PaperlayerView exact revision on arXiv

Source-linked AI summary

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 representative 3D electrode microstructures matters for multiphysics simulation and morphology optimization. This paper applies DC-GANs to synthetic three-phase microstructures and reports excellent agreement across assessed metrics, alongside lower variance in generated samples.

  • Problem

    Representative 3D microstructural volumes are needed for multiphysics simulations, and electrode morphology is important to performance and optimization.

    Source: §1 · pp. 1–2

  • Method

    The paper uses volumetric DC-GANs to generate synthetic three-dimensional microstructures composed of multiple material phases.

    Source: §4.2 · p. 5, §7 · p. 13

  • Results

    Across 100 real sub-volumes and 100 generated volumes, synthetic microstructures showed excellent agreement across all assessed metrics but lower variance than the training data.

    Source: §7 · p. 13

  • Takeaways & Limitations

    The authors identify periodic structure generation as potentially consequential for costly microstructural electrochemical simulations on representative volumes.

    Source: §6 · p. 12

    Generated samples, particularly for the Li-ion cathode, showed less property variation than training data, and visual or property accuracy alone cannot rule out mode collapse or memorization.

    Source: §6 · pp. 11–12

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.

1 Introduction

This work uses GANs to generate and assess three-phase 3D microstructures for two electrochemical electrodes, and modifies the generator to produce periodic structures. The approach addresses the challenge of creating synthetic volumes that retain real-material properties while supporting more efficient simulations.

1 Introduction

  • Porous electrodes need high surface area, percolating paths for electron and ion transport, and mechanical integrity, so their microstructure affects performance and optimisation.These requirements apply to electrodes in batteries, fuel cells, and supercapacitors.Source: §1 · p. 1
  • XCT enables researchers to view porous-material microstructures at useful resolution, but extracting their essential properties and modifying attributes without losing resemblance remain challenges.Source: §1 · pp. 1–2
  • Earlier reconstructions commonly used two-point correlation functions, while Mosser et al. [40,41] used GANs for two-phase media; this work extends GAN generation to three-phase electrodes.Source: §1 · p. 2
  • The work applies GANs to generate three-phase 3D microstructures for a Li-ion battery cathode and SOFC anode, then statistically compares generated and tomographic data.The comparisons cover morphological and transport properties and phase-wise two-point correlation functions; the trained generators are proposed as virtual representations of the electrodes.Source: §1 · pp. 2–3
  • By modifying generator input, the method produces periodic microstructures, and diffusion simulations illustrate the impact of periodic boundaries.The work also extends the approach to arbitrarily large multiphase periodic structures; such volumes could reduce representative simulation volume and computational cost.Source: §1 · p. 3

2 Generative Adversarial Networks

GANs learn an implicit approximation of a dataset’s probability distribution by training a generator and discriminator in competition. In this work, both are deep convolutional neural networks optimized in alternating steps.

2 Generative Adversarial Networks

  • GANs learn an implicit model distribution pmodel that can generate samples approximating those from the real dataset, pdata.GANs do not admit an explicit probability density; Goodfellow et al.15 developed this type of deep generative model.Source: §2 · p. 3
  • The generator maps a latent vector z to synthetic data, while the discriminator estimates whether an input comes from pdata; training aims to make pmodel close to pdata.This is a minimax game between the generator and discriminator.Source: §2 · p. 3
  • Both networks are deep convolutional neural networks, trained by stochastic gradient descent in two alternating optimization steps.The discriminator first maximizes its loss, then the generator minimizes its loss.Source: §2 · p. 4
  • Because the discriminator can initially outperform the generator and cause vanishing gradients, the generator instead maximizes the log-probability that the discriminator is mistaken.The discriminator is trained with a standard binary cross-entropy classifier loss.Source: §2 · p. 4
  • At Nash equilibrium, synthetic and real samples are indistinguishable, with pmodel = pdata and discriminator output D(x) = 1/2.The equilibrium is a solution in which each player achieves a local minimum.Source: §2 · p. 4

3 Microstructural image data

The study uses segmented open-source nano-tomography datasets for a three-phase Li-ion cathode and a three-phase SOFC anode. More than 10,000 overlapping sub-volumes were extracted from each dataset for training.

