Abstract
The generation of multiphase porous electrode microstructures is a critical step in the optimisation of electrochemical energy storage devices. This work implements a deep convolutional generative adversarial network (DCGAN) for generating realistic nphase microstructural data. The same network architecture is successfully applied to two very different threephase microstructures: A lithiumion battery cathode and a solid oxide fuel cell anode. A comparison between the real and synthetic data is performed in terms of the morphological properties (volume fraction, specific surface area, triplephase boundary) and transport properties (relative diffusivity), as well as the twopoint correlation function. The results show excellent agreement between datasets and they are also visually indistinguishable. By modifying the input to the generator, we show that it is possible to generate microstructure with periodic boundaries in all three directions. This has the potential to significantly reduce the simulated volume required to be considered “representative” and therefore massively reduce the computational cost of the electrochemical simulations necessary to predict the performance of a particular microstructure during optimisation.
Introduction
The geometrical properties of multiphase materials are of central importance to a wide variety of engineering disciplines. For example, the distribution of precious metal catalysts on porous supports; the structure of metallic phases and defects in highperformance alloys; the arrangement of sand, organic matter, and water in soil science; and the distribution of calcium, collagen and blood vessels in bone^{1,2,3}. In electrochemistry, whether we are considering batteries, fuel cells or supercapacitors, their electrodes are typically porous to maximise surface area but need to contain percolating paths for the transport of both electrons and ions, as well as maintaining sufficient mechanical integrity^{4,5}. Thus the microstructure of these electrodes significantly impacts their performance and their morphological optimisation is vital for developing the next generation of energy storage technologies^{6}.
Recent improvements in 3D imaging techniques such as Xray computed tomography (XCT) have allowed researchers to view the microstructure of porous materials at sufficient resolution to extract relevant metrics^{7,8,9,10}. However, a variety of challenges remain, including how to extract the key metrics or “essence“ of an observed microstructural dataset such that synthetic volumes with equivalent properties can be generated, and how to modify specific attributes of this microstructural data without compromising its overall resemblance to the real material.
A wide variety of methods that consist of generating synthetic microstructure by numerical means have been developed to solve these challenges^{6}. A statistical method for generating synthetic threedimensional porous media based on distance correlation functions was introduced by Quiblier et al.^{11}. Following this work, Torquato et al. implemented a stochastic approach based on the npoint correlation functions for generating reconstructions of heterogeneous materials.^{12,13,14,15,16} Jiao et al.^{17,18} extended this method to present an isotropypreserving algorithm to generate realisations of materials from their twopoint correlation functions (TPCF). Based on these previous works, the most widely used approach for reconstruction of microstructure implements statistical methods through the calculation of the TPCF^{2,19,20,21,22,23,24}.
In the area of energy materials, interest has recently surged for generating synthetic microstructure in order to aid the design of optimised electrodes. The threephase nature of most electrochemical materials adds an extra level of complexity to their generation compared to twophase materials. Suzue et al.^{20} implemented a TPCF from a twodimensional phase map to reconstruct a threedimensional microstructure of a porous composite anode. Baniassadi et al.^{25} extended this method by adding a combined Monte Carlo simulation with a kinetic growth model to generate threephase realisations of a SOFC electrode. Alternative algorithms for reconstruction of porous electrodes have been inspired by their experimental fabrication techniques. A stochastic algorithm based on the process of nucleation and grain growth was developed by Siddique et al.^{26} for reconstructing a threedimensional fuel cell catalyst layer. This process was later implemented by Siddique et al.^{27} to reconstruct a threedimensional threephase LiFePO4 cathode. A common approach for generating synthetic microstructure of SOFC electrodes involves the random packing of initial spheres or “seeds” followed by the expansion of such spheres to simulate the sintering process^{28,29,30,31}. Moussaoui et al.^{6} implement a combined model based on sphere packing and truncated Gaussian random field to generate synthetic SOFC electrodes. Additional authors have implemented plurigaussian random fields to model the threephase microstructure of SOFC electrodes and establish correlations between the microstructure and model parameters^{4,32,33}.
