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arxiv: 2602.05416 · v2 · submitted 2026-02-05 · 💻 cs.CE · cs.AI· cs.LG· physics.ao-ph· physics.flu-dyn

Reduced-Order Surrogates for Forced Flexible Mesh Coastal-Ocean Models

Freja H{\o}gholm Petersen , Jesper Sandvig Mariegaard , Rocco Palmitessa , Allan P. Engsig-Karup This is my paper

Pith reviewed 2026-05-16 07:28 UTC · model grok-4.3

classification 💻 cs.CE cs.AIcs.LGphysics.ao-phphysics.flu-dyn
keywords Koopman autoencoderreduced-order surrogatecoastal ocean modelingproper orthogonal decompositiontemporal stabilityhydrodynamic forecastingflexible mesh
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The pith

Koopman autoencoders with forcings deliver accurate year-long reduced-order surrogates for coastal-ocean models.

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

The paper establishes that a flexible Koopman autoencoder can incorporate meteorological forcings and boundary conditions to create stable reduced-order surrogates for flexible-mesh coastal-ocean models. It learns a linear temporal operator in latent space, regularizes the operator eigenvalues for stability, and applies temporal unrolling to reach long prediction horizons. Tested on three regimes spanning different dynamics at 30-minute resolution up to one year, the surrogates achieve relative root-mean-squared errors of 0.0068-0.14 and R-squared values of 0.61-0.995, with water-surface-elevation errors only a few centimeters above the full physics model. In two cases the Koopman version outperforms POD-based surrogates while delivering 300-1400x inference speed-ups.

Core claim

A Koopman autoencoder that maps the state to latent space and applies a learned linear temporal operator there, regularized for eigenvalue stability and trained on forcings and boundary conditions, produces bounded long-term trajectories whose accuracy reaches relative RMS errors of 0.0068-0.14 across three coastal regimes, matching or exceeding POD performance while remaining within practical tolerance of in-situ observations.

What carries the argument

Koopman autoencoder whose encoder-decoder pair produces a latent representation governed by a learned linear temporal operator that is regularized for eigenvalue stability and driven by meteorological forcings and boundary conditions.

If this is right

  • The 300-1400x inference speed-up enables ensemble forecasting and multi-decadal climate simulations that remain computationally infeasible with the full physics model.
  • Water-surface-elevation errors stay within a few centimeters of the physics-based model, keeping the surrogate usable for many operational coastal applications.
  • In two of the three tested regimes the Koopman formulation yields higher accuracy than POD-based surrogates under identical temporal-unrolling conditions.
  • Current-velocity fields show the largest relative errors while water-surface elevations remain the most accurate variable across all cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same latent-linear-operator construction could be applied to other externally forced fluid systems such as atmospheric or riverine models where full physics runs are too slow.
  • If eigenvalue regularization continues to bound trajectories under modest forcing changes, the surrogates could support real-time nowcasting without frequent retraining.
  • Systematic tests on extreme-event forcings would clarify whether the current accuracy levels hold when the underlying dynamics deviate strongly from the training distribution.

Load-bearing premise

The learned linear temporal operator in latent space remains stable and accurate when driven by real meteorological forcings outside the three tested regimes.

What would settle it

Long-term simulation under forcings from a fourth untested dynamical regime in which the surrogate's water-level or velocity trajectories diverge from the full physics model or in-situ data by amounts substantially larger than the reported 0.0068-0.14 relative RMS range.

Figures

Figures reproduced from arXiv: 2602.05416 by Allan P. Engsig-Karup, Freja H{\o}gholm Petersen, Jesper Sandvig Mariegaard, Rocco Palmitessa.

Figure 1
Figure 1. Figure 1: (a) Training of the Koopman-based Autoencoder in an end to end setting, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphical representation of autoencoders. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The domains and the bathymetries of the three cases, Øresund (a), Southern [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example timeseries of the surface elevation data used for training the surrogates. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction RMSEs for POD in the Adriatic test case. [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Prediction RMSE in a validation period for the Øresund case. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Prediction of surface elevations with the MIKE 21 and the linear KAE (TU) in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Test errors of the surface elevation relative to the range of values. The boxplots [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Time spent per epoch when training models. Total time for each model, in the [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: RMSE across the test period for each test case. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Scatter plots of the simulation data (x-axis) vs. the surrogate predictions (y [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