3 Microstructural image data

  • The datasets represent an NMC 532 cathode with conductive organic binder and pores, and a porous Ni-YSZ SOFC anode.Both datasets were open-source nano-tomography images already segmented into their three respective phases.Source: §3 · p. 4
  • The cathode ink is spread onto aluminium foil and dried, whereas the SOFC anode ink is deposited onto a ceramic substrate and sintered.Sample preparation, imaging, reconstruction, and segmentation details are cited as 59 for the cathode and 23 for the anode; dataset specifications appear in Table 1.Source: §3 · p. 4
  • More than 10,000 overlapping sub-volumes per dataset were cropped at an 8-voxel stride, with sizes chosen to fit at least two structuring elements.Cathode sub-volume sizing used particle size; because particle size is difficult to define in the sintered SOFC anode, its sizing followed stabilization of the two-point correlation function.Source: §3 · p. 4

4 Method

The method uses one-hot phase encoding and a volumetric DC-GAN to reconstruct multiphase electrode microstructures, then evaluates the generated and real volumes using morphological, transport, and correlation measures.

4.1 Pre-treatment of the training set

  • One-hot encoding represents each voxel with a separate binary channel per phase, avoiding ambiguous grayscale outputs and allowing uncertain labels to be decoded by selecting the maximum channel value.The encoding maps an n×n×n volume to c×n×n×n, where c is the number of phases.Source: §4.1 · pp. 4–5

4.2 GAN Architecture and Training

  • The volumetric DC-GAN uses fully convolutional networks; its five-layer generator mirrors the discriminator with transposed convolutions and a final Softmax, and can scale beyond training-set instance sizes.Circular padding is also explored to force periodic microstructures.Source: §4.2 · pp. 5–6
  • Training uses one-sided label smoothing with ε = 0.1, updates the generator twice per discriminator update, and applies ADAM with β1 = 0.5, β2 = 0.999, and learning rate 2 × 10−5.Sixteen hyperparameter combinations were compared statistically rather than using formal optimisation, and the selected architecture was chosen from those results.Source: §4.2 · p. 6

4.3 Microstructural characterisation parameters

  • Microstructural parameters were calculated from 100 real and 100 GAN-generated instances to assess how accurately the trained model captures the latent-space probability distribution.Source: §4.3 · p. 7

4.3.1 Morphological properties

  • Morphological characterization uses TauFactor to calculate phase volume fractions, specific surface areas, and triple-phase boundary densities.Source: §4.3.1 · p. 7

4.3.2 Relative diffusivities

  • Relative diffusivity is calculated for each phase along each of the three principal directions, with transport modeled between parallel faces held at Dirichlet boundaries.TauFactor obtains tortuosity factors by modeling steady-state diffusion.Source: §4.3.2 · p. 7

4.3.3 Two-point correlation function

  • The two-point correlation function gives the probability that two points separated by distance r belong to the same phase, and is calculated along the three Cartesian axes.It is used to characterize microstructural anisotropy and the volume-fraction representativeness of subvolumes.Source: §4.3.3 · p. 7

5 Results

The DC-GAN generated microstructures that qualitatively resembled the real cathode and anode data, with reported agreement in morphological and transport properties. Changing the generator input also enabled periodic microstructures for both electrodes.

5 Results

  • Visual comparisons showed that generated cathode particles and binder resembled the real data, while generated SOFC phase shapes displayed the typical sintering-related forms.The comparison included six real and synthetic instances from each electrode.Source: §5 · pp. 7–8, Figure 3 · p. 8

5.1 Lithium-ion cathode results

  • For the cathode, phase volume fraction, specific surface area, and relative diffusivity agreed well in their means, while generated TPB density was nearly 10% greater and property variance was smaller.Nearly all synthetic TPB values still fell within the training-set range.Source: §5.1 · pp. 8–9, Figure 4 · p. 9, Table 3 · p. 9
  • The cathode’s normalized two-point correlation functions followed the training-set trends; generated black- and grey-phase values deviated slightly but remained 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.Source: §5.1 · p. 9, Figure 5 · p. 9

5.2 SOFC anode results

  • For the SOFC anode, morphological properties and effective diffusivity had comparable means and distributions in training and synthetic data, but the synthetic properties showed lower variance.The mean values and standard deviations are reported in Table 4.Source: §5.2 · p. 10, Figure 6 · p. 10, Table 4 · p. 10
  • The SOFC anode’s averaged two-point correlation functions showed exponential decay in the black phase, a small hole effect in the grey phase, and a pronounced hole effect in the white phase.The functions were calculated along all three Cartesian directions.Source: Figure 7 · p. 10, §5.2 · p. 11

5.3 Periodic boundaries

  • Applying circular spatial padding to the generator’s first transposed convolutional layer produced periodic microstructures of arbitrary size for both electrodes.In steady-state diffusion simulations, periodic rather than mirror boundaries opened more transport paths and enabled larger flux by maintaining continuity at the edges.Source: §5.3 · p. 11, Figure 8 · p. 11

6 Discussion

The DC-GAN generates three-phase microstructures with morphological and transport properties that agree well with real data, while enabling periodic-volume generation with potential to reduce simulation costs.