In the area of Liion batteries some authors have implemented computational models to adhere a synthetic carbonbinder domain (CBD) (usually hard to image) into XCT threedimensional images of the NMC/pore phases^{34,35}. An analysis of transport properties such as tortuosity and effective electrical conductivity prove the effect of different CBD configurations in the electrode performance. Other authors have implemented physicsbased simulations to predict the morphology of threephase electrodes. Forouzan et al.^{36} developed a particlebased simulation that involves the superposition of CBD particles, to correlate the fabrication process of Liion electrodes with their respective microstructure. Srivastava et al.^{37} simulated the fabrication process Liion electrodes by controlling the adhesion of active material and CBD phases. Their physicsbased dynamic simulations are able to predict with accuracy the effect of microstructure in transport properties.
Recent work by Mosser et al.^{38,39} introduces a deep learning approach for the stochastic generation of threedimensional twophase porous media. The authors implement a type of generative model called Generative Adversarial Networks (GANs)^{40} to reconstruct the threedimensional microstructure of synthetic and natural granular microstructures. Li et al.^{41} extended this work to enable the generation of optimised sandstones, again using GANs. Compared to other common microstructure generation techniques, GANs are able to provide fast sampling of highdimensional and intractable density functions without the need for an a priori model of the probability distribution function to be specified^{39}. This work expands on the research of Mosser et al.^{38,39} and implements GANs for generating threedimensional, threephase microstructure for two types of electrode commonly used in electrochemical devices: a Liion battery cathode and an SOFC anode. A comparison between the statistical, morphological and transport properties of the generated images and the real tomographic data is performed. The twopoint correlation function is further calculated for each of the three phases in the training and generated sets to investigate the longrange properties. Due to the fully convolutional nature of the GANs used, it is possible to generate arbitrarily large volumes of the electrodes based on the trained model. Lastly, by modifying the input of the generator, structures with periodic boundaries were generated.
Performing multiphysics simulations on representative 3D volumes is necessary for microstructural optimisation, but it is typically very computationally expensive. This is compounded by the fact that the regions near the boundaries can show unrealistic behaviour due to the arbitrary choice of boundary condition. However, synthetic periodic microstructures (with all the correct morphological properties) enable the use of periodic boundary conditions in the simulation, which will significantly reduce the simulated volume necessary to be considered representative. This has the potential to greatly accelerate these simulations and therefore the optimisation process as a whole.
The main contributions of this work are listed below:

The implementation of a GANbased approach for the generation of multiphase 3D microstructural data.

Application of this method to two types of commonly used threephase electrodes, resulting trained generators that could be considered as “virtual representations” of these microstructures.

A statistical comparison of the microstructural properties between the real and generated microstructures, establishing the effectiveness of the approach.

The development of a method, based on the GAN approach, for generating periodic microstructures, and the implementation of a diffusion simulation on these volumes to illustrate the impact of periodic boundaries.

The extension of this approach for the generation of arbitrarily large, multiphase, periodic microstructures.
Generative Adversarial Networks are a type of generative model developed by Goodfellow et al.^{40} which learn to implicitly represent the probability distribution function (pdf) of a given dataset (i.e., p_{data}).^{42} Since p_{data} is unknown, the result of the learning process is an estimate of the pdf called p_{model} from which a set of samples can be generated. Although GANs by design do not admit an explicit probability density, they learn a function that can sample from p_{model}, which reasonably approximate those from the real dataset (p_{data}).
The training process consists of a minimax game between two functions, the generator G(z) and the discriminator D(x). G(z) maps an ddimensional latent vector \({\bf{z}} \sim {p}_{{\rm{z}}}({\bf{z}})\in {{\mathbb{R}}}^{d}\) to a point in the space of real data as G(z; θ^{(G)}), while D(x) represents the probability that x comes from p_{data}.^{40} The aim of the training is to make the implicit density learned by G(z) (i.e., p_{model}) to be close to the distribution of real data (i.e., p_{data}). A more detailed introduction to GANs can be found in Section A of the supplementary material.
