Helmholtz Scattering Eval

---
name: helmholtz-scattering-eval
description: Evaluates the accuracy and computational efficiency of numerical PDE solvers (BEM vs. PINNs) for 2D acoustic wave scattering. Probes generalization capability beyond the training domain and physical fidelity in far-field regions. Use when the user wants to benchmark on 2D Helmholtz Scattering Benchmark, or asks about evaluating this task. Reports relative error.
metadata:
  skill_kind: dataset_eval
  source_arxiv: 2509.12483
  bibtex_key: rinconcardeno2025wave
  confidence: high
---

# helmholtz-scattering-eval

> Comparative Analysis of Wave Scattering Numerical Modeling Using the Boundary Element Method and Physics-Informed Neural Networks — Rincon-Cardeño et al. (2025) (arXiv:2509.12483, 2025)

## What this evaluates

Evaluates the accuracy and computational efficiency of numerical PDE solvers (BEM vs. PINNs) for 2D acoustic wave scattering. Probes generalization capability beyond the training domain and physical fidelity in far-field regions.

## Datasets

- **2D Helmholtz Scattering Benchmark** — total ?; splits: train (-1), test (-1), generalization (-1)

## Metrics

- `relative error` **(primary)** — range: [0, 1]
  - Relative L2 error computed against the analytical solution: ||u_pred - u_exact||_2 / ||u_exact||_2. Evaluated on spatial grids inside and outside the training domain.
- `computational time` — range: seconds
  - Wall-clock time in seconds for BEM solution assembly/solving, PINN training, and PINN inference/evaluation.

## Input / output format

**Input**: Spatial coordinates (r, θ) or (x, y) on a 2D grid surrounding a circular obstacle of radius r_i = π/4, with incident plane wave wavenumber k = 3.

**Output**: Predicted total or scattered wave field amplitude and phase u(x, y) at each coordinate.

## Scoring recipe

```python
u_exact = analytical_helmholtz_solution(coords, k=3, r_i=np.pi/4)
u_pred = model.predict(coords)
rel_error = np.linalg.norm(u_pred - u_exact) / np.linalg.norm(u_exact)
train_time = time_bench(pinn_train)
eval_time = time_bench(pinn_infer)
bem_time = time_bem_solve(coords)
```

## Common pitfalls

- Confusing PINN training time with inference time; PINNs require ~42× longer training but ~200× faster evaluation than BEM.
- Assuming PINNs generalize well outside the training domain without explicit boundary condition enforcement; error increases drastically (from ~0.075 to 8.22) beyond the training region.
- Ignoring the continuous nature of the benchmark; results depend heavily on spatial discretization density and integration points.

## Evidence (verbatim from paper)

> For the computational time comparison, we selected the BEM configuration that achieved the closest accuracy to the optimal PINN solution. Specifically, the PINN with 3 layers and 75 neurons reached a relative error of 7.46 × 10⁻², while the closest BEM result was obtained with n = 15, yielding 7.25 × 10⁻².

## Citation

```bibtex
@misc{rinconcardeno2025wave,
  title={Comparative Analysis of Wave Scattering Numerical Modeling Using the Boundary Element Method and Physics-Informed Neural Networks},
  author={Rincon-Cardeño et al. (2025)},
  year={2025},
  note={arXiv:2509.12483}
}
```

- arXiv: 2509.12483