Four Gyre Rom Eval

---
name: four-gyre-rom-eval
description: Evaluates the ability of a machine learning closure model to stabilize reduced-order models for turbulent geophysical fluid dynamics. Specifically, it probes whether an extreme learning machine can predict mode-dependent eddy viscosities to maintain long-time integration accuracy and statistical steady-state behavior in coarse-grained ocean circulation simulations. Use when the user wants to benchmark on Four-gyre barotropic circulation problem, or asks about evaluating this task. Reports L2-norm error.
metadata:
  skill_kind: dataset_eval
  source_arxiv: 1803.00222
  bibtex_key: san2018extreme
  confidence: high
---

# four-gyre-rom-eval

> Extreme learning machine for reduced order modeling of turbulent geophysical flows — San et al. (2018) (arXiv:1803.00222, 2018)

## What this evaluates

Evaluates the ability of a machine learning closure model to stabilize reduced-order models for turbulent geophysical fluid dynamics. Specifically, it probes whether an extreme learning machine can predict mode-dependent eddy viscosities to maintain long-time integration accuracy and statistical steady-state behavior in coarse-grained ocean circulation simulations.

## Datasets

- **Four-gyre barotropic circulation problem** — total ?; splits: train (900), test (-1)

## Metrics

- `L2-norm error` **(primary)** — range: other
  - L2-norm of the difference between the ROM-predicted mean vorticity or streamfunction fields and the Full Order Model (FOM) reference fields over the assessment window.

## Input / output format

**Input**: Time-series snapshots of the barotropic vorticity equation (BVE) at Re=450, Ro=3.6e-3, used to construct POD basis and train the ELM closure.

**Output**: Predicted mode-dependent eddy viscosities and the resulting ROM state variables (mean vorticity and streamfunction fields) over the integration window.

## Scoring recipe

```python
def compute_l2_error(rom_pred, fom_ref):
    # rom_pred, fom_ref: arrays of mean vorticity or streamfunction fields
    return np.linalg.norm(rom_pred - fom_ref)
```

## Common pitfalls

- The FOM baseline uses a very small time step (Δt=2.5e-5) for stability, so CPU time comparisons heavily favor the stabilized ROM without normalization.
- Out-of-sample testing uses different physical parameters (Re=200, Ro=1.6e-3) than training, which tests generalization but requires separate L2 error reporting.
- The metric is computed on mean fields over a steady-state window, not instantaneous snapshots, which masks short-term transient errors.

## Evidence (verbatim from paper)

> Table 1: L2-norm errors of the reduced order models (with respect to FOM) for the mean vorticity and streamfunction fields. Note that the ROM-ANN retains only M=10 modes.

## Citation

```bibtex
@misc{san2018extreme,
  title={Extreme learning machine for reduced order modeling of turbulent geophysical flows},
  author={San et al. (2018)},
  year={2018},
  note={arXiv:1803.00222}
}
```

- arXiv: 1803.00222