Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure-property forecasting with physical guarantees

Heterogeneous materials are crucial to producing lightweight components, functional components, and structures composed of them. A crucial step in the design process is the rapid evaluation of their effective mechanical, thermal, or, in general, constitutive properties. The established procedure is to use forward models that accept microstructure geometry and local constitutive properties as inputs. The classical simulation-based approach, which uses, e.g., finite elements and FFT-based solvers, can require substantial computational resources. At the same time, simulation-based models struggle to provide gradients with respect to the microstructure and the constitutive parameters. Such gradients are, however, of paramount importance for microstructure design and for inverting the microstructure-property mapping. Machine learning surrogates can excel in these situations. However, they can lead to unphysical predictions that violate essential bounds on the constitutive response, such as the upper (Voigt-like) or the lower (Reuss-like) bound in linear elasticity. Therefore, we propose a novel spectral normalization scheme that a priori enforces these bounds. The approach is fully agnostic with respect to the chosen microstructural features and the utilized surrogate model. All of these will automatically and strictly predict outputs that obey the upper and lower bounds by construction. The technique can be used for any constitutive tensor that is symmetric and where upper and lower bounds (in the L\"owner sense) exist, i.e., for permeability, thermal conductivity, linear elasticity, and many more. We demonstrate the use of spectral normalization in the Voigt-Reuss net using a simple neural network. Numerical examples on truly extensive datasets illustrate the improved accuracy, robustness, and independence of the type of input features in comparison to much-used neural networks.

Hybrid machine-learned homogenization: Bayesian data mining and convolutional neural networks

Beyond the generally deployed features for microstructure property prediction this study aims to improve the machine learned prediction by developing novel feature descriptors. Therefore, Bayesian infused data mining is conducted to acquire samples containing characteristics inexplicable to the current feature set, and suitable feature descriptors to describe these characteristics are proposed. The iterative development of feature descriptors resulted in 37 novel features, being able to reduce the prediction error by roughly one third. To further improve the predictive model, convolutional neural networks (Conv Nets) are deployed to generate auxiliary features in a supervised machine learning manner. The Conv Nets were able to outperform the feature based approach. A key ingredient for that is a newly proposed data augmentation scheme and the development of so-called deep inception modules. A combination of the feature based approach and the convolutional neural network leads to a hybrid neural network: A parallel deployment of the both neural network archetypes in a single model achieved a relative rooted mean squared error below 1%, more than halving the error compared to prior models operating on the same data. The hybrid neural network was found powerful enough to be extended to predict variable material parameters, from a low to high phase contrast, while allowing for arbitrary microstructure geometry at the same time.

Data-Driven Microstructure Property Relations

An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data with the main emphasis being put on the 2-point spatial correlation function. This task is implemented using both unsupervised and supervised machine learning methods. First, a snapshot proper orthogonal decomposition (POD) is used to analyze big sets of random microstructures and thereafter compress significant characteristics of the microstructure into a low-dimensional feature vector. In order to manage the related amount of data and computations, three different incremental snapshot POD methods are proposed. In the second step, the obtained feature vector is used to predict the effective material property by using feed forward neural networks. Numerical examples regarding the incremental basis identification and the prediction accuracy of the approach are presented. A Python code illustrating the application of the surrogate is freely available.