In this work, both the generator \({G}_{{\theta }^{(G)}}({\bf{z}})\) and the discriminator \({D}_{{\theta }^{(D)}}({\bf{x}})\) consist of deep convolutional neural networks.^{43} Each of these has a cost function to be optimised through stochastic gradient descent in a twostep training process. First, the discriminator is trained to maximise its loss function J^{(D)}:
This is trained as a standard binary crossentropy cost in a classifier between the discriminator’s prediction and the real label. Subsequently, the generator is trained to minimise its loss function corresponding to minimising the logprobability of the discriminator being correct:
These concepts are summarised in Fig. 1. Early in training, the discriminator significantly outperforms the generator, leading to a vanishing gradient in the generator. For this reason, in practice instead of minimising \({\log}\,\left(1D\left(G({\bf{z}})\right)\right)\), it is convenient to maximise the logprobability of the discriminator being mistaken, defined as \({\log}\,\left(D(G({\bf{z}}))\right)\)^{42}.
The solution to this optimisation problem is a Nash equilibrium^{42} where each of the players achieves a local minimum. Throughout the learning process, the generator learns to represent the probability distribution function p_{model} which is as close as possible to the distribution of the real data p_{data}(x). At the Nash equilibrium, the samples of x = G(z) ~ p_{model}(z) are indistinguishable from the real samples x ~ p_{data}(x), thus p_{model}(z) = p_{data}(x) and \(D({\bf{x}})=\frac{1}{2}\) for all x since the discriminator can no longer distinguish between real and synthetic data.
Regarding the microstructural image data, opensource nanotomography data was used for both of the electrode materials considered in this study, which had already been segmented into each of their three respective phases (see Fig. 2). The first dataset is from a Liion battery cathode, comprising particles of a ceramic active material (nickel manganese cobalt oxide  NMC 532), a conductive organic binder (polymer with carbon black) and pores. This material is made by mixing the components in a solvent, thinly spreading this ink onto an aluminium foil and then drying. The second dataset is from an SOFC anode, comprising of a porous nickel/yttria stabilised zirconia (NiYSZ) cermet. This material is also made by mixing an ink, but this time it is deposited onto a ceramic substrate and then sintered at high temperature to bind the components together. Details of the sample preparation, imaging, reconstruction and segmentation approaches used can found in^{34} for the cathode and^{44} for the anode. The specifications of both datasets are shown in Table 1.
More than tenthousand overlapping subvolumes were extracted from each image dataset (Table 1), using a sampling function with a stride of 8 voxels. The spatial dimensions of the cropped volumes were selected based on the average size of the largest structuring element. The subvolume size was selected to guarantee that at least two structuring elements (i.e., particular geometric shape that we attempt to extract from the XCT image) could fit in one subvolume^{45}. In the case of the Liion cathode, this structuring element corresponds to the particle size. In the case of the SOFC anode, “particle size” is not easy to define once the sample is sintered, so the subvolume was selected based on the stabilisation of the twopoint correlation function (TPCF). A more detailed discussion of the representativeness of the subsamples chosen for training is given in Section G of the supplementary information.
Results
The two GANs implemented in this work, one for each microstructure, were trained for a maximum of 72 epochs (i.e., 72 complete iterations of the training set). The stopping criteria was established through the manual inspection of the morphological properties every two epochs. Supplementary Figs S9 and S10 of the supplementary data shows the visual reconstruction of both microstructures, beginning with Gaussian noise at epoch 0, and ending with a visually equivalent microstructure at epoch 50. The image generation improves with the number of iterations; however, as pointed out by Mosser et al.^{38}, this improvement cannot be observed directly from the loss function of the generator and so the morphological parameters described above are used instead.
Before introducing the quantitative comparison of the real and synthetic microstructural data, the excellent qualitative agreement between them can be seen in Fig. 3, which shows six instances of both real and synthetic data from both the anode and cathode samples. Each slice consists of 64^{2} voxels and is obtained from a 64^{3} generated volumetric image. This qualitative analysis consisted of visually comparing some key features of the data. In the case of the Liion cathode, the structuring element in the microstructure (i.e., the NMC particles – grey phase) shows round borders surrounded by a thin layer of binder (white phase), and the phases seem distributed in the same proportion in the real and generated images. In the case of the SOFC, no structuring element is clearly defined; however, the shapes of the white and grey phases of the generated set show the typical shape of the real data which is particular for the sintering process in the experimental generation of these materials^{44}.
In order to draw a comparison between the training and generated microstructures, 100 instance of each were randomly selected (64 × 64 × 64 voxels). The comparison results are presented in the following sections. Supplementary Figs S9 and S10 show 2D slices through generated volumes containing the max value in the phase dimension (rather than the label), which indicates the confidence with which it was labelled.
