# A deep learning driven pseudospectral PCE based FFT homogenization algorithm for complex microstructures

arXiv: https://arxiv.org/abs/2110.13440 · HTML: https://mlsanity.com/paper/2110.13440/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 67 references that are in the map.
- First posted on arXiv 26 Oct 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Dive into Deep Learning](https://mlsanity.com/paper/2106.11342/) (2021)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization](https://mlsanity.com/paper/2002.07600/) (2020)
- [Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet](https://mlsanity.com/paper/2104.01996/) (2021)

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

- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Three-dimensional microstructure generation using generative adversarial neural networks in the context of continuum micromechanics](https://mlsanity.com/paper/2206.01693/) (2022)
- [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022)

## Abstract (arXiv)

This work is directed to uncertainty quantification of homogenized effective properties for composite materials with complex, three dimensional microstructure. The uncertainties arise in the material parameters of the single constituents as well as in the fiber volume fraction. They are taken into account by multivariate random variables. Uncertainty quantification is achieved by an efficient surrogate model based on pseudospectral polynomial chaos expansion and artificial neural networks. An artificial neural network is trained on synthetic binary voxelized unit cells of composite materials with uncertain three dimensional microstructures, uncertain linear elastic material parameters and different loading directions. The prediction goals of the artificial neural network are the corresponding effective components of the elasticity tensor, where the labels for training are generated via a fast Fourier transform based numerical homogenization method. The trained artificial neural network is then used as a deterministic solver for a pseudospectral polynomial chaos expansion based surrogate model to achieve the corresponding statistics of the effective properties. Three numerical examples deal with the comparison of the presented method to the literature as well as the application to different microstructures. It is shown, that the proposed method is able to predict central moments of interest while being magnitudes faster to evaluate than traditional approaches.