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Pores for thought: The use of generative adversarial networks for the stochastic reconstruction of 3D multi-phase electrode microstructures with periodic boundaries

Andrea Gayon-Lombardo, Lukas Mosser, Nigel P. Brandon, Samuel J. Cooper

arXiv:2003.11632v2cs.NEcs.CV

TL;DR

Generating realistic multiphase electrode microstructures is important for electrochemical-device optimisation, but representative 3D simulations are computationally expensive and boundary effects complicate volume selection. This work uses volumetric DC-GANs to generate and compare synthetic three-phase cathode and anode structures, finding excellent agreement across metrics while identifying reduced variance and demonstrating periodic-generation potential.

  • Problem

    Representative 3D multiphysics simulations are computationally expensive, while boundary effects complicate the volume needed for microstructural optimisation.

  • Method

    The study applies volumetric fully convolutional DC-GANs to generate three-dimensional multiphase microstructures and compares synthetic volumes with real electrode data.

  • Results

    The generated structures showed excellent agreement across all evaluated metrics for lithium-ion cathode and SOFC anode datasets.

  • Takeaways & Limitations

    Periodic synthetic microstructures could reduce representative simulation volumes and accelerate electrochemical optimisation, while latent-space continuity supports future morphology optimisation.

  • Takeaways & Limitations

    Synthetic structures showed smaller variance than the training data, and low variation can reflect mode collapse that visual and property accuracy do not rule out.

Abstract

from arXiv · show

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 (DC-GAN) for generating realistic n-phase microstructural data. The same network architecture is successfully applied to two very different three-phase microstructures: A lithium-ion 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, triple-phase boundary) and transport properties (relative diffusivity), as well as the two-point correlation function. The results show excellent agreement between for 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.

PERIODIC BOUNDARIES

The supplied passages identify the work as a preprint on electrode microstructure reconstruction using GANs and machine learning, but do not provide substantive information about periodic boundaries.

  • The paper is identified as a preprint dated May 6, 2020.
  • The listed authors are Andrea Gayon-Lombardo, Lukas Mosser, Nigel P. Brandon, and Samuel J. Cooper.
  • The keywords identify microstructure, electrodes, GANs, reconstruction, and machine learning as the paper's topical terms.

1 Introduction

Multiphase electrode microstructure generation supports optimisation but must preserve complex structure while reducing the cost and boundary artefacts of representative-volume simulations. The work proposes GAN-based generation, statistical validation, and periodic, scalable reconstructions.

  • GANs offer fast sampling of high-dimensional, intractable density functions for three-dimensional microstructure reconstruction.
  • Multiphysics simulations require representative 3D volumes, but they are computationally expensive and boundary conditions can create unrealistic edge behaviour.
  • This work implements GAN-based generation of multiphase 3D microstructural data and applies it to two three-phase electrode types.
  • The approach statistically compares real and generated microstructures to establish reconstruction effectiveness.
  • The method develops periodic microstructure generation and demonstrates its impact with diffusion simulations.
  • Arbitrarily large multiphase periodic microstructures can be generated using the GAN-based approach.

2 Generative Adversarial Networks

GANs learn an implicit approximation of a dataset’s probability distribution through adversarial training between a generator and discriminator. Deep convolutional networks implement this process for microstructural data.

  • GANs learn an implicit model distribution from a dataset and generate samples that approximate the real data distribution.
  • The generator maps a latent vector z to synthetic data while the discriminator estimates whether x comes from the real-data distribution.
  • Training alternates between discriminator maximisation and generator minimisation through stochastic-gradient optimisation.
  • The generator objective is commonly implemented by maximising the discriminator’s probability of being mistaken, avoiding vanishing gradients early in training.
  • At Nash equilibrium, generated and real samples become indistinguishable, with the model and data distributions matching.

3 Microstructural image data

The study uses segmented open-source nano-tomography datasets from a lithium-ion cathode and an SOFC anode. Over ten-thousand overlapping sub-volumes were extracted from each dataset for training.

  • The datasets represent three phases in a lithium-ion cathode and an SOFC anode.The cathode contains NMC 532, conductive binder, and pores; the anode is a porous Ni-YSZ cermet.
  • More than ten-thousand overlapping sub-volumes were extracted from each image dataset using a stride of 8 voxels.

4 Method

The method prepares labeled three-phase electrode volumes for a volumetric DC-GAN, evaluates generated structures against real data using morphology, transport, and correlation metrics, and explores periodic padding.

