# Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization

arXiv: https://arxiv.org/abs/2610.12801 · HTML: https://mlsanity.com/paper/2610.12801/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 55 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 13 of 13)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [Neural networks meet hyperelasticity: A guide to enforcing physics](https://mlsanity.com/paper/2302.02403/) (2023)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Neural networks meet anisotropic hyperelasticity: A framework based on generalized structure tensors and isotropic tensor functions](https://mlsanity.com/paper/2410.03378/) (2024)
- [Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases](https://mlsanity.com/paper/2008.02891/) (2020)
- [Hyper-reduction for Petrov-Galerkin reduced order models](https://mlsanity.com/paper/2309.16267/) (2023)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)
- [Advances in polyconvex anisotropic hyperelasticity](https://mlsanity.com/paper/2605.27011/) (2026)
- [Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights](https://mlsanity.com/paper/2609.03068/) (2026)

## Abstract (arXiv)

Approximating the effective anisotropic hyperelastic response of a representative volume element (RVE) requires constitutive expressiveness, mechanical consistency, and computational efficiency. This work develops a polyconvex physics-augmented neural network (PANN) construction based on paired directional features whose orientations, stretch powers, and determinant exponents adapt to the RVE response. The features are composed with convex, componentwise nondecreasing cores, realized as an input-convex neural network (ICNN) or an input-convex Kolmogorov-Arnold network (ICKAN). The complete energies satisfy objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth conditions by construction, and nonnegative energy is certified separately for the selected models. On a multicavity and a single-cavity periodic cell, these energies are compared with an Unconstrained energy, trained identically without the curvature and growth constraints, and with three hyperreduced projection-based reduced-order models (HPROMs), which serve as physics-based references. On the first cell, learning the features lowers the stress error sevenfold, and six features suffice on both. The constrained energies remain below one percent in stress error and in the displacement error of a tensile-coupon FE2 application with a full-order reference, whereas the Unconstrained energy and the HPROMs are more accurate inside the data. Beyond the data, however, the Unconstrained energy loses rank-one convexity, and under confined compression it stops below half of the 500 MPa that the constrained energies reach. These results quantify what the guarantees cost in accuracy and what they secure beyond the data.