Voigt-Reuss net : A universal approach to microstructure‐property forecasting with physical guarantees

Surrogates for microstructure–property linkages that inherently fulfill the Voigt-Reuss bounds.

This repository contains the code and acts as an extension to the article : "Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure‐property forecasting with physical guarantees" published in GAMM Mitteilungen by Sanath Keshav, Felix Fritzen, and Julius Herb - https://doi.org/10.1002/gamm.70005.

Summary

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: It can be linked to neural networks, kernel methods, or combined schemes. 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 Löwner sense) exist, i.e., for permeability, thermal conductivity, linear elasticity, and many more.

In this GitHub repository, 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.

Installation

The project uses Pixi for dependency management. If you don't have Pixi installed, you can install it using:

curl -fsSL https://pixi.sh/install.sh | sh

To set up your isolated environment with all the required packages installed:

# Clone the repository
git clone https://github.com/DataAnalyticsEngineering/VoigtReussNet.git
cd VoigtReussNet

# Create and activate the environment with all dependencies
pixi install
pixi shell

Download Datasets

The datasets used in the examples are made publicly available on DaRUS. You can download them using the following script:

bash data/download_data.sh

Examples

The examples/ contains a collection of scripts and Jupyter notebooks demonstrating the implementation and evaluation of different neural network approaches for thermal problems:

2D Thermal Problem

• train_vrnn_therm2d.py : Voigt-Reuss network training script for 2D thermal homogenization problem.
• train_vann_therm2d.py : Vanilla neural network training script for 2D thermal homogenization problem.
• eval_therm2d.ipynb : Evaluation and comparison of the trained models.

3D Thermal Problem

• train_vrnn_therm3d.py : Voigt-Reuss network training script for 3D thermal homogenization problem.
• train_vann_therm2d.py : Vanilla neural network training script for 3D thermal homogenization problem.
• eval_therm3d.ipynb.py : Evaluation and comparison of the trained models.

Acknowledgments

• Contributions by Sanath Keshav are supported by the consortium NFDI-MatWerk, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the National Research Data Infrastructure – NFDI 38/1 – project number 460247524.

• Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2075 – 390740016. Contributions by Felix Fritzen are funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Heisenberg program DFGFR2702/8 - 406068690 and DFG-FR2702/10 - 517847245.

• Contributions of Julius Herb are partially funded by the Ministry of Science, Research and the Arts (MWK) Baden-Württemberg, Germany, within the Artificial Intelligence Software Academy (AISA).

• We acknowledge the support by the Stuttgart Center for Simulation Science (SimTech).