  • 4.1 Pre-treatment of the training set: Three-phase voxel labels are converted from greyscale material values into one-hot c×n×n×n volumes, preventing ambiguous intermediate labels during generation.Each voxel contains a 1 for its material phase and 0 for all others; decoding selects the maximum value.
  • 4.2 GAN Architecture and Training: The volumetric DC-GAN uses fully convolutional generator and discriminator networks, allowing generated instances to exceed the training-set dimensions.The generator expands spatial dimensions with transposed convolutions, while the discriminator outputs a real-versus-generated probability.
  • 4.2 GAN Architecture and Training: The generator produces phase-labeled volumes from a length-100 latent vector, with a final Softmax layer assigning material probabilities.The architecture comprises five-layer generator and discriminator networks; the Softmax operates on the one-hot encoded material vector.
  • 4.2 GAN Architecture and Training: Circular padding is tested alongside zero padding to force generated microstructures to have periodic boundaries.Training stability is also addressed through generator-to-discriminator update ratio k:1 with k=2, plus one-sided label smoothing.
  • 4.3 Evaluation: Model quality is assessed from 100 real and generated instances using volume fractions, specific surface areas, triple-phase-boundary densities, relative diffusivity, and correlation functions.Relative diffusivity is calculated separately for each phase and principal direction using TauFactor-based finite-difference diffusion modeling.

5 Results

The DC-GAN generated visually corresponding three-phase cathode and anode microstructures, with morphological, transport, and correlation properties generally agreeing with the training data. Synthetic samples showed lower diversity, while periodic structures were generated for both electrode types by modifying the generator input.

  • Qualitative comparison: The GAN produced visually corresponding cathode and anode microstructures, including realistic phase proportions and characteristic particle or sintering shapes.Cathode NMC particles had rounded borders with binder layers; SOFC phases reproduced shapes associated with experimental sintering.
  • Lithium-ion cathode: The cathode’s volume fraction, specific surface area, and relative diffusivity showed good agreement between real and synthetic data, while TPB density was nearly 10% higher synthetically.Nearly all synthetic TPB values remained within the training-data range.
  • Lithium-ion cathode: Synthetic cathode realisations had smaller variance than the real datasets across the calculated microstructural properties.The same reduced diversity was visible when relative diffusivity was plotted against phase volume fraction.
  • Lithium-ion cathode: Synthetic cathode TPCFs followed the training-data trends, with slight black- and grey-phase deviations remaining within the real-data standard deviation.The black phase showed near-exponential decay, the grey phase a small hole effect, and the white phase exponential decay.
  • SOFC anode: SOFC anode synthetic samples showed comparable morphological-property and effective-diffusivity means and distributions, but lower variance than training samples.The SOFC TPCF exhibited exponential decay in the black phase, a small grey-phase hole effect, and a pronounced white-phase hole effect.
  • Periodic boundaries: Periodic microstructures were generated for both electrodes by applying circular spatial padding to the generator’s first transposed convolutional layer.The resulting structures supported comparisons of mirror and periodic boundary conditions using local scalar flux maps.

6 Discussion

The DC-GAN reconstructs three-phase electrode microstructures with strong agreement across morphological, transport, and correlation metrics, while revealing variability and stability limitations. Its latent representation also supports scalable and periodic microstructure generation, with further optimisation and periodic-boundary analyses left for future work.

  • Reconstruction accuracy: Morphological metrics, relative diffusivities, and two-point correlation functions show excellent agreement between real and generated microstructures.TPB density differs from the averaged real value but remains within the real dataset’s confidence interval.
  • Diversity and limitations: Generated Li-ion cathode samples exhibit less property variation than the training set, indicating moderate mode collapse.Small variance appears in the calculated generated properties, although latent-space interpolation indicates the generator is not memorising training data.
  • Future directions: Future work proposes improved GAN loss functions, latent-space optimisation, and physics-conditioned generation for more stable and property-directed reconstructions.A Conditional GAN would add a physical-property input y so the generator produces images associated with that property.
  • Generation and scalability: The DC-GAN can generate larger volumes by increasing the latent input size, although training is computationally expensive.Once trained, the generator produces image data inexpensively.
  • Periodic boundaries: Periodic-boundary generation could reduce the representative simulated volume and accelerate computationally expensive electrochemical simulations.Non-periodic tomographic data and arbitrary mirror boundaries can produce unrealistic boundary-region behaviour.

7 Conclusions

The study demonstrates DC-GAN reconstruction of three-phase electrode microstructures across battery and fuel-cell datasets, with strong metric agreement but reduced synthetic variance. It highlights arbitrary-size and periodic generation as important capabilities while identifying GAN instability and mode collapse as unresolved issues.

  • Method: DC-GANs represent statistical and morphological properties of three-dimensional microstructures through trained generator and discriminator weights.The method supports microstructures composed of any number of distinct material phases.
  • Results: Across 100 real and 100 generated sub-volumes, the evaluated microstructural properties show excellent agreement, although synthetic structures have smaller variance.The comparison uses open-source tomographically derived lithium-ion cathode and solid oxide fuel-cell anode datasets.
  • Limitations: The study identifies training instability and moderate mode collapse as issues likely attributable to the GAN loss function.Alternative loss-function solutions are proposed for future implementation.
  • Implications: Arbitrarily large synthetic volumes and periodic boundaries are highlighted as capabilities of particular interest to electrochemical modelling.The impact of periodic boundaries on simulation-time reduction remains under study.
  • Future work: Future work will optimise morphological and electrochemical properties through the continuous latent space to seek improved battery and fuel-cell electrode microstructures.The proposed optimisation relies on the continuity and differentiability of the GAN representation.