Lithiumion cathode results
The results of the calculation of the microstructural characterisation parameters (i.e., phase volume fraction, specific surface area and TPB) for the three phases are presented in Fig. 4. For ease of comparison in a single figure, the results of the specific surface area analysis are presented in terms of the percentage deviation from the maximum mean area value among the three phases, Δ(SSA). In the case of the cathode, the maximum mean area, \({A}_{\max ,{\rm{mean}}}\), corresponds to the mean area of the white phase (binder) of the training set (\({A}_{\max ,{\rm{mean}}}=0.72 {\rm{\mu}} {{\rm{m}}}^{1}\)) and all other areas, A_{i}, are normalised against this.
The phase volume fraction, specific surface area and relative diffusivity show good agreement between the real and the synthetic data, particularly in the mean values of both distributions. These mean values and standard deviations are reported in Table 2. The distribution of relative diffusivity in the white phase is very close to zero due to the low volume fraction and resulting low percolation of this phase. For the TPB density, the mean of the generated set is nearly 10% greater than that of the training data; however, nearly all of the values for the synthetic data do still fall within the same range as the training set.
From Fig. 4 it is clear that the synthetic realisations show a smaller variance in all of the calculated microstructural properties compared to the real datasets. This is also shown in Supplementary Fig. S2, where the relative diffusivity is plotted against the volume fraction of the three phases.
The averaged S_{2}(r)/S_{2}(0) along three directions is shown in Fig. 5 for each of the three phases present in a Liion cathode. Since S_{2}(0) represents the volume fraction ϕ_{i} of each phase, S_{2}(r)/S_{2}(0) is a normalisation of the TPCF that ranges between 0 and 1. In this expression, S_{2}(r)/S_{2}(0) stabilised at the value of ϕ_{i}. In all cases, the average values of S_{2}(r)/S_{2}(0) of the synthetic realisations follow the same trend as the training data. The black phase shows a near exponential decay, the grey phase presents a small hole effect and the white phase shows an exponential decay. A hole effect is present in a twopoint correlation function when the decay is nonmonotonic and presents peaks and valleys. This property indicates a form of pseudoperiodicity and in most cases is linked to anisotropy^{46,47}. For the black and grey phases, the S_{2}(r)/S_{2}(0) values of the generated images show a slight deviation from the training set, however this value falls within the standard deviation of the real data.
SOFC anode results
Figure 6 presents the results of the SOFC anode microstructural characterisation parameters calculated for the training data and for the synthetic realisations generated with the GAN model. The Δ(SSA) was calculated using the same approach as described in the previous section, but in the case of the anode, the maximum mean area was the mean area of the white phase (YSZ) of the training set (\({A}_{\max ,{\rm{mean}}}=3.98\,\upmu {{\rm{m}}}^{1}\))
The results in Fig. 6 show a comparable mean and distributions for the morphological properties calculated, as well as for the effective diffusivity of the training images and the synthetic realisations. Once again, the synthetic images show lower variance in the calculated properties than the training set. These mean values and standard deviations are reported in Table 3.
Once again, the difference in the diversity of synthetic images with respect to the training set can be clearly seen in Supplementary Fig. S3 where the effective diffusivity averaged over the three directions for each phase is plotted against its respective volume fraction.
The TPCF was calculated for the three phases along the three Cartesian directions. Figure 7 shows the value of S_{2}(r)/S_{2}(0) for the three phases present in the SOFC anode. The averaged results show an exponential decay in the black phase, a small hole effect^{47} in the grey phase and a pronounced hole effect in the white phase.
Periodic boundaries
Once the generator parameters have been trained, the generator can be used to create periodic microstructures of arbitrary size. This is simply achieved by applying circular spatial padding to the first transposed convolutional layer of the generator (although other approaches are possible). Figure 8 shows generated periodic microstructures for both the cathode and anode, arranged in an array to make their periodic nature easier to see. Additionally, local scalar flux maps resulting from steadystate diffusion simulations in TauFactor^{48} are shown for each microstructure. In both cases, the upper flux map shows the results of the simulation with mirror (i.e., zero flux) boundaries on the vertical edges, and the lower one shows the results of the simulation with periodic boundaries on the vertical edges. Comparing the results from the two boundary conditions, it is clear that using periodic boundaries opens up more paths that enable a larger flux due to the continuity of transport at the edges. Furthermore, this means that the flow effectively does not experience any edges in the horizontal direction, which means that, unlike the mirror boundary case, there are no unrealistic regions of the volume due to edge effects.
