MICRO2D: A Large, Statistically Diverse, Heterogeneous Microstructure Dataset

Abstract

The availability of large, diverse datasets has enabled transformative advances in a wide variety of technical fields by unlocking data scientific and machine learning techniques. In Materials Informatics for Heterogeneous Microstructures capitalization on these techniques has been limited due to the extreme complexity of generating or curating sizeable heterogeneous microstructure datasets. Historically, this difficulty can be attributed to two main hurdles: quantification (i.e., measuring microstructure diversity) and curation (i.e., generating diverse microstructures). In this paper, we present a framework for curating large, statistically diverse mesoscale microstructure datasets composed of 2-phase microstructures. The framework generates microstructures which are statistically diverse with respect to their n-point statistics—the primary emphasis is on diversity in their 2-point statistics. The framework’s foundation is a proposed set of algorithms for synthesizing salient 2-point statistics and neighborhood distributions. We generate statistically diverse microstructures by using the outputs of these algorithms as inputs to a statistically conditioned Local-Global Decomposition generation procedure. Finally, we demonstrate the proposed framework by curating MICRO2D, a diverse, large-scale, and open source heterogeneous microstructure dataset comprised of 87, 379 2-phase microstructures. The contained microstructures are periodic and $256 \times 256$ pixels. The dataset also contains salient homogenized elastic and thermal properties computed across a range of constituent contrast ratios for each microstructure. Using MICRO2D, we analyze the statistical and property diversity achievable via the proposed framework. We conclude by discussing important areas of future research in microstructure dataset curation.

Appendices

Appendix A: Real Space Mixtures

The mixture method adopted in this paper differs slightly from the approach adopted historically in Spectral Mixture-based Gaussian Process Regression modeling [97, 98]. Specifically, instead of constructing the kernel function via a mixture model approximation to a probability density in the frequency space (e.g., a Symmetric Gaussian mixture model [97]), we approximate the kernel using a real-space symmetric Gaussian mixture model and, subsequently, enforce the spectral requirements of the kernel function via two linear projections. This approach takes the following path. The approximate kernel function, ${\hat{k}}(\varvec{\tau })$, is constructed via a mixture of symmetric Gaussians.

$$\begin{aligned} {\hat{k}}(\varvec{\tau })&= \sum _{i=1}^M \frac{\alpha _i}{2} \left[ \phi (\varvec{\tau }; \varvec{\mu }_i, \varvec{\Sigma }_i) + \phi (\varvec{\tau }; -\varvec{\mu }_i, \varvec{\Sigma }_i) \right] \end{aligned}$$

(11)

$$\begin{aligned} \phi (\varvec{\tau }; \varvec{\mu }, \varvec{\Sigma })&= \left( 2^k \pi ^k |\varvec{\Sigma }| \right) ^{-1/2} \exp \left( -0.5 \left( \varvec{\tau } - \varvec{\mu } \right) ^T \varvec{\Sigma }^{-1} \left( \varvec{\tau } - \varvec{\mu } \right) \right) \end{aligned}$$

(12)

Here, $\varvec{\mu }$ and $\varvec{\Sigma }$ are the mean and covariances of each mixture. $|\cdot |$ is the determinant operator. The mixture weights, $\alpha _i$, are selected to add to unity.

An approximate kernel structure of this form produces the following expression when transformed into frequency space [142].

$$\begin{aligned} {\mathcal {F}}({\hat{k}}(\varvec{\tau }))(\varvec{\xi })&\propto \sum _{i=1}^M \left[ \exp \left( -i \varvec{\mu }_i^T \varvec{\xi } \right) + \exp \left( i \varvec{\mu }_i^T \varvec{\xi } \right) \right] \exp \left( - \varvec{\xi }^T \varvec{\Sigma }_i \varvec{\xi } \right) \end{aligned}$$

(13)

$$\begin{aligned}&\propto \sum _{i=1}^M \cos \left( \varvec{\mu }_i^T \varvec{\xi } \right) \exp \left( - \varvec{\xi }^T \varvec{\Sigma }_i \varvec{\xi } \right) \end{aligned}$$

(14)

Here, superscripts refer to exponentiation not indexing. Note that this produces the inverse of the kernel structure proposed by Wilson and Adams [97]—with the cosine fluctuations in the frequency space instead of the real space. This kernel structure meets only one of the minimum requirements outlined in Sects. 2.2 and 2.1: it is real valued. The presence of the cosine fluctuations introduces negative values in the spectrum. These fluctuations are removed by zeroing the negative values [92].

$$\begin{aligned} {\mathcal {F}}(k(\varvec{\tau }))(\varvec{\xi }) = \max \left( {\mathcal {F}}({\hat{k}}(\varvec{\tau }))(\varvec{\xi }), \epsilon \right) \end{aligned}$$

(15)

Here, $\epsilon$ is a near zero, positive value added for computational stability of the subsequent steps. Finally, the generated kernel function is produced by applying the inverse cosine transform, ${\mathcal {C}}^{-1}[\cdot ]$. This operation returns the real space equivalent of the kernel without introducing spurious imaginary components in either the real or Fourier space representation of the kernel. In practice, we discretely sample the approximate covariance kernel, ${\hat{k}}(\varvec{\tau })$, in real space to a discrete covariance kernel, ${\hat{k}}_r$, and, subsequently, apply the two identified projections discretely. This procedure produces the following set of expressions.

$$\begin{aligned} {\hat{k}}(\varvec{\tau })&= \sum _{i=1}^M \frac{\alpha _i}{2} \left[ \phi (\varvec{\tau }; \varvec{\mu }_i, \varvec{\Sigma }_i) + \phi (\varvec{\tau }; -\varvec{\mu }_i, \varvec{\Sigma }_i) \right] \end{aligned}$$