6 Discussion

  • Morphological metrics, relative diffusivities, and two-point correlations agree excellently between real and generated data; TPB density differs from the mean but remains within real-data confidence intervals.The authors describe the reconstruction as as accurate as state-of-the-art methods and note that the GAN learns voxel-wise phase distributions without requiring physical parameters as input.Source: §6 · p. 11
  • Generated Li-ion cathode samples show reduced property variation, consistent with slight mode collapse, while smooth latent interpolation suggests learned features rather than memorization.Figures 4 and 6 show small variance; unique-sample tests remain necessary [3,21]. Suggested loss alternatives for variation and stability include WGAN/WGAN-GP [2,16], LSGAN [37], and SN-GAN [65].Source: §6 · pp. 11–12
  • A trained generator can produce volumes larger than the 64^3 training sub-volumes by increasing latent input size, with image generation inexpensive after computationally costly training.The trained generator parameters total 55 MB; proposed uses include larger samples, periodic microstructures, and optimization through latent space.Source: §6 · p. 12
  • Periodic boundaries make smaller simulation volumes possible, and diffusion results show transport parameters closer to bulk response than with non-periodic boundaries.The authors note that further analysis of periodic generator architectures and generated-versus-real properties is ongoing.Source: §6 · pp. 12–13
  • Future work proposes optimizing latent space for desired 3D morphology, potentially adding physics-specific loss [47] or conditioning the generator on physical properties through G(z, y) [24].Related approaches include latent-space optimization [34] and 3D in-painting of details from 2D conditioning data, with EBSD information proposed as a possible addition [41,62].Source: §6 · p. 13

7 Conclusions

The DC-GAN generated n-phase 3D microstructures whose measured properties closely matched two real electrode datasets, while enabling arbitrarily large, periodically bounded volumes.

7 Conclusions

  • DC-GANs can generate 3D microstructures with any number of material phases, representing real microstructures’ statistical and morphological properties.These properties are captured in the trained discriminator and generator weights.Source: §7 · p. 13
  • Synthetic volumes showed excellent agreement with real data across all metrics, but had smaller variance than the training data.The authors note that reduced variance is commonly reported for DC-GANs; mitigation will be explored in future work.Source: §7 · p. 13
  • Training encountered instability, likely from vanishing gradients, and moderate mode collapse, both attributed to the GAN loss function.Solutions from the literature will be explored in future work.Source: §7 · p. 13
  • The method can generate arbitrarily large synthetic microstructures and periodic boundaries, both of interest to electrochemical modelling.A detailed study of periodic boundaries’ impact on simulation times is underway.Source: §7 · p. 13
  • Future work will use latent-space continuity and GAN differentiability to optimise morphological and electrochemical properties for improved battery and fuel-cell electrodes.Source: §7 · p. 13

Supplementary information · A Introduction to GANs

The introduction presents GANs as an adversarial two-player framework in which a generator creates synthetic samples from latent variables while a discriminator distinguishes generated from real data.

Supplementary information

A Introduction to GANs

  • The generator maps a d-dimensional latent vector of independent standard-normal variables into the image domain using parameters θ(G).The latent vector is drawn from prior distribution pz, corresponding to N(0, 1).Source: A Introduction to GANs · p. 18
  • The discriminator Dθ(D)(x) receives both real-data samples and synthetic samples generated by G(z).Source: A Introduction to GANs · p. 19
  • The discriminator aims to label real data 1 and generated data 0, while the generator tries to make D(G(z)) equal 1 in the minimax game.The passage identifies V(G, D)2 as the objective function to optimize.Source: A Introduction to GANs · p. 19

B Sample details … D.2 Relative diffusivities

The appendices describe the two three-phase electrode datasets and outline the encoding and definitions used to characterise morphology and relative diffusivity.

B.1 Li-ion cathode

  • The Li-ion cathode has NMC532, a carbon/binder domain of C45 carbon and PVDF, and a porous phase; the carbon/binder phase is generated over the X-ray CT geometry because it is not directly visible.The material was generated at Argonne National Laboratory's CAMP facility.Source: §B.1 · p. 19

B.2 SOFC anode

  • The SOFC anode sample has three phases and yields 45,492 overlapping 64^3-voxel training sub-volumes, cropped at spacing 16 from an image of approximately 4 × 108 voxels.The authors state that the image size is sufficient to represent the whole sample's two-point correlation function.Source: §B.2 · p. 19

C One-hot encoding

D.1 Morphological parameters

  • The morphological parameters are phase volume fraction, specific surface area, and TPB density, defined respectively by phase voxel share, interface area, and three-phase intersection length per domain volume.For a cuboid lattice, a TPB is an edge whose four connecting voxels include three different phases.Source: §D.1 · p. 20