While proper orthogonal decomposition (POD)-based surrogates are widely explored for hydrodynamic applications, the use of Koopman autoencoders for real-world coastal-ocean modelling remains relatively limited. This paper introduces a flexible Koopman autoencoder formulation that incorporates meteorological forcings and boundary conditions, and systematically compares its performance against POD-based surrogates. The Koopman autoencoder employs a learned linear temporal operator in latent space, enabling eigenvalue regularization to promote temporal stability. This strategy is evaluated alongside temporal unrolling techniques for achieving stable and accurate long-term predictions. The models are assessed on three test cases spanning distinct dynamical regimes, with prediction horizons up to one year at 30-minute temporal resolution. Across all cases, the reduced order surrogates with temporal unrolling achieve high accuracy with relative root-mean-squared-errors of 0.0068-0.14 and $R^2$-values of 0.61-0.995, where prediction errors are largest for current velocities, and smallest for water surface elevations. In two of the three cases, the Koopman Autoencoder have higher accuracy than the POD-based surrogates. Comparing to in-situ observations, the surrogate yields -0.64% to 12% increase in water surface elevation prediction error when compared to prediction errors of the physics-based model. These error levels, corresponding to a few centimeters, are acceptable for many practical applications, while inference speed-ups of 300-1400x enables workflows such as ensemble forecasting and long climate simulations for coastal-ocean modelling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a flexible Koopman autoencoder formulation for reduced-order surrogates of forced flexible-mesh coastal-ocean models. The approach learns a linear temporal operator in latent space with eigenvalue regularization for stability, incorporates meteorological forcings and boundary conditions, and employs temporal unrolling for long-term rollouts. It systematically compares performance to POD-based surrogates on three distinct dynamical regimes with prediction horizons up to one year at 30-minute resolution, reporting rRMSE values of 0.0068–0.14 and R² values of 0.61–0.995 (best for water surface elevation, worst for currents). In two of three cases the Koopman model outperforms POD; against in-situ observations the surrogate increases water-level error by –0.64% to 12% relative to the full physics model while delivering 300–1400× speed-ups.

Significance. If the reported accuracy and stability hold under broader forcing conditions, the work would be significant for operational coastal modeling. It supplies concrete, reproducible error metrics against both physics-based output and real observations, demonstrates practical speed-ups that enable ensemble forecasting and long climate runs, and shows that a regularized Koopman operator can match or exceed POD accuracy in forced regimes. These elements directly address the computational bottleneck in coastal-ocean applications.

major comments (2)
  1. The headline claim that the regularized Koopman operator produces bounded long-term trajectories under real meteorological forcings rests on the three tested regimes spanning the relevant variability. No explicit out-of-distribution sensitivity analysis (e.g., scaling wind or pressure amplitudes beyond the training envelope) is described; if the latent eigenvalues drift under distribution shift, the 1-year rollouts would diverge even if short-term training metrics remain good. This assumption is load-bearing for the operational utility asserted in the abstract.
  2. The manuscript states that temporal unrolling is combined with eigenvalue regularization to achieve stable predictions, yet the precise interaction between the two mechanisms (e.g., how the regularization term is weighted during unrolled training) is not quantified in the results. Without this, it is unclear whether the reported rRMSE and R² values are attributable to the Koopman operator itself or to the unrolling procedure.
minor comments (2)
  1. Clarify the exact architecture of the encoder/decoder (number of layers, latent dimension) and the form of the learned linear operator matrix; these details are needed for reproducibility even if the abstract reports aggregate metrics.
  2. The abstract reports R² = 0.61–0.995; the lower end of this range should be discussed explicitly—does it correspond to a particular variable or regime, and does it still support the claim of “high accuracy” for practical use?

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify important aspects of the stability claims and training procedure that warrant clarification and additional analysis. We address each point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: The headline claim that the regularized Koopman operator produces bounded long-term trajectories under real meteorological forcings rests on the three tested regimes spanning the relevant variability. No explicit out-of-distribution sensitivity analysis (e.g., scaling wind or pressure amplitudes beyond the training envelope) is described; if the latent eigenvalues drift under distribution shift, the 1-year rollouts would diverge even if short-term training metrics remain good. This assumption is load-bearing for the operational utility asserted in the abstract.

    Authors: We agree that the long-term boundedness claim would be more robust with explicit out-of-distribution testing. In the revised manuscript we will add a dedicated sensitivity subsection that scales wind speed and pressure amplitudes by factors of 1.5× and 2.0× relative to the training envelope, reports the resulting shifts in latent eigenvalues, and quantifies rollout divergence over the one-year horizon. These results will be presented alongside the existing three-regime evaluation. revision: yes

  2. Referee: The manuscript states that temporal unrolling is combined with eigenvalue regularization to achieve stable predictions, yet the precise interaction between the two mechanisms (e.g., how the regularization term is weighted during unrolled training) is not quantified in the results. Without this, it is unclear whether the reported rRMSE and R² values are attributable to the Koopman operator itself or to the unrolling procedure.

    Authors: We will revise the methods and results sections to quantify the interaction. The revised text will explicitly state the weighting coefficient λ_reg used for the eigenvalue regularization term within the unrolled loss, provide the full composite loss expression, and include an ablation table comparing (i) unrolling alone, (ii) regularization alone, and (iii) the combined formulation on the same three test cases. This will isolate the contribution of each component to the reported accuracy metrics. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; accuracy claims rest on independent physics-based and observational benchmarks

full rationale

The paper trains Koopman autoencoders (with learned linear latent operator and eigenvalue regularization) and POD surrogates on data generated by the full physics-based coastal-ocean model, then reports relative RMSE and R² on held-out temporal segments against the same model's outputs plus in-situ observations. These metrics are external to the fitted parameters; the central result is an empirical performance comparison across three regimes, not a derivation that reduces to its own inputs by construction. No self-definitional equations, fitted-input-as-prediction, or load-bearing self-citations appear in the described workflow. Eigenvalue regularization is a standard training constraint, not a tautology that forces the reported accuracy numbers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The approach implicitly assumes that a learned linear operator in latent space can represent forced nonlinear coastal dynamics over year-long horizons.

pith-pipeline@v0.9.0 · 5607 in / 1168 out tokens · 27722 ms · 2026-05-16T07:28:27.104955+00:00 · methodology

discussion (0)

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