Quantifying microstructures of Earth materials: Reconstructing higher-order correlation functions using deep generative adversarial networks

It is well established that effective and macroscopic properties of geological materials are controlled by the geometry and physical properties at small scales, i.e., by their microstructures. Recent progress in imaging technology has enabled us to visualize and characterize the microstructures at different length scales and dimensions. As Earth materials are often heterogeneous with a certain degree of randomness, such a characterization must be of statistical nature – and one approach to this end is performed by computing n-point correlation functions known as statistical microstructural descriptors. These microstructural descriptors can, in principle, then be directly employed in upscaling to predict the macroscopic behaviours of the system as a whole. Alternatively, once microstructural descriptors are inferred from one or more samples, they can be used to generate new, statistically-equivalent structures having a larger size and additional dimension – this process is known as reconstruction. While several approaches have been proposed in the past decades, advanced machine-learning based image processing methods have shown to be promising for reconstructing microstructures of chosen representative sizes. Here, we train a deep-convolutional generative adversarial network (GAN) to reconstruct two-dimensional electron microscopy images of two chemically-altered rock samples. We show that employing a Wasserstein-loss with a gradient penalty, instead of common binary cross entropy, results in improved training stability and high-quality reconstructed microstructures. To quantitatively evaluate how reconstruction performs in retrieving patterns with high-order spatial correlations, n-point polytope functions are calculated in both reconstructed and original microstructures, and mean square error (MSE) between them is used as a quality metric. These n-point polytope functions, which are a subset of n-point correlation functions, provide statistical information about symmetric higher-order geometrical patterns in microstructures. Furthermore, we compare our model with a benchmark reconstruction method based on a two-point correlation function and stochastic optimization by simulated annealing (SA). Our findings indicate that although showing the same two-point statistics, two microstructures can be morphologically and structurally different, emphasizing the need for coupling higher-order correlation functions with reconstruction methods. We also show that GANs are naturally able to capture higher-order correlation functions at short and long range scales due to the convolutional layers which can learn to extract complex structural features, leading to realistic image reconstructions. This is of critical importance for future schemes that aim to exceed the limits of current imaging techniques by reconstructing the higher-order geometry in complex heterogeneous systems and couple microstructures to macroscopic phenomena.