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‎docs/assets/images/zenodo-icon-blue.svg‎

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‎docs/benchmarks.md‎

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!!! info
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Page under construction
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<div style="text-align:center;">
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<img src="../assets/images/plaid_benchmarks.png" class="align-center" width="60%"
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alt="PLAID Benchmarks" />
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</div>
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Six interactive benchmarks are provided in by the [PLAID competitions on Hugging Face](https://huggingface.co/PLAIDcompetitions).
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We provide interactive benchmarks hosted on Hugging Face, in which
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anyone can test their own SciML method. These benchmarks involve
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regression problems posed on datasets provided in PLAID format. Some of
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these datasets have been introduced in the MMGP (Mesh Morphing Gaussian
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Process) paper [@casenave2023mmgp], and the PLAID paper
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[@casenave2025plaid]. A ranking is automatically updated based on a score
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computed on the testing set of each dataset. For the benchmarks to be
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meaningful, the outputs on the testing sets are not made public.
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The relative RMSE is the considered metric for comparing methods. Let
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$\{ \mathbf{U}^i_{\rm ref} \}_{i=1}^{n_\star}$ and
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$\{ \mathbf{U}^i_{\rm pred} \}_{i=1}^{n_\star}$ be the test observations
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and predictions, respectively, of a given field of interest. The
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relative RMSE is defined as
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$$
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\mathrm{RRMSE}_f(\mathbf{U}_{\rm ref}, \mathbf{U}_{\rm pred}) = \left( \frac{1}{n_\star}\sum_{i=1}^{n_\star} \frac{\frac{1}{N^i}\|\mathbf{U}^i_{\rm ref} - \mathbf{U}^i_{\rm pred}\|_2^2}{\|\mathbf{U}^i_{\rm ref}\|_\infty^2} \right)^{1/2},
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$$
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where $N^i$ is the number of nodes in the mesh $i$, and
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$\max(\mathbf{U}^i_{\rm ref})$ is the maximum entry in the vector
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$\mathbf{U}^i_{\rm ref}$. Similarly for scalar outputs:
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$$
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\mathrm{RRMSE}_s(\mathbf{w}_{\rm ref}, \mathbf{w}_{\rm pred}) = \left( \frac{1}{n_\star} \sum_{i=1}^{n_\star} \frac{|w^i_{\rm ref} - w_{\rm pred}^i|^2}{|w^i_{\rm ref}|^2} \right)^{1/2}.
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$$
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## Interactive benchmark applications
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<div class="grid cards">
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<div class="card">
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<p><strong>Tensile2d</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/Tensile2dBenchmark"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="Tensile2d_Be" style="height:30px;"/></a>
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</div>
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</div>
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<div class="card">
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<p><strong>2D_MultiScHypEl</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/2DMultiscaleHyperelasticityBenchmark"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="2D_MultiScHypEl_Be" style="height:30px;"/></a>
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</div>
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</div>
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<div class="card">
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<p><strong>2D_ElPlDynamics</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/2DElastoPlastoDynamics"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="2D_ElPlDynamics_Be" style="height:30px;"/></a>
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</div>
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</div>
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<div class="card">
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<p><strong>Rotor37</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/Rotor37Benchmark"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="Rotor37_Be" style="height:30px;"/></a>
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</div>
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</div>
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<div class="card">
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<p><strong>2D_profile</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/2DprofileBenchmark"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="2D_profile_Be" style="height:30px;"/></a>
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</div>
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</div>
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<div class="card">
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<p><strong>KI-LS59</strong></p>
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<div class="card-badges">
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<a href="https://huggingface.co/spaces/PLAIDcompetitions/VKILS59Benchmark"><img src="https://huggingface.co/datasets/huggingface/badges/resolve/main/open-in-hf-spaces-sm-dark.svg" alt="VKI-LS59_Be" style="height:30px;"/></a>
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</div>
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</div>
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</div>
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## Benchmark results
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As of August 5, 2025
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| <div style="text-align:center;"><p>Benchmark<p></div> | <div style="text-align:center;"><p>MGN<p></div> | <div style="text-align:center;"><p>MMGP<p></div> | <div style="text-align:center;"><p>Vi-Transf.<p></div> | <div style="text-align:center;"><p>Augur<p></div> | <div style="text-align:center;"><p>FNO<p></div> | <div style="text-align:center;"><p>MARIO<p></div> |
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|------------------------------------------------|--------|--------|------------|--------|--------|--------|
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| <span class="title-ref">Tensile2d</span> | 0.0673 | 0.0026 | 0.0116 | 0.0154 | 0.0123 | 0.0038 |
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| <span class="title-ref">2D_MultiScHypEl</span> | 0.0437 | ❌ | 0.0325 | 0.0232 | 0.0302 | 0.0573 |
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| <span class="title-ref">2D_ElPlDynamics</span> | 0.1202 | ❌ | 0.0227 | 0.0346 | 0.0215 | 0.0319 |
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| <span class="title-ref">Rotor37</span> | 0.0074 | 0.0014 | 0.0029 | 0.0033 | 0.0313 | 0.0017 |
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| <span class="title-ref">2D_profile</span> | 0.0593 | 0.0365 | 0.0312 | 0.0425 | 0.0972 | 0.0307 |
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| <span class="title-ref">VKI-LS59</span> | 0.0684 | 0.0312 | 0.0193 | 0.0267 | 0.0215 | 0.0124 |
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❌: Not compatible with topology variation
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!!! note
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- MMGP does not support variable mesh topologies, which limits its
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applicability to certain datasets and often necessitates custom
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preprocessing for new cases. However, when morphing is either
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unnecessary or inexpensive, it offers a highly efficient solution,
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combining fast training with good accuracy (e.g., `Tensile2d` and
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`Rotor37`).
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- MARIO is computationally expensive to train but achieves consistently
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a very strong performance across most datasets. Its result on
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`2D_MultiScHypEl` is slightly worse than other tested methods, which
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may reflect the challenge of capturing complex shape variability in
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these cases.
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- Vi-Transformer and Augur perform well across all datasets, showing
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strong versatility and generalization capabilities.
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- FNO suffers on datasets featuring unstructured meshes with pronounced
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anisotropies, due to the loss of accuracy introduced by projections to
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and from regular grids (e.g., `Rotor37` and `2D_profile`).
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Additionally, the use of a 3D regular grid on `Rotor37` results in
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substantial computational overhead.
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## References
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\bibliography

‎docs/css/extra.css‎

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.md-content {
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max-width: 1200px; /* default ~780px */
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/* Each card flex layout */
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.grid.cards .card {
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display: flex;
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flex-direction: column;
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align-items: center; /* horizontal center */
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justify-content: flex-start;
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padding: 1em;
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text-align: center;
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}
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/* Top image in card */
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.card-top-img {
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max-width: 100%; /* scale to fit card width */
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height: auto;
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margin-bottom: 0.5em;
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border-radius: 4px; /* optional: rounded corners */
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}
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/* Horizontal badges */
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.card-badges {
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display: flex;
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gap: 0.4em; /* slightly reduce spacing */
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flex-wrap: wrap;
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justify-content: center;
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margin-top: 0.5em;
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}
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/* Reduce badge size slightly */
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.card-badges img {
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height: 24px; /* slightly smaller than before */
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object-fit: contain;
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}
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/* Center the first column of all tables */
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table td:first-child,
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table th:first-child {
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text-align: center;
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}
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/* Optional: center badges inside first column */
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table td:first-child img {
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vertical-align: middle;
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}

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