Deep learning for microstructural characterization

The digitization of material microstructure via X-ray CT has necessitated robust algorithms for quantifying topology. Early data-driven approaches relied heavily on CNNs to extract features from voxelized volumes. For instance, Elmorsy et al. [11] employed multi-scale convolutional architectures to predict effective properties directly from raw tomograms, while Alqahtani et al. [12] provided a comprehensive review of 2D/3D CNN applications in porous media. However, these Euclidean architectures struggle with the curse of dimensionality. As noted by Li et al. [13], increasing the resolution of 3D-CNNs to capture critical interfacial features like the ITZ results in cubic growth in memory requirements. This often forces researchers to employ aggressive downsampling or sliding-window techniques, which obscure the global connectivity governing transport phenomena.

Moreover, Euclidean architectures inherently lack rotational invariance. To predict the behavior of an isotropic porous medium, a standard CNN must be trained on all possible rotations of the input, necessitating massive data augmentation [5]. To mitigate this, recent work has explored spectral analysis and topological data analysis (TDA) to extract invariant descriptors. Salehi et al. [14] demonstrated that partitioning-based schemes could improve efficiency, but such methods typically break the continuity of the fracture network. Consequently, the field is shifting toward discrete topological models, specifically graphs and unstructured meshes, which offer a sparse, rotation-invariant description of material geometry that aligns more closely with physical intuition.

Surrogate modeling of transport phenomena

The simulation of fluid transport in porous media is a primary bottleneck in materials design. While LBM provides ground-truth accuracy, its computational cost is prohibitive for iterative optimization. This has driven the development of GNN-based surrogates. Alzahrani [9] introduced “Pore-GNN,” a framework where message-passing operations along pore throats approximate Navier-Stokes solutions. By representing the pore space as a network of nodes, they achieved high-fidelity permeability predictions using a fraction of the parameters required by voxel-based solvers. Similarly, I’Anson et al. [15] recently validated that GNNs trained on LBM data could generalize across varying porosity regimes, effectively acting as high-speed inference engines for flow properties.

Despite these successes, most graph-based flow solvers rely on simplified skeletonization techniques that may discard geometric nuances critical to capillary dynamics. Zhao et al. [16] addressed this by integrating GNNs with traditional pore network modeling (PNM) to enhance the prediction of multiphase flow. Their results highlight that while pure graph learning captures topology, hybrid approaches that encode geometric attributes (such as throat radii and local roughness) into the node feature vectors yield superior convergence. This insight is central to our method, which enriches the graph representation with physical metadata derived from the raw X-ray CT scans.

Graph-based fracture mechanics

Parallel to fluid dynamics, advancements in computational solid mechanics have leveraged mesh-based learning to model deformation and failure [17]. The MeshGraphNets architecture proposed by Pfaff et al. [10] established a paradigm for learning Lagrangian dynamics by encoding the simulation state into graph edges. This approach has been successfully extended to complex contact dynamics and plasticity. Maurizi et al. [18] demonstrated that GNNs could model elastic-plastic deformation in heterogeneous materials by learning the constitutive laws directly from displacement data, bypassing the need for explicit formulation of the yield surface.

However, modeling fracture propagation presents a distinct challenge due to the dynamic change in topology. Zheng et al. [19] and Liu et al. [20] employed physics-informed deep learning to predict crack paths in brittle materials, utilizing energy minimization principles within the loss function. While these models excel at single-physics fracture, the interaction between mechanical damage and hydraulic transport, which is the defining characteristic of self-healing systems, remains largely unexplored. Existing architectures typically treat solid mechanics and fluid flow as separate domains, whereas the effective design of resilient composites requires a coupled multiphysics solver that accounts for how stress concentrations (solid) drive crack opening (geometry) and subsequent healing agent flow (fluid).

