snapshots of wave patterns after propagating over a [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Conical shoal setup. Left: Two-dimensional plan view of bathymetry with [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: Top: Two-dimensional plan view of wave patterns when solitary wave passes [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Numerical results comparison of results using UM2N on coarse mesh (red solid [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: Conical island setup. Left: Two-dimensional plan view of bathymetry with [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Conical island: Two-dimensional plan view snapshots of wave patterns (Fig [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: Conical island: Two-dimensional plan view snapshots of elevation field when [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

**Figure 12.** Figure 12: Comparisons of numerical results between adaptive mesh approaches and [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]

**Figure 13.** Figure 13: Monai Valley test case. Left: Bottom topography and land. Right: Setup [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]

**Figure 14.** Figure 14: Monai Valley: Two-dimensional plan view snapshots of elevation field (left [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

**Figure 15.** Figure 15: Comparisons of free-surface elevation time-series at three gauges in the Monai [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 5 canonical work pages

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Introduction 1 Accurate modeling of coastal tsunami has been a challenging research2 topic for many decades, with a goal of effective hazard assessments in the3 arXiv:2603.06152v1 [physics.ao-ph] 6 Mar 2026 faceofcomplexcoastaltopography, changeablemulti-scaleflowdynamicsand4 the impacts of flooding and drying processes. Unlike hydrostatic models that5 ar...

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Governing equations of non-hydrostatic model117 In this study, a single-layer non-hydrostatic free surface model is adopted118 for our simulations

Mathematical formulation 116 2.1. Governing equations of non-hydrostatic model117 In this study, a single-layer non-hydrostatic free surface model is adopted118 for our simulations. The total water depth, denoted byH[m], satisfies:119 H=η+b,(1) whereη[m] is the free surface elevation andb=b(x)[m] is the bathymetry.120 Although the flow is assumed to be hy...

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Following the pioneering work and relevant extensions of layer-206 averaged non-hydrostatic approach (Stelling and Zijlema, 2003; Wei and Jia,207

Discretization methods and implementation in Thetis and Fire-201 drake 202 The non-hydrostatic model is solved using the unstructured-mesh finite203 element coastal ocean modelling system, Thetis (Kärnä et al., 2018), which204 is built upon the automated code generation framework, Firedrake (Ham205 et al., 2023). Following the pioneering work and relevant...

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Model Verification and Validation412 5.1. N-wave strip source 413 This problem was introduced in Kanoğlu et al. (2013) as an idealised414 model for tsunami propagation studies, and is based on a finite strip source415 over a constant bathymetry solved using the linear shallow water equation.416 This test case is used here to investigate the convergence, r...

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Conclusion 620 In this paper, the use of machine learning based mesh movement for non-621 hydrostatic shallow water modelling has been investigated for the simulation622 of complex wave interactions and tsunami run-up on irregular bathymetries623 and changeable boundary conditions. 624 Results indicate that mesh movement via the UM2N network not only625 o...

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provide alternative movement strategies that could also be applied to645 non-hydrostatic models. Considering mesh regularity and uniformity during646 movement, the mesh adaptation algorithms can be further improved (e.g.,647 allowing boundary node movement), thus helping to generate meshes with648 31 superior quality. 649 Acknowledgements 650 YL would lik...

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Pith tools

Reviewed July 15, 2026 · model on record in the stance chip above.

[1] [1]

Migrating Trench

Introduction 1 Accurate modeling of coastal tsunami has been a challenging research2 topic for many decades, with a goal of effective hazard assessments in the3 arXiv:2603.06152v1 [physics.ao-ph] 6 Mar 2026 faceofcomplexcoastaltopography, changeablemulti-scaleflowdynamicsand4 the impacts of flooding and drying processes. Unlike hydrostatic models that5 ar...

arXiv 2026

[2] [2]

Governing equations of non-hydrostatic model117 In this study, a single-layer non-hydrostatic free surface model is adopted118 for our simulations

Mathematical formulation 116 2.1. Governing equations of non-hydrostatic model117 In this study, a single-layer non-hydrostatic free surface model is adopted118 for our simulations. The total water depth, denoted byH[m], satisfies:119 H=η+b,(1) whereη[m] is the free surface elevation andb=b(x)[m] is the bathymetry.120 Although the flow is assumed to be hy...

2019

[3] [3]

Following the pioneering work and relevant extensions of layer-206 averaged non-hydrostatic approach (Stelling and Zijlema, 2003; Wei and Jia,207

Discretization methods and implementation in Thetis and Fire-201 drake 202 The non-hydrostatic model is solved using the unstructured-mesh finite203 element coastal ocean modelling system, Thetis (Kärnä et al., 2018), which204 is built upon the automated code generation framework, Firedrake (Ham205 et al., 2023). Following the pioneering work and relevant...

2018

[4] [4]

(2019) integrated the layer-209 averaged non-hydrostatic approach into DG finite element implementation in210 Thetis

and non-hydrostatic pressure correction (Marshall et al., 1997; Stansby208 and Zhou, 1998; Lai et al., 2010), Pan et al. (2019) integrated the layer-209 averaged non-hydrostatic approach into DG finite element implementation in210 Thetis. In this formulation the 3D incompressible Navier–Stokes (NS) equa-211 tions are vertically integrated to obtain a dept...

1997

[5] [5]

Monge–Ampère mesh movement 249 Following the work of McRae et al

Universal Mesh Movement Network (UM2N)248 4.1. Monge–Ampère mesh movement 249 Following the work of McRae et al. (2018), Wallwork (2021) and Zhang250 et al. (2024), the mesh movement process can be defined as the optimization251 process to find the transformation between a computational domain,ΩC, and252 a physical domain,ΩP. The aim of mesh movement is t...

2018

[6] [6]

N-wave strip source 413 This problem was introduced in Kanoğlu et al

Model Verification and Validation412 5.1. N-wave strip source 413 This problem was introduced in Kanoğlu et al. (2013) as an idealised414 model for tsunami propagation studies, and is based on a finite strip source415 over a constant bathymetry solved using the linear shallow water equation.416 This test case is used here to investigate the convergence, r...

arXiv 2013

[7] [7]

Conclusion 620 In this paper, the use of machine learning based mesh movement for non-621 hydrostatic shallow water modelling has been investigated for the simulation622 of complex wave interactions and tsunami run-up on irregular bathymetries623 and changeable boundary conditions. 624 Results indicate that mesh movement via the UM2N network not only625 o...

2005

[8] [8]

provide alternative movement strategies that could also be applied to645 non-hydrostatic models. Considering mesh regularity and uniformity during646 movement, the mesh adaptation algorithms can be further improved (e.g.,647 allowing boundary node movement), thus helping to generate meshes with648 31 superior quality. 649 Acknowledgements 650 YL would lik...

work page doi:10.1016/j.compfluid.2017.10.026.659 2018