Sublinear Neural Networks for Convex Shape Optimization

Implementation of the paper: “Parametrizing Convex Sets Using Sublinear Neural Networks”

This repository provides code to reproduce the numerical experiments from the paper, including:

• Shape reconstruction
• PDE-constrained shape optimization
• Mahler volume minimization
• Torsion-based optimization (MFS, Galerkin, PINNs)
• Minkowski problem

The method relies on sublinear neural networks to parameterize convex sets and uses automatic differentiation for computing geometric and PDE-related quantities .

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📦 Installation

Clone the repository and install dependencies:

git clone https://github.com/EloiMartinet/SublinearNet.git
cd SublinearNet
pip install -e .

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🚀 Running Experiments

All experiments are located in the scripts/ folder. Each experiment corresponds to a single script, except for the torsion benchmark.

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1. Shape reconstruction

📍 Paper: Section 4.1 – Learning convex sets with noisy boundary samples

python scripts/fit_noisy_single.py

This reproduces the optimization of shapes minimizing:

$$ L(\theta) = \sum_{i=1}^n |p_\theta(y_i) - 1|^2 $$

where $p_\theta$ is the gauge function of a convex and the $y_i$ are noisy measurements of the boundary of a convex set.

The statistical experiments (Appendix F) can be run with

python scripts/fit_noisy_number_of_points.py

and

python scripts/fit_noisy_different_noises.py

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2. Poisson Shape Optimization

📍 Paper: Section 4.2 – Optimization of a Poisson problem

python scripts/poisson_galerkin.py

This reproduces the optimization of shapes minimizing:

$$ J(\Omega) = \int_\Omega u_\Omega $$

where $u_\Omega$ solves a Poisson equation.

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3. Mahler Volume Minimization

📍 Paper: Appendix E – Minimization of the Mahler volume

python scripts/mahler_volume.py

This experiment minimizes:

$$ \mathrm{Vol}(\Omega)\mathrm{Vol}(\Omega^\circ) $$

among shapes satisfying certain symmetries.

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4. Optimization of the gradient of the torsion function

📍 Paper: Section 4.3 – Maximization of the gradient of the torsion function

python scripts/max_torsion.py

This script explores torsion-based optimization objectives (related to Section 4.3).

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5. Minkowski Problem

📍 Paper: Section 4.4 – Minkowski problem

python scripts/minkovski_problem.py

Solves a curvature prescription problem via neural parametrization.

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6. Comparison with PINNs

📍 Paper: Appendix D – Comparison with PINNs

Maximizes the torsion function by three different methods:

• Method of Fundamental Solutions (MFS)
• Mesh-free Galerkin
• Physics-Informed Neural Networks (PINNs)

✅ Run all experiments

python scripts/run_torsion_experiments.py

This will automatically:

1. Run torsion_MFS.py

2. Run torsion_Galerkin.py

3. Run torsion_PINN.py with:

   • config_1
   • config_2
   • config_3

4. Generate plots via plot_torsion_experiments.py

⚙️ Run individually

MFS

python scripts/torsion_MFS.py

Galerkin

python scripts/torsion_Galerkin.py

PINNs

python scripts/torsion_PINN.py --config config_1
python scripts/torsion_PINN.py --config config_2
python scripts/torsion_PINN.py --config config_3

👉 The three configs correspond to different hyperparameter settings used in the PINN comparison (see paper).

📊 Plotting results

python scripts/plot_torsion_experiments.py

⚠️ Requires running the experiments first.

🔍 Hyperparameter Search (optional)

python scripts/torsion_parameter_search.py

This reproduces the PINN hyperparameter search described in the paper.

📁 Output Files

Results are saved in:

res/
    csvs/                # numerical results
    torsion_Galerkin/    # Galerkin outputs
    torsion_MFS/         # MFS outputs

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📖 Citation

If you use this code in you own research, please cite:

@misc{martinet2026parametrizingconvexsetsusing,
      title={Parametrizing Convex Sets Using Sublinear Neural Networks}, 
      author={Eloi Martinet},
      year={2026},
      eprint={2605.03520},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2605.03520}, 
}