Generative models for material synthesis

The inverse aspect of materials design requires not just analyzing existing structures, but generating new ones. Generative models, such as Variational Autoencoders (VAEs) and Denoising Diffusion Probabilistic Models (DDPMs), have revolutionized this space. Yang et al. [21] applied diffusion models to the inverse design of mechanical metamaterials, generating complex lattice structures that match target stress-strain curves. Merchant et al. [3] further demonstrated the scalability of deep learning for discovering stable inorganic crystals, suggesting that generative AI can navigate the vast chemical space of materials.

Nevertheless, a critical limitation of image-based generative models in this context is their tendency to produce visually realistic but functionally suboptimal structures. Yazdani et al. [22] noted that while GANs can reproduce the texture of concrete, they often fail to enforce long-range connectivity constraints required for percolation. Furthermore, standard generative models operate in Euclidean space. Generating valid graphs that satisfy physical constraints (e.g., non-overlapping aggregates, connected pore networks) is significantly harder than generating pixel grids. This necessitates the use of graph-generative approaches or constructive algorithms that can build valid topologies from the ground up, ensuring that every generated candidate is physically realizable.

Active learning and bayesian optimization

To close the loop between generation and simulation, Bayesian Optimization (BO) has become the standard for navigating high-dimensional design spaces. By utilizing Gaussian Processes (GPs) to model uncertainty, BO sequentially selects candidates that balance exploration and exploitation. Lookman et al. [23] provided the foundational information-theoretic perspective for this approach in materials science. Recent applications have been highly effective: Thomas et al. [24] utilized BO to optimize the fiber orientation in reinforced composites, significantly outperforming genetic algorithms in convergence speed. Similarly, Kim et al. [25] employed an active learning loop to accelerate the discovery of stable perovskites, reducing the number of required experiments by an order of magnitude.

In the specific domain of cementitious materials, Pessoa et al. [26] and Huang et al. [27] applied machine learning to predict healing efficiency. However, these studies largely operated in a forward prediction mode or utilized pre-defined datasets. They lacked a truly autonomous cycle where the system creates novel microstructural designs based on feedback from a surrogate physics model. The GTAL framework differentiates itself by integrating the GNN surrogate directly into the acquisition function, allowing the optimizer to query the physics of the material at minimal computational cost and thereby enabling the exploration of complex, non-intuitive design strategies that maximize microcapsule survival and healing efficacy.

Positioning relative to graph-based pore surrogates

To clarify the incremental contribution of the present work, Table S1 in S1 Appendix provides a structured comparison of LSE-MGN against three representative graph-based approaches along five axes: graph type, edge physics, long-range propagation, multiphysics coupling, and closed-loop optimization. In brief, Pore-GNN [9] operates on a single-type pore-to-pore graph for permeability prediction only; MeshGraphNets [10] use local physical contact edges on simulation meshes with uniform edge types and no spectral augmentation; Reiser et al.’s graph-based fluid surrogates [7] predict permeability from homogeneous pore graphs. None of these methods handles coupled multiphysics (simultaneous transport and fracture) on a heterogeneous substrate, nor integrates spectral shortcuts for long-range fracture dynamics, nor embeds the surrogate within a closed-loop Bayesian optimization system.

Synthesizing these developments reveals a critical methodological gap. While GNNs have proven effective for isolated physical domains and BO has accelerated parameter tuning, no existing framework successfully couples topological representation with multiphysics surrogate modeling to navigate the survival-versus-healing conflict in composite design. Current approaches either simplify the geometry to the point of losing critical ITZ dynamics or rely on computationally expensive LBM/DEM solvers that preclude extensive exploration. Consequently, there is an urgent need for a closed-loop system capable of learning the complex mapping between stochastic graph topologies and competing macroscopic properties. This research addresses this necessity by proposing GTAL, a unified paradigm that bridges the divide between geometric deep learning and autonomous materials discovery.

Topological discretization via heterogeneous multigraphs

Euclidean representations (voxels) introduce prohibitive sparsity and lack rotation invariance. We fundamentally recast the material volume as a permutation-invariant heterogeneous graph . This mapping preserves the topological genus of the pore network and the contact topology of the solid phase.

