Neural Network Layers for Prediction of Positive Definite Elastic Stiffness Tensors

TL;DR

This paper introduces transformation layers based on Cholesky and eigendecomposition to ensure neural network predictions of elasticity tensors remain symmetric positive definite, improving accuracy in modeling homogenized stiffness tensors.

Contribution

It develops SPD-enforcing transformation layers for machine learning models, demonstrating their effectiveness across various models in predicting elasticity tensors.

Findings

SPD layers improve model accuracy

Different positivity functions impact performance

Neural networks benefit from SPD enforcement

Abstract

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.