Generative adversarial framework to calibrate excursion set models for the 3D morphology of all-solid-state battery cathodes

TL;DR

The paper tackles calibrating 3D multiphase morphologies from limited 2D microscopy data by coupling excursion-set stochastic geometry with generative-adversarial calibration to create digital twins of all-solid-state battery cathodes. It develops isotropic and cylindrically isotropic three-phase models built from Gaussian and chi-square random fields, including high- and low-parametric variants, and validates them against 2D plane sections and 3D data. A combined training approach that integrates two-point coverage statistics with a neural discriminator improves realism and transferability for virtual materials testing. This framework enables systematic exploration of morphologies and their macroscopic properties, with potential applicability to other materials sharing similar three-phase microstructures and to accelerated materials design workflows using tools like GeoDict.

Abstract

This paper presents a computational method for generating virtual 3D morphologies of functional materials using low-parametric stochastic geometry models, i.e., digital twins, calibrated with 2D microscopy images. These digital twins allow systematic parameter variations to simulate various morphologies, that can be deployed for virtual materials testing by means of spatially resolved numerical simulations of macroscopic properties. Generative adversarial networks (GANs) have gained popularity for calibrating models to generate realistic 3D morphologies. However, GANs often comprise of numerous uninterpretable parameters make systematic variation of morphologies for virtual materials testing challenging. In contrast, low-parametric stochastic geometry models (e.g., based on Gaussian random fields) enable targeted variation but may struggle to mimic complex morphologies. Combining GANs with advanced stochastic geometry models (e.g., excursion sets of more general random fields) addresses these limitations, allowing model calibration solely from 2D image data. This approach is demonstrated by generating a digital twin of all-solid-state battery (ASSB) cathodes. Since the digital twins are parametric, they support systematic exploration of structural scenarios and their macroscopic properties. The proposed method facilitates simulation studies for optimizing 3D morphologies, benefiting not only ASSB cathodes but also other materials with similar structures.

Figures (8)

• Figure 1: 2D sections of segmented (experimentally measured) 3D image data, parallel to the $y$--$z$ plane (a) and the $x$--$y$ plane (b). Both sub-figures use the same length scale. The pore space, active material and the solid electrolyte are represented by black, red and gray color, respectively.
• Figure 2: Computational scheme for mapping the parameter vector $\theta$ onto a realization $\Xi_\theta$ and a corresponding estimate of the two-point coverage probability functions $\widehat{C}_{ij,\theta}$ (first and second rows). Arrows visualized in black indicate operations that are differentiable with respect to $\theta$. The red arrow indicates a non-differentiable operation, i.e., the thresholding performed in Eq. (\ref{['eq:XiOneXiTwo']}) for computing excursion sets is non differentiable. As an alternative a differentiable approximation is proposed (third row).
• Figure 3: 2D model realizations of a model trained with Algorithm \ref{['alg:training_procedure']} (a), a model trained with Algorithm \ref{['alg:training_procedure_gan']} (b) and a model trained with Algorithm \ref{['alg:training_procedure_gan_combined']} (c). The pore space, active material and the solid electrolyte are represented by black, red and gray color, respectively. All figures use the same length scale.
• Figure 4: Visualization of the discriminator's network architecture. The $\alpha$ parameter of the LeakyReLU layers is set to 0.2, see Maas2013RectifierNI for further details on LeakyReLu layers. The labels above convolutional layers (Conv) indicate the kernel size (k), the number of feature maps (n) and the stride (s), see Goodfellow2016 for more details on the deployed layers and their parameters. For example, the label k4n64s2 indicates a convolutional layer with a kernel size of 9, 64 feature maps and a stride of 1.
• Figure 5: 2D visualization of tomographic image data (a), and of microstructures generated by $\Xi^{\mathrm{HP}}$ (b) and $\Xi^{\mathrm{LP}}$ (c). The pore space, active material and the solid electrolyte are represented by black, red and gray color, respectively. All figures use the same length scale.