Manifold-aware node feature engineering

The node set is semantically partitioned into (pores, aggregates, capsules). The segmentation employs a marker-controlled watershed transform [28] on the binarized solid phase (Otsu threshold). Markers are seeded at local maxima of the distance transform with a minimum spacing of  voxels. Over-segmentation is suppressed by merging regions whose shared boundary gradient is below a threshold (normalized units). Isolated components with volume  voxels³ are pruned as segmentation noise, and edges exceeding (three standard deviations above the mean edge length) are removed as physically implausible. Full pipeline pseudocode is provided in Algorithm S1 (S1 Appendix). A systematic sensitivity analysis over watershed marker spacing (), pore-throat detection threshold (), and synthetic segmentation noise (SNR) is reported in S3 Appendix; performance varies by in R² across these perturbations, confirming robustness. We recommend  voxels and as defaults.

Following semantic segmentation via the marker-controlled watershed transform, we extract a feature vector for each node :

(1)

Here, is the centroid, is the volume, and is the sphericity. Crucially, to encode the rotational anisotropy of microcapsules, we compute the inertia tensor and perform an eigendecomposition . The flattened eigenvector matrix serves as a rotation-equivariant descriptor of orientation. and represent the phase-specific Young’s modulus and Poisson’s ratio, respectively.

Physics-based edge construction

The edge set is a union of two disjoint sets, , enabling the graph to switch dynamic modes between fluid and solid simulations.

Hydraulic Connectivity ().

We define edges between pore nodes based on the Maximal Ball Algorithm. An edge exists if the two pores share a constriction (throat) that limits non-wetting phase entry. The edge attribute explicitly encodes the Hagen-Poiseuille conductance:

(2)

where is the throat radius, is the Euclidean distance, is fluid viscosity, and is a shape factor accounting for throat cross-sectional irregularity. Following Mason and Morrow [29], , where and are the cross-sectional area and perimeter of the minimum throat cross-section, extracted by planimetry at the throat centroid. The Hagen-Poiseuille conductance is valid under laminar, incompressible, Newtonian flow (Re 1, Kn 1) and the hydraulic radius approximation.

Mechanical Force Chains ().

Connectivity for solid phases () is established via Weighted Delaunay Triangulation. To model the Interfacial Transition Zone (ITZ), the primary nucleation site for cracks, we introduce a novel interface weakness metric . We compute by integrating the X-ray attenuation gradient along the path connecting centroids:

(3)

High gradient accumulation indicates a sharp material boundary (weak ITZ), whereas low accumulation implies a diffuse, stronger bond. The path is the shortest Euclidean path connecting the centroids of and within the solid phase, computed by Dijkstra’s algorithm on the voxel graph. The integral is approximated as a discrete sum , where are voxels along ,  voxel, and is computed by a Sobel operator. The raw values are z-score normalized over the training set: . We note that is a relative proxy for phase boundary density, not independently calibrated against ITZ fracture toughness measurements; absolute calibration via nano-indentation is identified as a future direction in Sec. 5. This scalar is appended to the edge features alongside the equilibrium distance .

Long-Short-Edge MeshGraphNet (LSE-MGN)

Standard GNNs suffer from the “oversmoothing” phenomenon when network depth increases, which hinders the modeling of global fracture propagation. We propose the Long-Short-Edge MeshGraphNet (LSE-MGN), which introduces spectral shortcuts to decouple local deformation from global stress redistribution.

Fig 2 illustrates the LSE-MGN architecture.

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Fig 2. Architecture of the LSE-MGN.

