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Parametrizing Convex Sets Using Sublinear Neural Networks

arXiv 2605.03520 ↗First posted 5 May 2026math.OC cs.AI cs.LGcited by 0

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Abstract (arXiv)

We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 20 of its 46 references that are in the map. First posted on arXiv 5 May 2026.

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References · 20

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2021Convex optimizationcited by 2,479
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  4. 2016Input Convex Neural Networkscited by 407
  5. 2013Maxout Networkscited by 186
  6. 2019CvxNet: Learnable Convex Decompositioncited by 173
  7. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  8. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  9. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  10. 2022Topology Optimization via Machine Learning and Deep Learning: A Reviewcited by 22
  11. 2023On convex decision regions in deep network representationscited by 7
  12. 2018Parametric Shape Optimization using the Support Functioncited by 3
8 more
  1. 2024Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networkscited by 3
  2. 2018Numerical approximation of optimal convex shapescited by 2
  3. 2022Numerical shape optimization among convex setscited by 2
  4. 2024Optimization of Neumann Eigenvalues under convexity and geometric constraintscited by 2
  5. 2025Improved description of Blaschke--Santal'o diagrams via numerical shape optimizationcited by 2
  6. 2025Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphscited by 2
  7. 2025Geometric properties of optimizers for the maximum gradient of the torsion functioncited by 1
  8. 2026Numerical exploration of the range of shape functionals using neural networkscited by 1

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