D.2 Relative diffusivities

  • The tortuosity factor τ measures geometric deviation from a straight conductive path; τ = 1 indicates direct flow, and τ ≥ 1 for all systems.Source: §D.2 · p. 21
  • Tortuosity is inversely related to diffusivity; equation 8 relates it to conductive-phase volume fraction φi, intrinsic diffusivity Do, and effective diffusivity Deff.Source: §D.2 · p. 21
  • The tortuosity factor is obtained by dividing steady-state diffusive flow through the conductive phase by flow through a fully dense volume of equal size.Source: §D.2 · p. 21

E Relative diffusivity extended results · E.1 Lithium-ion cathode · E.2 SOFC anode

The extended results present relative diffusivity with respect to phase volume fraction for the three phases in both the lithium-ion cathode and SOFC anode.

E.1 Lithium-ion cathode

  • Figure S2 represents relative diffusivity with respect to the volume fraction of the three phases for the lithium-ion cathode.Source: §E.1 · p. 22, Figure S2 · p. 22

E.2 SOFC anode

F TPCF extended results · F.1 Lithium ion cathode · F.2 SOFC anode

For both the Li-ion cathode and SOFC anode, synthetic normalized two-point correlation functions follow the real-data trends and represent anisotropy in some phases. Tables describe TPCF decay and stabilization along three axes for each phase.

F.1 Lithium ion cathode

  • The Li-ion cathode synthetic TPCFs represent anisotropy in some phases along the three directions.Source: §F.1 · p. 23, Figure S4 · p. 23
  • For the Li-ion cathode, Table 1 details normalized TPCF decay and stabilization along three axes for pores, NMC, and binder.Source: §F.1 · p. 23, Table 1 · p. 23
  • In the Li-ion cathode, synthetic normalized TPCFs for pores, NMC-532, and binder follow the real-data trends.Figure S4 presents S2(r)/S2(0) along the three directions.Source: Figure S4 · p. 23, §F.1 · p. 23

F.2 SOFC anode

  • The SOFC anode synthetic TPCFs represent anisotropy in some phases along the three directions.Source: §F.2 · p. 24, Figure S5 · p. 24
  • For the SOFC anode, Table 2 details normalized TPCF decay and stabilization along three axes for pores, Ni, and YSZ.Source: §F.2 · p. 24, Table 2 · p. 24
  • In the SOFC anode, synthetic normalized TPCFs for pores, Ni, and YSZ follow the real-data trends.Figure S5 presents S2(r)/S2(0) along the three directions.Source: Figure S5 · p. 24, §F.2 · p. 24

G Representativity … K Uncertainty maps

The sections assess training-volume representativity and describe how trained generators produce larger samples, alongside checks of computational scaling, mode collapse, and voxel-level uncertainty. They report that 64^3 training sub-volumes fairly represent volume fraction by the two-point statistic criterion, while uncertainty falls through training and latent-space interpolation shows no total mode collapse.

G Representativity

  • The 64^3 training size was selected to capture an average 35-voxel NMC particle and neighbouring particles, while the SOFC anode size was based on volume-fraction representativity.The anode's three phases form continuous networks without an observed structuring element, so a standard 64^3 training size was used.Source: G Representativity · p. 25
  • Choosing training size is difficult for irregular, highly porous, or continuously fibrous media, and assessing specific-surface-area or TPB representativity experimentally is exhaustive and microstructure-specific.The authors recommend research on long-range functions that can assess representativity for properties beyond volume fraction.Source: G Representativity · pp. 25–26
  • For the two materials analysed, S2(r) stabilised at an average of 40 voxels along three directions and across three phases, showing 64^3 training sub-volumes fairly represent microstructure at least in volume fraction.Source: G Representativity · p. 25

H Generating larger volumes

  • 320^3-voxel samples were generated from generators trained on 64^3 sub-volumes, with sample size increased by enlarging the input z.The minimum generated sample size matches the training sub-volume size, and can be increased arbitrarily by increasing z.Source: H Generating larger volumes · p. 26, Figure S6 · p. 26

I Computational expense

J Mode collapse

  • Smooth transitions between interpolated latent-space samples indicate that the generator did not memorise the training set, avoiding total mode collapse in both microstructures.The authors interpret the transitions as evidence that the generator learned a lower-dimensional representation of z.Source: J Mode collapse · p. 28, Figure S8 · p. 28

K Uncertainty maps

  • By epoch 50, most uncertainty-map voxels were white, with higher uncertainty at interphases than in bulk and uncertainty decreasing during training.White denotes high phase-assignment certainty and black denotes low certainty; the maps cover both the Li-ion cathode and SOFC anode.Source: K Uncertainty maps · p. 29, Figure S9 · p. 30, Figure S10 · p. 31
Loading 2003.11632v2…