The model processes inputs via two parallel streams: a local stream operating on physical edges (Mesh) and a global stream operating on spectral edges (Long-Range). A Multi-Head Attention mechanism fuses these streams, allowing the network to dynamically prioritize stress concentrations (local) or load balancing (global).

https://doi.org/10.1371/journal.pone.0353692.g002

Spectral long-range edge generation

To facilitate rapid information transfer, we augment with virtual edges . We compute the normalized graph Laplacian and extract the k lowest-frequency eigenvectors . Nodes are connected in if their spectral distance is minimized, linking topologically central nodes regardless of Euclidean separation. We use k = 16 eigenvectors in all experiments, selected via an ablation over (S4 Appendix). Each node connects to its k most spectrally proximate neighbors, increasing the mean node degree from (local only) to . Performance plateaus at k = 16 () and degrades slightly at k = 64 due to over-connectivity. The eigendecomposition via ARPACK requires 180 ms per graph () and is performed once during preprocessing, not at inference time. Spectral proximity in the Laplacian eigenbasis corresponds to topological similarity: spectrally close nodes tend to occupy similar structural environments (e.g., load-bearing aggregate clusters or high-porosity zones), enabling the network to aggregate information from topologically analogous regions. The MHA gate (Eq. 5) learns to assign appropriate weights; at fracture nucleation, local ITZ edges dominate (mean ), whereas during linear elastic loading, long-range attention is high (mean ).

Dual-stream message passing

The node state at layer l is updated via asynchronous message passing. We define two aggregation functions: Agg_short for physical contact forces and Agg_long for global continuity constraints.

(4)

To fuse these streams, we employ a Multi-Head Attention (MHA) mechanism:

(5)

where the attention coefficients determine the relative importance of local interlocking versus global stiffness. This allows the network to learn that during linear elastic loading, global edges dominate, whereas during fracture nucleation, local edges (ITZ) govern the response.

For full reproducibility, the multi-head attention and training configuration is fixed as follows across all experiments: hidden dimension d = 128, L_mp = 6 message-passing layers, 4 attention heads (head dimension 32), attention dropout 0.1, ReLU activations, and LayerNorm residual fusion (Algorithm S2). Training uses AdamW (learning rate 10⁻³, weight decay 10⁻⁵), batch size 16, cosine-annealing schedule, and early stopping on a 10% validation split (patience 20 epochs). The exact discrete formulation of the interface weakness metric (Sobel-gradient line integral along the Dijkstra solid-phase path, with z-score normalization) is given in the Methods and Algorithm S1 in S1 Appendix, and the inlet/outlet/displacement boundary-condition implementation for and is specified in S1 Appendix and illustrated in Fig S1.

Physics-informed objective function

Training is regularized by physical constraints to ensure thermodynamic consistency. The total loss is a weighted sum:

(6)

The optimization of the proposed architecture is governed by a composite loss function that rigorously balances empirical accuracy with physical consistency. The primary data fidelity term, , minimizes the Huber loss between the predicted macroscopic properties, namely permeability and survival rate, and their ground-truth counterparts derived from high-fidelity LBM and DEM simulations. The Huber loss is specifically selected for its robustness to outliers, offering a stable gradient response even when the stochastic nature of the microstructure induces high variance in the training labels.

To ensure that the learned representations adhere to fundamental conservation laws, we incorporate domain-specific regularization terms. For the hydraulic prediction stream, we introduce a continuity constraint, , which strictly enforces Kirchhoff’s law at every pore node . By minimizing the divergence of the flux field, defined as the squared norm of the net flux entering each node, the network is compelled to satisfy local mass balance. This acts as a soft constraint that guides the surrogate model toward physically valid steady-state flow solutions without requiring explicit PDE integration:

(7)

Complementing the hydraulic constraints, the mechanical predictions are regularized via a strain energy term, . The surrogate predicts, for each node , a displacement vector and a stress tensor . The energy residual penalizes violation of strain-energy compatibility:

(8)

where is the symmetric strain tensor approximated by least-squares over [30], is the known stiffness tensor of the phase at node , and denotes the Frobenius norm.

For , the surrogate predicts a pressure scalar at each pore node . The inter-node flux is computed from the predicted pressures via the hydraulic conductance edge attribute: , where . Boundary conditions (prescribed inlet/outlet pressure) are enforced by fixing at boundary nodes to the training labels’ inlet/outlet values; these nodes are excluded from the continuity loss sum.

The weights and were determined by coarse grid search followed by Bayesian optimization (Optuna, 50 trials) over . A sensitivity analysis (S5 Appendix) shows that is relatively insensitive to ( variation), while removing () decreases by , confirming its decisive role for mechanical prediction.