Discussion
This work presents a technique for generating synthetic threephase, threedimensional microstructure images through the use of deep convolutional generative adversarial networks. The main contributions of this methodology are mentioned as follows: The results from comparing the morphological metrics, relative diffusivities and twopoint correlation functions all show excellent agreement between the real and generated microstructure. One of the properties that is different from the averaged value in both cases is the TPB density. Nevertheless, its value falls within the confidence interval of the real dataset. This comparison demonstrates that the stochastic reconstruction developed in this work is as accurate as the stateoftheart reconstruction methods reported in the introduction of this article. One variation from previous methodologies is that GANs do not require additional physical information from the microstructure as input data. The methodology was developed to approximate the probability distribution function of a real dataset, so it learns to approximate the voxelwise distribution of phases, instead of directly inputting physical parameters, which is significant; although inputting physical parameters in additional may be beneficial^{49}.
Despite the accurate results obtained in terms of microstructural properties, a number of questions still need to be addressed. One of which involves the diversity in terms of properties of the generated data. Particularly in the case of the Liion cathode microstructure, the generated samples present less variation than the training set. This issue was already encountered by Mosser et al.^{39} and low variation in the generated samples is a much discussed issue in the GAN literature. The typical explanation for this is based on the original formulation of the GAN objective function, which is set to represent unimodal distributions, even when the training set is multimodal^{39,40,42}. This behaviour is known as “mode collapse” and is observed as a low variability in the generated images. A visual inspection of the generated images as well as the accuracy in the calculated microstructural properties do not provide a sufficient metric to guarantee the inexistence of mode collapse or memorisation of the training set.
Figures 4 and 6 show some degree of mode collapse given by the small variance in the calculated properties of the generated data. Nevertheless, further analysis of the diversity of the generated samples is required to evaluate the existence of mode collapse based on the number of unique samples that can be generated^{50,51}. Following the work of Radford et al.^{52}, an interpolation between two points in the latent space is performed to test the absence of memorisation in the generator. The results shown in Supplementary Fig. S8 present a smooth transformation of the generated data as the latent vectors is progressed along a straight path. This indicates that the generator is not memorising the training data but has learned the meaningful features of the microstructure.
The presence of mode collapse and vanishing gradient remain the two main issues with the implementation of GANs. As pointed out by^{53}, these problems are not necessarily related to the architecture of GANs, but potentially to the original configuration of the loss function. This work implements a DCGAN architecture with the standard loss function; however, recent improvements of GANs have focused on reconfiguring the loss function to enable a more stable training and more variability in the output. Some of these include WGAN (and WGANGP) based on the Wasserstein or Earthmover distance^{54,55}, LSGAN which uses least squares and the Pearson divergence^{56}, SNGAN that implement a spectral normalisation^{57}, among others^{53}. Therefore, an improvement of the GAN loss function is suggested as future work in order to solve the problems related to low variability (i.e., slight mode collapse) and training stability.
The applicability of GANs can be extended to transfer the learned weights of the generator (i.e., G_{θ}(z)) into (a) generating larger samples of the same microstructure, (b) generating microstructure with periodic boundaries, (c) performing an optimisation of the microstructure according to a certain macroscopic property based on the latent space z. As such, G_{θ} can be thought of as a powerful “virtual representation” of the real microstructure and it interested to note that the total size of the trained parameters, θ^{(G)} is just 55 MB.
The minimum generated samples are the same size as the training data subvolumes (i.e., 64^{3} for both cases analysed in this work), but can be increased to any arbitrarily large size by increasing the size of the input z. Although the training process of the DCGAN is computationally expensive, once a trained generator is obtained, it can produce image data inexpensively. The relation between computation time and generated image size is shown in Supplementary Fig. S7.
