# HyperShape: Hyperelasticity Across Diverse Shapes

arXiv: https://arxiv.org/abs/2608.09938 · HTML: https://mlsanity.com/paper/2608.09938/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 16 of 16)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Feature Pyramid Networks for Object Detection](https://mlsanity.com/paper/1612.03144/) (2016)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [Simulation of hyperelastic materials in real-time using Deep Learning](https://mlsanity.com/paper/1904.06197/) (2019)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Soft Tissue Simulation and Force Estimation from Heterogeneous Structures using Equivariant Graph Neural Networks](https://mlsanity.com/paper/2509.10125/) (2025)

## Abstract (arXiv)

Hyperelastic deformations are highly sensitive to domain geometry and boundary conditions, making generalization across both a critical capability for neural operators applied to these problems. However, existing benchmarks for neural operators on hyperelasticity rely on simple or few geometries, which makes it difficult to assess this capability rigorously. To address this gap, we introduce HyperShape, an extensible framework designed to generate synthetic shapes and their corresponding hyperelastic simulation data, producing a suite of 2D and 3D datasets with adjustable complexity and controllable shape variations. This design enables systematic assessment of generalization across in-distribution, out-of-distribution, and synthetic-to-real transfer settings. Using this framework, we evaluated the performance of several state-of-the-art neural operators over diverse shape distributions. Our findings reveal that neural operators perform well on simple shapes but struggle as shape complexity, geometric diversity, and boundary condition variability increase, requiring large amounts of training data in such regimes. Performance degrades consistently and predictably with geometric complexity highlighting the need for further model development. As an open and extensible benchmark, HyperShape is designed to grow alongside the field: new geometries, material models, loading conditions, and evaluation settings can be easily incorporated to validate hyperelastic surrogate models.