# Deep Convolutional Neural Networks Predict Elasticity Tensors and their Bounds in Homogenization

arXiv: https://arxiv.org/abs/2109.03020 · HTML: https://mlsanity.com/paper/2109.03020/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 65 references that are in the map.
- First posted on arXiv 4 Sep 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [A Survey of the Recent Architectures of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1901.06032/) (2019)
- [Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization](https://mlsanity.com/paper/2002.07600/) (2020)
- [Predicting Porosity, Permeability, and Tortuosity of Porous Media from Images by Deep Learning](https://mlsanity.com/paper/2007.02820/) (2020)
- [Predicting Effective Diffusivity of Porous Media from Images by Deep Learning](https://mlsanity.com/paper/1912.05532/) (2019)
- [Computational Homogenization of Concrete in the Cyber Size-Resolution-Discretization (SRD) Parameter Space](https://mlsanity.com/paper/2103.08957/) (2021)

## Abstract (arXiv)

In the present work, 3D convolutional neural networks (CNNs) are trained to link random heterogeneous, two-phase materials of arbitrary phase fractions to their elastic macroscale stiffness thus replacing explicit homogenization simulations. In order to reduce the uncertainty of the true stiffness of the synthetic composites due to unknown boundary conditions (BCs), the CNNs predict beyond the stiffness for periodic BC the upper bound through kinematically uniform BC, and the lower bound through stress uniform BC. This work describes the workflow of the homogenization-CNN, from microstructure generation over the CNN design, the operations of convolution, nonlinear activation and pooling as well as training and validation along with backpropagation up to performance measurements in tests. Therein the CNNs demonstrate the predictive accuracy not only for the standard test set but also for samples of the real, two-phase microstructure of a diamond-based coating. The CNN that covers all three boundary types is virtually as accurate as the separate treatment in three different nets. The CNNs of this contribution provide through stiffness bounds an indicator of the proper RVE size for individual snapshot samples. Moreover, they enable statistical analyses for the effective elastic stiffness on ensembles of synthetical microstructures without costly simulations.