The ability of the DCGAN to generate periodic structures has potentially profound consequences for the simulation of electrochemical processes at the microstructural scale. Highly coupled, multiphysics simulations are inherently computational expensive^{58,59}, which is exacerbated by the need to perform them on volumes large enough to be considered representative. To make matters worse, the inherent nonperiodic nature of real tomographic data, combined with the typical use of “mirror” boundary conditions means that regions near the edges of the simulated control volume will behave differently from those in the bulk. This leads to a further increase in the size of the simulated volume required, as the impact of the “near edge” regions need to be eclipsed by the bulk response. Already common practice in the study of turbulent flow^{60,61}, the use of periodic boundaries enables much smaller volumes to be used, which can radically accelerate simulations. The flux maps shown in Fig. 8 highlight the potential impact even for a simple diffusion simulation and the calculated transport parameters of these small volumes are much closer to the bulk response when periodic boundaries are implemented.
Examples of generated periodic (and similar nonperiodic) volumes for both the Liion cathode and SOFC anode can be found in the supplementary materials accompanying this paper and the authors encourage the community to investigate their utility. A detailed exploration of the various methods for reconfiguring the generator’s architecture for the generation of periodic boundaries, as well as an analysis of the morphological and transport properties of the generated microstructures compared to the real ones are ongoing and will be presented in future work.
An additional benefit of the use of GANs in microstructural generation lies in the ability to interpolate in the continuous latent space to generate more samples of the same microstructure. The differentiable nature of GANs enables the latent space that parametrises the generator to be optimised. Li et al.^{41} have implemented a Gaussian process to optimise the latent space in order to generate an optimum 2D twophase microstructure. Other authors^{62} have used an inpainting technique to imprint over the threedimensional image some microstructural details that are only available in twodimensional conditioning data. This process is performed by optimising the latent vector with respect to the mismatch between the observed data and the output of the generator^{39,62}. A potential implementation of the inpainting technique could involve adding information from electron backscatter diffraction (EBSD), such as crystallographic structure and orientation, into the already generated 3D structures, which would be of great interest to the electrochemical modelling community.
Future work will aim to extend the study by Li et al.^{41} to perform an optimisation of the 3D threephase microstructure based on desired morphological properties by optimising the latent space. One proposed pathway to improve these optimisation process would involve providing physical parameters to the GAN architecture. This could be achieved by adding a physicsspecific loss component to penalise any deviation from a desired physical property^{49}. It could also involve giving a physical meaning to the z space through the implementation of a Conditional GAN algorithm^{63}. With this, apart from the latent vector, the Generator has a second input y related to a physical property. Thus, the Generator becomes G(z, y) and produces a realistic image with its corresponding physical property
Concluding remarks
This work presents a method for generating synthetic threedimensional microstructures composed of any number of distinct material phases, through the implementation of DCGANs. This method allows the model to represent the statistical and morphological properties of the real microstructure, which are captured in the weights of the trained discriminator and generator networks.
A pair of opensource, tomographically derived, threephase microstructural datasets were investigated: a lithiumion battery cathode and a solid oxide fuel cell anode. Various microstructural properties were calculated for 100 subvolumes of the real data and these were compared to 100 instances of volumes created by the trained generator. The results showed excellent agreement across all metrics, although the synthetic structures showed a smaller variance compared to the training data, which is a commonly reported problem for DCGANs and mitigation strategy will be reported in future work.
Two issues encountered when training the DCGANs in this study were instability (likely due to a vanishing gradient) and moderate mode collapse. Both issues can be attributed to the GANs loss function and solutions have been suggested in the literature, the implementation of which will be explored in future work.
Two particular highlights of this work include the ability to generate arbitrarily large synthetic microstructural volumes and the generation of periodic boundaries, both of which are of high interest to the electrochemical modelling community. A detailed study of the impact of periodic boundaries on the reduction of simulation times is already underway.
Future work will take advantage of the continuity of the latent space, as well as the differentiable nature of GANs, to perform optimisation of certain morphological and electrochemical properties in order to discover improved electrode microstructures for batteries and fuel cells.
Methods
This section outlines the architecture that constitutes the two neural networks in the GAN, as well as the methodology followed for training. It also describes the microstructural properties used to analyse the quality of the microstructural reconstruction when compared to the real datasets. The codes developed for training the GANs are opensource.