(16)

$$\begin{aligned} {\hat{k}}_r&= {\hat{k}}(\varvec{\tau }_r) \end{aligned}$$

(17)

$$\begin{aligned} k_r&= {\mathcal {C}}^{-1} [ \max \left( {\mathcal {F}}[{\hat{k}}_r]_t, \epsilon \right) ]_r \end{aligned}$$

(18)

Here, $\varvec{\tau }_r$ is the value of $\varvec{\tau }$ at the center of pixel r. The autocorrelation is derived from the kernel function via addition of the mean squared [63].

$$\begin{aligned} f^{\beta \beta }_r = k_r + (v_{f}^{\beta })^2 \end{aligned}$$

(19)

As noted in the main body of the paper, we used this alternative structure instead of the traditional method for two important reasons. Most importantly, the traditional Fourier-space mixture model produces spatially compact real-space kernels. This means that it cannot easily produce kernels with multiple modes. Mathematically, this is clear in the original real-space expression provided by Wilson and Adams [97] (here, for simplicity, we reproduce the 1D single mixture expression).

$$\begin{aligned} k(\tau ) = \exp (-2 \pi ^2 \tau ^2 \sigma ^2) \cos (2 \pi \tau \mu ) \end{aligned}$$

(20)

Clearly, the dominant exponential term has zero mean. As a result, this type of kernel cannot easily reproduce the important longer range peaks (for example, secondary peaks in layered composites that statistically represent the repetition of the layering [63]) that are present in 2-point statistics maps [90, 94, 123]. In contrast, our real-space formulation can directly construct these secondary peaks via the direct placement of the means of the individual symmetric mixtures. The second reason is practical and an extension of the first: in real space, we can use our expert knowledge of 2-point statistics [21, 72, 88, 90, 92, 93, 123] to guide the placement of the symmetric mixtures into common regions. The unfamiliar nature of the Fourier representation makes it challenging to embed domain knowledge into the construction of the kernel function (and, by extension, the autocorrelation). Of course, this second reason would be irrelevant if one was using an optimization-based placement strategy for the mixtures instead of an expert driven one.

Appendix B: PCA Truncation for MaxPro Filtering

We used PCA to perform distance preserving dimensionality reduction of the initial candidate autocorrelation set. This facilitated the framework’s spacefilling filtering operation by significantly decreasing the computational expense of the Min–Max optimization central to the MaxPro algorithm [113]. Importantly, the extracted latent space must be a good approximation of the original 2-point statistics. Therefore, it must have sufficient representational capacity to recreate the original autocorrelations’ salient features with high fidelity. Recent work by Generale et al. observed that PCA is relatively inefficient for generative tasks like this one [21]. As a result, we expect the number of necessary principal components to be quite high. We selected the truncation level for the number of principal components by tracking the reconstruction error (the relative $L_2$ error—i.e., the $L_2$ distance between a proposed autocorrelation and the reconstruction from the principal component basis normalized by the $L_2$ magnitude of the original autocorrelation) of the original candidate autocorrelation dataset. Figure 14 summarizes the relative error. We selected a truncation level of 750, achieving an average reconstruction error of approximately $0.75 \%$.

Fig. 14

Trend of the average reconstruction error (the relative $L_2$ error) as a function of the number of retained principal components

Fig. 15

Trend of the expected feature size as a function of the principal component index

Additionally, we also explored the average feature size of stochastic microstructure functions differentiated by each eigenvector. Here, our aim was to ensure that the selected principal component basis was sensitive to every feature lengthscale in the initial candidate autocorrelation set. This is important to check because PCA is known to filter out short lengthscale (i.e., high frequency) features [112, 143,144,145]. For several eigenvectors, we identified a set of diverse autocorrelations with respect to the selected eigenvector. We did so by performing spacefilling in just the subspace defined by that eigenvector using the described MaxPro procedure. Then, we used Berryman’s method [63, 146, 147] to estimate the feature size of microstructures corresponding to the identified autocorrelations. Figure 15 depicts the trend. Importantly, the decay largely stabilizes well before the 750th eigenvector. Therefore, we expect this selected cutoff to create a subspace which is sufficiently sensitive to all salient lengthscales in the candidate autocorrelation set. We also note that the trend displays largely monotonic decay—i.e., lower index eigenvectors correspond to larger features.

Appendix C: Examples from MICRO2D

Here, we simply display a collection of randomly selected examples from each class in the MICRO2D dataset: GRF—Fig. 16, NBSA—Fig. 17, AngEllipse—Fig. 18, RandomEllipse—Fig. 19, VoidSmall—Fig. 20, VoidSmallBig—Fig. 21, VoronoiLarge—Fig. 22, VoronoiMedium—Fig. 23, VoronoiMediumSpaced—Fig. 24, and VoronoiSmall—Fig. 25.

Fig. 16

Eighty randomly selected microstructures corresponding to the GRF class

Fig. 17

Eighty randomly selected microstructures corresponding to the NBSA class

Fig. 18

Eighty randomly selected microstructures corresponding to the AngEllipse class

Fig. 19

Eighty randomly selected microstructures corresponding to the RandomEllipse class

Fig. 20

Eighty randomly selected microstructures corresponding to the VoidSmall class

Fig. 21

Eighty randomly selected microstructures corresponding to the VoidSmallBig class

Fig. 22

Eighty randomly selected microstructures corresponding to the VoronoiLarge class

Fig. 23

Eighty randomly selected microstructures corresponding to the VoronoiMedium class

Fig. 24

Eighty randomly selected microstructures corresponding to the VoronoiMediumSpaced class

Fig. 25

Eighty randomly selected microstructures corresponding to the VoronoiSmall class