Two points raised in review merit explicit treatment. First, on the choice of soft penalty terms over hard constraint enforcement: because the DEM survival labels are themselves stochastic (the high-shear rupture process is sensitive to packing realization), a hard projection onto the strain-energy-compatible manifold would force the network to fit label noise that is not physically meaningful at the level of an individual capsule, and in preliminary trials a projection-layer variant reduced by relative to the soft penalty while roughly doubling training time. The soft penalty instead acts as a Tikhonov-type regularizer whose strength is tuned to the signal-to-noise ratio of the labels, which is precisely the quantity that the sensitivity sweep in S5 Appendix calibrates. Second, the small optimal magnitudes () are consistent with this interpretation: the residuals and enter as auxiliary consistency objectives that shape the loss landscape and prune non-physical minima, rather than as dominant terms that would override the empirical data fit in the presence of noisy DEM/LBM labels.

This component penalizes latent configurations that violate the principle of minimum potential energy, thereby ensuring that the predicted stress field remains thermodynamically consistent with the resulting displacement field. By embedding these first-principles constraints directly into the optimization landscape, the framework effectively prunes the solution space of non-physical local minima, ensuring that the discovered microstructures are not only high-performing but also physically realizable.

Bayesian optimization with GNN surrogates

The LSE-MGN acts as a deterministic surrogate . To enable active learning, we model the residuals using a Gaussian Process (GP) prior. Let the dataset at iteration t be . We approximate the objective function as:

(9)

We utilize an Anisotropic Matérn-5/2 kernel to handle the varying sensitivity of the objective to different synthesis parameters. The kernel length scales for each of the d = 5 design parameters are initialized to and jointly optimized by maximizing the log marginal likelihood via L-BFGS-B with 20 random restarts at each BO iteration. The next experimental candidate is selected by maximizing the Expected Improvement (EI):

(10)

The acquisition function balances exploration (high variance regions) and exploitation (high mean regions). EI is maximized using Sobol quasi-random initialization (10⁴ candidates) followed by multi-start L-BFGS-B local optimization (50 restarts from the top 50 Sobol candidates), following the standard BoTorch [31] protocol. Upon selection, if the surrogate’s epistemic uncertainty exceeds a confidence threshold , the system triggers the high-fidelity oracle (LBM/DEM), generates new ground truth, and updates . The complete BO loop executed exactly 30 high-fidelity LBM + DEM evaluations in addition to 50 warm-start samples (total oracle budget: 80 runs, 496 CPU-hours). By comparison, a full factorial grid over the 5-parameter design space would require 21,600 evaluations, representing a reduction in oracle cost. The iteration-by-iteration protocol is formalized in Algorithm 1.

We adopt a GP on the surrogate residual rather than propagating the GNN’s intrinsic epistemic uncertainty into the acquisition function, and we make the rationale and its limitation explicit. In the present low-dimensional setting (d = 5 continuous synthesis parameters), a residual GP is the standard, well-conditioned choice: it yields closed-form posterior variance and an analytic Expected Improvement, and it isolates exploration in the 5-dimensional parameter space from the much higher-dimensional graph-feature space in which a GNN-native uncertainty (deep ensembles or Monte Carlo dropout) would operate. We verified that the residual GP is not the bottleneck by comparing it against a 5-member deep-ensemble acquisition on the warm-start pool; the two strategies selected overlapping candidates and reached the same optimum within the run-to-run variance of Fig 4, so we retain the simpler GP. Regarding kernel conditioning, with d = 5 the anisotropic Matérn-5/2 length scales remained well separated (condition number of the fitted covariance <10⁴ throughout), and we did not observe the premature convergence that ill-conditioning would produce. We agree, however, that for substantially higher-dimensional synthesis spaces () a GNN-native uncertainty integrated directly into the acquisition function would be the more principled choice, and we flag this as a direction in Sec. 5.

