🧠 SPhyR-Quick-Start

🦾 Code
📄 Paper
🧰 Prompt Template

Prompt Template:

You are given a structural material distribution represented as a grid. Each cell can have one of the following states:
- 'L' indicates applied load.
- 'V' indicates void.
- 'S' indicates support.

The goal is to predict the correct material distribution by filling in all {FILL_INSTRUCTION}, based on the surrounding structure and implicit physical reasoning (such as load paths, supports, and forces).

Important: The completed structure should use as little material as possible while remaining stable and plausible for carrying the applied forces. Minimize material usage unless necessary for structural support.

Below is the input grid with masked regions:

{GRID}

Please output the completed grid by replacing all {FILL_INSTRUCTION}.
Maintain the same format as the input: one row per line, cells separated by spaces, and the total number of rows and columns unchanged.
Return only the completed grid without any additional explanation.

For easy difficulty use {FILL_INSTRUCTION}: 'V' cells with either '1' (solid) or '0' (empty)
or for hard difficulty use {FILL_INSTRUCTION}: 'V' cells with a floating point number between 0 and 1, with one decimal place (e.g., 0.0, 0.1, 0.2, ..., 1.0)

Replace {GRID} with data from the subject respective column in the dataset for example 1_random_cell_easy:

L L L 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 V
V 1 1 0 0 0 0 0 0 V
1 1 1 0 0 0 0 V 0 0
1 1 1 0 0 0 0 0 V 0
1 1 1 0 V 0 0 0 0 V
1 1 1 0 0 0 0 0 0 0
1 1 1 0 0 0 0 V 0 0
0 1 0 0 0 0 V 0 0 0
V S S 0 0 0 0 0 0 0

Evaluation

There is more than one right answer to a topology optimization problem, so SPhyR is scored dynamically: a completion is simulated as a structure and compared against a topology optimizer re-run on the same masked region, rather than against the reference answer cell by cell.

Cell-by-cell comparison asks the wrong question. Across the 24,325 valid completions in the published results, 72.7% are structures that carry their load correctly but that string matching marks wrong — and the converse also happens, as in the example above where 99 of 100 cells match while the one missed cell severs the only load path.

How a completion is scored

1. Rebuild the load case. The L, S and density cells of the sample become a plane-stress finite element model: a unit load spread over the load cells, the support cells clamped, one bilinear (Q4) element per grid cell, and stiffness interpolated from density with the SIMP law E(ρ) = E_min + ρ^p (E_0 − E_min).
2. Simulate the completion. Solving K u = f gives its compliance — the work done by the load, i.e. the inverse of stiffness. A broken load path shows up as a compliance orders of magnitude larger than a solid grid, so whether a structure carries its load is a physical fact rather than a graph heuristic.
3. Re-optimize the same hole. A SIMP optimizer (density filter, optimality criteria update, penalization continuation, discrete swap polish) is re-run on exactly the masked cells, with the rest of the sample pinned and with the reference material budget. The search is seeded with the dataset answer, so the design a completion is measured against is never worse than the answer the dataset ships.
4. Score. How close the completion gets to that design, discounted by any material it spent beyond the reference budget.

Metrics

Metric	Meaning
topology_score	Headline. structural_efficiency × material_efficiency
structural_efficiency	optimal_compliance / compliance, clipped to [0, 1]
material_efficiency	min(1, reference_volume / used_volume)
load_carrying	Does the load actually reach the supports?
compliance	Raw simulated compliance (lower is better)
optimal_compliance	Compliance of the re-optimized structure
compliance_efficiency_vs_ground_truth	Stiffness relative to the dataset answer
volume_ratio	Material used, relative to the dataset answer

Both halves of the headline score are needed: stiffness alone is maximised by filling the grid solid, thrift alone by building nothing. Filling everything solid earns a perfect structural_efficiency and is then discounted to roughly the reference volume fraction; an empty grid scores zero.

Scoring your own completions

pip install numpy scipy

from sphyr.metrics.structural import get_structural_metrics

metrics = get_structural_metrics(
    output_grid=completion_grid,          # list[list[str]], the model's answer
    gt_grid=sample["ground_truth"],
    input_grid=sample["input_grid"],
)

print(metrics.topology_score)   # headline, higher is better
print(metrics.load_carrying)    # does it stand up at all
print(metrics.compliance)       # raw stiffness measure, lower is better

The evaluation is in src/sphyr/metrics/structural.py, and the physics engine behind it is standalone and reusable: fea.py (finite elements), simp.py (topology optimization), boundary_conditions.py (grid → load case).

To check that the metric ranks known-good and known-bad completions the way it should — and that the reference optimizer is never beaten by the answer it is meant to bound:

python -m sphyr.evaluate_dynamic validate --subject full_easy --samples 20

Reconstruction metrics

The cell-by-cell metrics are still computed and reported alongside the dynamic ones, as a measure of how closely a completion reproduces the reference answer — which is a different question from whether it is a sound structure:

Metric	Meaning
exact_match	The completion equals the reference answer everywhere
difference_ratio	Differing cells, relative to the material in the answer
relative_difference_ratio	As above, weighted by how far each cell is off
difficulty_weighted_difference_ratio	As above, weighted by per-cell difficulty

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SPhyR Dataset Card

SPhyR is a benchmark dataset for evaluating the physical and spatial reasoning capabilities of Large Language Models (LLMs) through topology optimization tasks. Given 2D design conditions—boundaries, loads, and supports—models must predict optimal material distributions without physics engines. Tasks include masked region completion and full-structure prediction, testing models’ ability to infer structural stability and material flow.

Dataset Details

Dataset Sources

• Repository: https://github.com/philippds/SPhyR
• Paper [optional]: https://arxiv.org/pdf/2505.16048

Dataset Structure

Legend

• L - Load
• S - Support
• V - Void

Subjects

Easy

Note: Here we use 0 and 1 for material distribution

1_random_cell_easy
5_random_cell_easy
10_random_cell_easy
1_random_row_easy
3_random_row_easy
1_random_column_easy
3_random_column_easy
full_easy

Hard

Note: Here we use floating point numbers 0-1 for material distribution

1_random_cell_hard
5_random_cell_hard
10_random_cell_hard
1_random_row_hard
3_random_row_hard
1_random_column_hard
3_random_column_hard
full_hard

Dataset Creation

Please refer to the dataset repository on GitHub if you want to re-generate the dataset or interested in how this has been done: https://github.com/philippds/SPhyR. We used Rhinoceros with Grasshopper and Milipede plugin to design the structural scenarios and simulated topology optimization.

Citation

BibTeX:

@misc{siedler2025sphyr,
  title        = {SPhyR: Spatial-Physical Reasoning Benchmark on Material Distribution},
  author       = {Philipp D. Siedler},
  year         = {2025},
  eprint       = {2505.16048},
  archivePrefix= {arXiv},
  primaryClass = {cs.AI},
  doi          = {10.48550/arXiv.2505.16048},
  url          = {https://arxiv.org/abs/2505.16048}
}

APA:

Siedler, P. D. (2025). SPhyR: Spatial-Physical Reasoning Benchmark on Material Distribution. arXiv. https://doi.org/10.48550/arXiv.2505.16048

Dataset Card Authors

Philipp D. Siedler

Dataset Card Contact

p.d.siedler@gmail.com