# Neural Network Layers for Prediction of Positive Definite Elastic Stiffness Tensors

arXiv: https://arxiv.org/abs/2203.13938 · HTML: https://mlsanity.com/paper/2203.13938/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 25 Mar 2022.

## Builds on (most cited first, 5 of 5)

- [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)
- [Learning-based Robust Optimization: Procedures and Statistical Guarantees](https://mlsanity.com/paper/1704.04342/) (2017)

## Built on by (most cited first, 3 of 3)

- [Energy-conserving equivariant GNN for elasticity of lattice architected metamaterials](https://mlsanity.com/paper/2401.16914/) (2024)
- [Gaussian-Mixture-Model Q-Functions for Policy Iteration in Reinforcement Learning](https://mlsanity.com/paper/2512.18763/) (2025)
- [Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learning](https://mlsanity.com/paper/2608.24386/) (2026)

## Abstract (arXiv)

Machine learning models can be used to predict physical quantities like homogenized elasticity stiffness tensors, which must always be symmetric positive definite (SPD) based on conservation arguments. Two datasets of homogenized elasticity tensors of lattice materials are presented as examples, where it is desired to obtain models that map unit cell geometric and material parameters to their homogenized stiffness. Fitting a model to SPD data does not guarantee the model's predictions will remain SPD. Existing Cholsesky factorization and Eigendecomposition schemes are abstracted in this work as transformation layers which enforce the SPD condition. These layers can be included in many popular machine learning models to enforce SPD behavior. This work investigates the effects that different positivity functions have on the layers and how their inclusion affects model accuracy. Commonly used models are considered, including polynomials, radial basis functions, and neural networks. Ultimately it is shown that a single SPD layer improves the model's average prediction accuracy.