Dataset generation and simulation protocols

To circumvent the scarcity of public benchmarks for self-healing cementitious composites, we generated a comprehensive synthetic dataset comprising 5,000 stochastic microstructures. The generation process employed a random sequential packing algorithm to place ellipsoidal aggregates and microcapsules within a 100³ voxel domain, ensuring no physical overlap between solid phases. Following packing, a Diamond-Square fractal algorithm was applied to synthesize realistic, spatially correlated pore networks. The resulting dataset exhibits a diverse range of porosity (15% to 35%) and capsule volume fractions (2% to 10%). This wide parametric variance ensures a rigorous test of the model’s interpolation capabilities, forcing the surrogate to learn the global mapping rather than memorizing local geometric motifs.

We acknowledge that synthetic generation cannot fully reproduce the complexity of real cement paste, and we make the intended correspondence explicit. The porosity envelope (15% to 35%) brackets the capillary porosity of ordinary Portland cement paste over the practically relevant range of water-to-cement ratios ( to 0.55), and the ellipsoidal-aggregate plus fractal-pore construction reproduces the first- and second-order statistics that most strongly govern transport and force-chain percolation, namely the phase-volume fractions, the two-point spatial correlation length, and the aggregate-size distribution. The generator does not reproduce several features of hydrating cement: the curing-induced evolution of the C-S-H gel pore network, the chemically graded interfacial transition zone, capsule polydispersity beyond the sampled aspect-ratio range, and realistic segmentation/partial-volume artifacts. These omissions are precisely the sources of the synthetic-to-real gap, and they motivate both the segmentation-noise robustness study (S3 Appendix) and the experimental-scan validation (S6 Appendix). The synthetic dataset is therefore intended as a controlled, physics-labeled training corpus for the surrogate, not as a substitute for experimental microstructures.

Ground-truth labeling was performed using coupled high-fidelity physics engines, creating a “digital twin” for each microstructure. For fluid dynamics, we utilized the OpenLB library [32] to solve the Lattice Boltzmann equation (D3Q19 scheme), calculating the effective permeability tensor for each sample. Concomitantly, the mechanical response was simulated using LIGGGHTS [33], a Discrete Element Method (DEM) solver. We simulated a high-shear mixing environment with a shear rate of 50 s^-1, quantifying the survival rate as the ratio of intact microcapsules post-processing. The capsule rupture criterion in the DEM model is a maximum-stress threshold: a capsule is recorded as ruptured when the peak von Mises stress on its shell exceeds  MPa, a value representative of urea-formaldehyde microcapsule shells reported in the self-healing literature. Capsule-matrix bonding is modeled via a cohesive contact with normal/tangential stiffness  N m⁻¹ and a shell-to-diameter thickness ratio of 0.08. These parameters are held fixed across the dataset; we emphasize that they are nominal literature-derived values rather than measurements on a specific cement chemistry, and that the survival rate is therefore a mechanical-robustness proxy under a single representative parameter set. This dual-physics labeling protocol is critical; by training on both transport (permeability) and failure (survival) manifolds simultaneously, the surrogate model is forced to learn a representation that reconciles the conflicting geometric requirements of fluid connectivity and mechanical stiffness.

Surrogate model benchmarking

We benchmark the proposed LSE-MGN against three distinct architectures representing the state-of-the-art in geometric deep learning: a 3D-ResNet-50 operating on voxel grids [34], PointNet++ operating on centroid point clouds [35], and a standard Graph Attention Network (GAT) [36] without spectral edges. These baselines were selected to isolate the benefits of specific representational choices: ResNet tests the efficacy of Euclidean grids, PointNet++ tests the necessity of explicit connectivity, and GAT tests the impact of long-range message passing.

As presented in Table 1, the LSE-MGN achieves a decisive performance advantage. The 3D-ResNet struggles significantly with the sparsity of the pore network, exhibiting a high Normalized Root Mean Squared Error (NRMSE) of 0.18 for permeability. This result confirms that Euclidean grids induce significant computational waste and feature blurring in porous media, as the convolution kernels spend the majority of their capacity processing empty void space. Similarly, PointNet++ fails to accurately predict survival rates (R² = 0.68), yielding the poorest performance on the mechanical task. This failure is attributable to the architecture’s reliance on global pooling of point features, which discards the explicit edge connectivity required to model the force chains and inter-particle friction that govern granular failure.

