# Polyconvex anisotropic hyperelasticity with neural networks

arXiv: https://arxiv.org/abs/2106.14623 · HTML: https://mlsanity.com/paper/2106.14623/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 88 references that are in the map.
- First posted on arXiv 20 Jun 2021.

## Builds on (most cited first, 12 of 12)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity](https://mlsanity.com/paper/2001.04292/) (2020)
- [Data-driven fracture mechanics](https://mlsanity.com/paper/2006.03133/) (2020)
- [Integrating Machine Learning with Physics-Based Modeling](https://mlsanity.com/paper/2006.02619/) (2020)
- [Log-sum-exp neural networks and posynomial models for convex and log-log-convex data](https://mlsanity.com/paper/1806.07850/) (2018)
- [The exponentiated Hencky energy: Anisotropic extension and case studies](https://mlsanity.com/paper/1702.00394/) (2017)
- [Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling](https://mlsanity.com/paper/1904.01521/) (2019)

## Built on by (most cited first, 30 of 55)

- [Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics](https://mlsanity.com/paper/2304.14369/) (2023)
- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Neural Material Adaptor for Visual Grounding of Intrinsic Dynamics](https://mlsanity.com/paper/2410.08257/) (2024)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics](https://mlsanity.com/paper/2310.03652/) (2023)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulation](https://mlsanity.com/paper/2505.16971/) (2025)
- [Geometric deep learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity](https://mlsanity.com/paper/2208.00246/) (2022)
- [Evolution TANN and the identification of internal variables and evolution equations in solid mechanics](https://mlsanity.com/paper/2209.13269/) (2022)
- [Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen](https://mlsanity.com/paper/2308.11080/) (2023)
- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Symmetry-enforcing neural networks with applications to constitutive modeling](https://mlsanity.com/paper/2312.13511/) (2023)
- [Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches](https://mlsanity.com/paper/2503.00575/) (2025)
- [Improving the performance of Stein variational inference through extreme sparsification of physically-constrained neural network models](https://mlsanity.com/paper/2407.00761/) (2024)
- [Neural Networks for Generating Better Local Optima in Topology Optimization](https://mlsanity.com/paper/2407.17957/) (2024)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [Neural Network Layers for Prediction of Positive Definite Elastic Stiffness Tensors](https://mlsanity.com/paper/2203.13938/) (2022)
- [Consistent machine learning for topology optimization with microstructure-dependent neural network material models](https://mlsanity.com/paper/2408.13843/) (2024)
- [Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies](https://mlsanity.com/paper/2502.08534/) (2025)
- [Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteria](https://mlsanity.com/paper/2509.16040/) (2025)

## Abstract (arXiv)

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.