Supplementary information

The supplementary material introduces GANs as an adversarial generator–discriminator framework that maps latent variables into synthetic data and trains both networks through competition. The generator seeks realistic samples while the discriminator distinguishes generated from real data.

  • Adversarial framework: GANs formulate generation as a two-player game between a generator that creates samples and a discriminator that distinguishes real from generated data.Both players improve through continuous competition.
  • Generator: The generator maps a normally distributed latent vector z into the image domain using parameters θ(G).The latent vector has dimensionality d and consists of independent normally distributed real variables.
  • Discriminator: The discriminator receives real data and generator-produced samples and uses parameters θ(D) to classify them.Its role is to distinguish samples originating from the real dataset from synthetic outputs.
  • Training: Training alternates discriminator maximisation and generator minimisation within the adversarial objective.The discriminator learns binary classification, while the generator minimises the probability that the discriminator correctly identifies synthetic data.

B Sample details

The study analyses three-phase lithium-ion cathode and solid oxide fuel-cell anode microstructures, including their phase representations and training sub-volumes. It defines the morphological and transport metrics used to evaluate these materials.

  • Li-ion cathode: The lithium-ion cathode contains active NMC532, carbon/binder, and porous phases.Because carbon/binder is not directly visible in X-ray CT images, it is generated over the reconstructed geometry.
  • SOFC anode: The SOFC anode contains YSZ, nickel, and porous phases, with approximately 4 × 10^8 voxels in the full sample.The voxelised image is cropped into 45,492 overlapping 64^3 sub-volumes for training.
  • Data representation: One-hot encoding is used to represent the material phases in the microstructure images.The supplementary figure provides a visual representation of this encoding process.
  • Morphological metrics: Phase volume fraction is calculated from the volume of phase i relative to the total microstructure volume.The phase volume is obtained from the percentage of voxels assigned to that phase.
  • Morphological metrics: Specific surface area measures the interface area between phase i and the remaining phases.The metric is based on the total surface area of the relevant phase interface.
  • Morphological metrics: TPB density measures the three-phase intersection length normalised by total microstructure volume.On a cuboid lattice, a TPB is an edge whose four connecting voxels include three different phases.
  • Transport metrics: The tortuosity factor quantifies geometric resistance to diffusive transport through the porous medium.It can be expressed using effective and intrinsic diffusivities or as the ratio of effective pathway length to sample length.

E Relative diffusivity extended results

The synthetic Li-ion cathode and SOFC anode reproduce the real datasets’ two-point correlation trends along three directions, including phase anisotropy. Relative diffusivity is reported for all three phases in both microstructures.

  • Relative diffusivity: Relative diffusivity is evaluated against phase volume fraction for all three phases in the Li-ion cathode and SOFC anode.The supplied figures present the corresponding phase-wise relative-diffusivity relationships.
  • Li-ion cathode: Synthetic Li-ion cathode data follows the real dataset’s normalised TPCF trend for pores, NMC-532, and binder along three directions.The comparison uses S2(r)/S2(0), with decay and stabilisation described in Table 1.
  • SOFC anode: Synthetic SOFC anode data follows the real dataset’s normalised TPCF trend for pores, Ni, and YSZ along three directions.The comparison includes phase anisotropy and decay or stabilisation descriptions in Table 2.

G Representativity

Representativity is assessed through training-volume selection, generated-volume scaling, latent-space interpolation, and uncertainty maps. The 64^3 training sub-volume provides a fair volume-fraction representation, while larger generation scales without retraining and uncertainty concentrates at interphases.

  • Training-volume selection: A 64^3 training sub-volume was selected for both microstructures, with the Li-ion cathode size tied to particle-scale structure and the SOFC anode lacking a characteristic structuring element.The cathode’s average particle diameter is 35 voxels; the SOFC anode uses a standard 64^3 training size.
  • Training-volume selection: Choosing an ideal training size is challenging for irregular, highly porous, or continuously fibrous structures such as the SOFC anode.Training size is closely related to the representativeness of the training set.
  • Representative volume: Two-point statistics can inform representative-volume size, but complex samples may require additional analysis of specific surface area and triple-phase boundary.The passage identifies specific surface area and TPB as potentially more representative morphology measures when S2(r) does not stabilise.
  • Representative volume: The representativity of specific surface area and TPB is not conclusively characterised, and experimental sub-volume analysis is exhaustive and microstructure-specific.The authors recommend further work on long-range functions that can estimate representative volumes for additional properties.
  • Generation scale: Generated samples can exceed the 64^3 training size without further training; samples as large as 320^3 voxels were generated, with generation time increasing linearly with image size.The larger images are produced by increasing the latent-input size.
  • Mode collapse: Smooth latent-space interpolation between generated endpoints indicates that the generator did not memorise the training set or undergo total mode collapse.The interpolation uses points between z_start and z_end, with smooth transitions in both microstructures.
  • Uncertainty: Uncertainty maps show higher uncertainty at interphases and lower uncertainty within phase regions during training for both microstructures.White denotes high certainty and black denotes low certainty in the grayscale maps.
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