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Table 1. Performance Comparison of Surrogate Models on Test Set (meanstd, 5 seeds). NRMSE. All models trained on 4,000 samples; tested on 1,000 held-out samples (80/20 stratified split). All models tuned via Bayesian hyperparameter search (Optuna 3.4, 100 trials each).

https://doi.org/10.1371/journal.pone.0353692.t001

In contrast, our LSE-MGN achieves an R² > 0.94 across both tasks, validating that the heterogeneous multigraph representation successfully captures the topological nuances of the composite. The graph structure naturally encodes the pore throat network (for permeability) and the contact network (for mechanics) within a unified framework. Furthermore, the low NRMSE (0.03 and 0.04) suggests that the model has not only learned the mean trends but also captures the high-frequency variations induced by stochastic aggregate placement.

To isolate the contribution of each novel component, we include three additional baselines. An SE(3)-equivariant E3NN [37] point cloud network achieves and , outperforming PointNet++ but underperforming LSE-MGN. We attribute this gap to the fact that our composite system is not globally isotropic: the shear direction, gravity axis, and casting direction define preferred orientations, and enforcing global rotational symmetry as a hard constraint is suboptimal. Two ablation baselines further decompose the gains: LSE-MGN without spectral edges () and LSE-MGN without physics-informed losses (). Spectral edges contribute primarily to permeability () by facilitating long-range pressure field propagation, while physics losses contribute primarily to survival rate () by enforcing energy consistency in the mechanical response. This orthogonality confirms that the two components are complementary rather than redundant.

Ablation study: the role of spectral edges

To isolate the contribution of the spectral long-range connections, we conducted an ablation study comparing the full LSE-MGN against a variant operating solely on physical mesh edges (Mesh-Only). Fig 3 illustrates the distribution of prediction error as a function of microcapsule aspect ratio. The Mesh-Only model exhibits a sharp degradation in performance as the aspect ratio increases, indicating an inability to model the global bending moments and stress concentrations associated with highly anisotropic inclusions. In purely local message-passing schemes, information regarding a stress concentration at one tip of a long capsule must propagate node-by-node to affect the global state; for high aspect ratios, this path length becomes prohibitive, leading to signal attenuation.

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Fig 3. Error Distribution Heatmap.

The relative error in survival rate prediction is plotted against the microcapsule aspect ratio. The “Mesh-Only” ablation (left) shows significant error accumulation for highly anisotropic capsules, whereas the proposed LSE-MGN (right) maintains robust performance, demonstrating the necessity of spectral edges for capturing global mechanics.

https://doi.org/10.1371/journal.pone.0353692.g003

Conversely, the full LSE-MGN maintains a low error variance regardless of capsule geometry. This stability confirms that the spectral shortcuts effectively short-circuit the message-passing distance, allowing the network to propagate stress information across the domain instantly. By connecting topologically central nodes regardless of Euclidean distance, the spectral edges act as “highways” for gradient flow. This architectural feature is critical for modeling stiff, heterogeneous solids where the macroscopic failure mode is determined by non-local interactions between disparate microstructural features.

Active learning convergence analysis

The core utility of the GTAL framework lies in its ability to navigate the design space efficiently. We executed a closed-loop optimization targeting a scalar utility function , initializing the process with 50 random samples. We compared our Bayesian acquisition strategy against a Genetic Algorithm (NSGA-II) and Random Search. The convergence trajectories, depicted in Fig 4, reveal that GTAL reaches the optimal utility plateau within 30 iterations. In contrast, the Genetic Algorithm requires over 85 iterations to achieve comparable performance, while Random Search fails to converge. This acceleration is a direct consequence of the Gaussian Process (GP) acquisition function, which intelligently balances exploration (sampling high-uncertainty regions) and exploitation (refining high-performance regions), drastically reducing the number of expensive ground-truth evaluations required.

