# Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen

arXiv: https://arxiv.org/abs/2308.11080 · HTML: https://mlsanity.com/paper/2308.11080/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 71 references that are in the map.
- First posted on arXiv 21 Aug 2023.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity](https://mlsanity.com/paper/2001.04292/) (2020)
- [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)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [Discovering plasticity models without stress data](https://mlsanity.com/paper/2202.04916/) (2022)
- [A machine learning based plasticity model using proper orthogonal decomposition](https://mlsanity.com/paper/2001.03438/) (2020)
- [On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling](https://mlsanity.com/paper/2109.11028/) (2021)
- [Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties](https://mlsanity.com/paper/2203.07422/) (2022)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [Exploring the 3D architectures of deep material network in data-driven multiscale mechanics](https://mlsanity.com/paper/1901.04832/) (2019)
- [Automatic generation of interpretable hyperelastic material models by symbolic regression](https://mlsanity.com/paper/2206.15099/) (2022)
- [Tensor Basis Gaussian Process Models of Hyperelastic Materials](https://mlsanity.com/paper/1912.10872/) (2019)
- [A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation](https://mlsanity.com/paper/1903.04307/) (2019)
- [Model-Free Data-Driven Methods in Mechanics: Material Data Identification and Solvers](https://mlsanity.com/paper/1903.07983/) (2019)

## Built on by (most cited first, 9 of 9)

- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics](https://mlsanity.com/paper/2310.03652/) (2023)
- [Structure-Preserving Digital Twins via Conditional Neural Whitney Forms](https://mlsanity.com/paper/2508.06981/) (2025)
- [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)
- [Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)](https://mlsanity.com/paper/2604.07746/) (2026)
- [Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response](https://mlsanity.com/paper/2404.17584/) (2024)
- [Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models](https://mlsanity.com/paper/2601.03367/) (2026)
- [Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling](https://mlsanity.com/paper/2607.20339/) (2026)

## Abstract (arXiv)

Data-driven constitutive modeling frameworks based on neural networks and classical representation theorems have recently gained considerable attention due to their ability to easily incorporate constitutive constraints and their excellent generalization performance. In these models, the stress prediction follows from a linear combination of invariant-dependent coefficient functions and known tensor basis generators. However, thus far the formulations have been limited to stress representations based on the classical Rivlin and Ericksen form, while the performance of alternative representations has yet to be investigated. In this work, we survey a variety of tensor basis neural network models for modeling hyperelastic materials in a finite deformation context, including a number of so far unexplored formulations which use theoretically equivalent invariants and generators to Finger-Rivlin-Ericksen. Furthermore, we compare potential-based and coefficient-based approaches, as well as different calibration techniques. Nine variants are tested against both noisy and noiseless datasets for three different materials. Theoretical and practical insights into the performance of each formulation are given.