Pretreatment of the training set
The image data used in this study is initially stored as 8bit greyscale elements, where the value of each voxel (3D pixel) is used as a label to denote the material it contains. For example, as in the case of the anode in this study, black (0), grey (127), and white (255), encode pore, metal, and ceramic respectively.
In several previous studies where GANs were used to analyse materials, the samples in question had only two phases and as such, the materials information could be expressed by a single number representing the confidence it belongs to one particular phase. However, in cases where the material contains three or more phases, this approach can be problematic, as a voxel in which the generator had low confidence deciding between black or white may end up outputting grey (i.e., a value halfway between black and white), which means something entirely different.
The solution comes from what is referred to as “onehot” encoding of the materials information. This means that an additional dimension is added to the dataset (i.e., three spatial dimensions, plus one materials dimension), so an initially 3D cubic volume of n × n × n, is encoded to a 4D c × n × n × n volume, where c is the number of material phases present. This materials dimension contains a ‘1’ in the element corresponding to the particular material at that location and a ‘0’ in all other elements (hence, “onehot”). So, what was previous black, white, and grey, would now be encoded as [1,0,0], [0,1,0], and [0,0,1] respectively. This concept is illustrated in Supplementary Fig. S1 for a threematerial sample, but it could trivially be extended to samples containing any number of materials. It is also easy to decode these 4D volumes back to 3D greyscale, even when there is uncertainty in the labelling, as the maximum value can simply be taken as the label.
GAN Architecture and Training
The GAN architecture implemented in this work is a volumetric version based on the deep convolutional GAN (DCGAN) model proposed by Radford et al.^{52}. Both generator and discriminator are represented by fully convolutional neural networks. In particular, the convolutional nature of the generator allows it to be scalable, thus it can generate instances from the p_{model} larger than the instances in the original training set, which is useful.
The discriminator is composed of five layers of convolutions, each followed by a batch normalisation. In all cases, the convolutions cover the full length of the materials dimension, c but the kernels within each layer are of smaller spatial dimension than the respective inputs to these layers . The first four layers apply a “leaky” rectified linear unit (leaky ReLU) activation function and the last layer contains a sigmoid activation function that outputs a single scalar constrained between 0 and 1, as it is a binary classifier. This value represents an estimated probability of an input image to belong to the real dataset (output = 1) or to the generated sample (output = 0).
The generator is an approximate mirror of the discriminator, also composed of five layers, but this time transposed convolutions^{64} are used to expand the spatial dimensions in each step. Once again, each layer is followed by a batch normalization and all layers use ReLU as their activation function, except for the last layer which uses a Softmax function, given by Eq. (4)
where x_{j} represents the jth element of the onehot encoded vector x at the last convolutional layer.
It is well known that the hyperparmeters that define the architecture of the neural networks have significant impact on the quality of the results and the speed of training. In this work, although a formal hyperparameter optimisation was not performed due to computational expense, a total of 16 combinations between four hyperparmeters was performed. A statistical analysis between the real and generated microstructures was performed (as described in section 2), and the optimum architecture was chosen based on these results.
The generator requires a latent vector z as its input in order to produce variety in its outputs. In this study, the input latent vector z is of length 100. Table 4 summarises all of the GAN’s layers configuration described above, as well as the size, stride and number of kernels applied between each layer, and the padding applied around the volume when calculating the convolutions. As will be discussed later in this paper, although zeros were initially used for padding, the study also explores the use of circular padding, which forces the microstructure to become periodic.
In theory, a Nash equilibrium is achieved after sufficient training; however, in practice this is not always the case. GANs have shown to present instability during training that can lead to mode collapse^{42}. Mescheder et al.^{65} present an analysis of the stability of GAN training, concluding that instance noise and zerocentred gradient penalties lead to local convergence. Another proposed stabilisation mechanism, which was implemented effectively in this work, is called onesided label smoothing^{42}. Through this measure, the label 1 corresponding to real images is reduced by a constant ε, such that the new label has the value of 1 − ε. For all cases in this work, ε has a value of 0.1.
An additional source of instability during training is attributed to the fact that the discriminator learns faster than the generator, particularly at the early stages of training. To stabilise the alternating learning process, it is convenient to set a ratio of network optimisation for the generator and discriminator to k: 1. In other words, the generator is updated k times while the discriminator is updated once. In this work k has a value of 2. In both cases (i.e., cathode and anode data) stochastic gradient descent is implemented for learning using the ADAM optimiser^{66}. The momentum constants are β_{1} = 0.5, β_{2} = 0.999 and the learning rate is 2 × 10^{−5}. All simulations are performed on a GPU (Nvidia TITAN Xp) and the training process is limited to 72 epochs (c. 48 h).
Microstructural characterisation parameters
The electrode microstructures can be characterised by a set of parameters calculated from the 3D volumes. These parameters include morphological properties, transport properties, and statistical correlation functions. To evaluate the ability of the trained model to accurately capture the pdf that describes the microstructure within the latent space, parameters were calculated from 100 instances of both the real and GAN generated data.
Morphological properties
Three morphological properties are considered in this work, each computed using the opensource software TauFactor^{48}. These consist of the volume fractions and specific surface areas, as well as the triplephase boundary (TPB) densities. More information about these parameters can be found in the supplementary information.
Relative diffusivities
The relative diffusivity, D^{rel}, is a dimensionless measure of the ease with which diffusive transport occurs through a system held between Dirichlet boundaries applied to two parallel faces. In this study, it is calculated for each of the three material phases separately, as well as in each of the three principal directions in a cubic volume. It is directly related to the diffusive tortuosity factor of phase i, τ_{i}, as can be seen from the following equation,
where ϕ_{i} is the volume fraction of phase i, \({D}_{i}^{0}\) is the intrinsic diffusivity of the bulk material (arbitrarily set to unity), and \({D}_{i}^{{\rm{eff}}}\) is the calculated effective diffusivity given the morphology of the system. The tortuosity factors were obtained with the opensource software TauFactor^{48}, which models the steadystate diffusion problem using the finite difference method and an iterative solver.
Twopoint correlation function
According to Lu et al.^{12}, the morphology of heterogeneous media can be fully characterised by specifying one of the various statistical correlation functions. One of such correlations is the npoint probability function S_{n}(x^{n}), defined as the probability of finding n points with positions x_{n} in the same phase^{12,13,14}. Based on this, the socalled twopoint correlation function (TPCF), S_{2}(r), allows the first and secondorder moments of a microstructure to be characterised^{14,39}. Assuming stationarity (i.e., the mean and variance have stabilised), the TPCF is defined as the noncentred covariance, which is the probability P that two points x_{1} = x and x_{2} = x + r separated by a distance r belong to the same phase p_{i},
At the origin, S_{2}(0) is equal to the phase volume fraction ϕ_{i}. The function S_{2}(r) stabilised at the value of \({\phi }_{i}^{2}\) as the distance, r, tends to infinity. This function is not only valuable for analysing the anisotropy of the microstructure, but also to account for the representativeness in terms of volume fraction of subvolumes taken from the same microstructure sample. In this work, the TPCF of the three phases is calculated along the three Cartesian axes.
Data Availability
The study used openaccess training data available from references listed in the manuscript. Many instances of the generated data are available at https://github.com/agayonlombardo/pores4thought. All other generated data used is available from the authors on request.
Code Availability
All the codes used in this manuscript can be accessed via the following link https://github.com/agayonlombardo/pores4thought.
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Acknowledgements
This work was supported by funding from both the CONACYTSENER fund and the EPSRC Faraday Institution MultiScale Modelling project (https://faraday.ac.uk/; EP/S003053/1, grant number FIRG003). The Titan Xp GPU used for this research was kindly donated by the NVIDIA Corporation through their GPU Grant program. The authors would also like to thank various colleagues for their valuable comments: Prof. Stephen Neethling, Antonio Bertei, Tilman Roeder, Steven Kench, Vitaly Levdik, Donal Finegan and Harry Abernathy.
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A.G.L. wrote the code. S.J.C. supervised the project. S.J.C. and A.G.L. conceived the idea and performed the analysis. L.M. provided the theoretical foundations and expertise in GANs. N.P.B. offered supervision and guidance on the wider impact of the work. A.L.G. and S.J.C. wrote the manuscript.
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GayonLombardo, A., Mosser, L., Brandon, N.P. et al. Pores for thought: generative adversarial networks for stochastic reconstruction of 3D multiphase electrode microstructures with periodic boundaries. npj Comput Mater 6, 82 (2020). https://doi.org/10.1038/s4152402003407
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