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Fig 4. Optimization Convergence Rates.

The GTAL framework (blue) rapidly maximizes the composite utility function, outperforming the Genetic Algorithm (orange) and Random Search (grey). Shaded regions indicate the standard deviation over 5 independent optimization runs.

https://doi.org/10.1371/journal.pone.0353692.g004

The active learning agent successfully identified a non-intuitive design region: microcapsules with an aspect ratio of oriented at to the shear plane. This configuration represents a sophisticated physical trade-off that human intuition might overlook. The orientation minimizes the projected cross-sectional area perpendicular to the primary shear flow during mixing, thereby enhancing the survival rate. Simultaneously, the moderate aspect ratio of 2.5 maximizes the probability that the capsule will span across propagating microcracks, ensuring effective release of the healing agent. The ability of the framework to autonomously converge on this specific geometric “sweet spot” highlights its potential for genuine materials discovery. To verify that this optimum reflects a genuine resolution of the competing constraints rather than a statistical artifact of the synthetic dataset, we derive the two competing geometric quantities analytically in S7 Appendix: the orientation-dependent projected shear area (governing survival) and the expected crack-interception cross-section (governing healing). The analytical product is independently maximized in the neighborhood of , , in agreement with the surrogate-discovered optimum and confirming that the two objectives are co-optimized rather than coincidentally aligned. We nonetheless emphasize that this is a geometric/mechanical argument; direct experimental confirmation via capsule fabrication and crack-interception testing remains future work (Sec. 5).

Interpretability and attention dynamics

Beyond predictive accuracy, the LSE-MGN offers interpretability through its attention mechanism, providing transparency into the model’s decision-making process. By visualizing the learned attention weights on the mechanical edges, we can identify which microstructural features the network deems critical for failure prediction. As shown in Fig 5, the model assigns the highest attention weights (depicted in red and with increased thickness) to the edges within the Interfacial Transition Zone (ITZ) surrounding the microcapsules and aggregates.

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Fig 5. Attention Weight Visualization.

A cross-section of the graph where edge thickness and color intensity correspond to the learned attention weights . The network focuses intensely on the Interfacial Transition Zone (ITZ) (red clusters), autonomously identifying the regions physically responsible for crack nucleation. (b) Scatter plot of mean attention weight versus the DEM-derived stress concentration factor (). (c) Histogram of attention stability (coefficient of variation of the top-10 edge attention weights) across 200 test microstructures.

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This visualization aligns perfectly with classical fracture mechanics theory, which identifies the ITZ as the “weakest link” in cementitious composites where stress concentrations arise and cracks nucleate. Crucially, the network was never explicitly supervised to focus on the ITZ; it autonomously learned that these interfacial regions contain the highest information density for minimizing the survival rate loss. This validates that the graph representation is capturing true physical causality rather than spurious correlations, effectively rediscovering the mechanics of composite failure from raw data.

To quantify this observation, we computed Spearman rank correlations between mean attention weights and two physical metrics across 200 test microstructures: the DEM-derived stress concentration factor and the ITZ thickness . The results show with (higher stress concentration → higher attention) and with (thicker ITZ → lower attention, consistent with weaker but more diffuse interfaces). The coefficient of variation of attention weights for the top-10 edges is across test microstructures, indicating stable but geometry-dependent attention patterns. Fig 5 now includes a scatter plot of versus and a histogram of attention stability (Fig 5b and c).

Computational complexity and inference speed

For a surrogate model to drive an active learning loop effectively, it must offer orders-of-magnitude acceleration over ground-truth solvers without incurring prohibitive memory costs. We scrutinized the computational tractability of the proposed LSE-MGN against the baselines and the full-physics simulation (LBM coupled with DEM). Table 2 details the parameter count, GPU memory consumption during inference, and the average time required to evaluate a single microstructural design. The full-physics simulation requires approximately 6.2 hours per sample on a high-performance CPU cluster, rendering it infeasible for real-time optimization. While the 3D-ResNet-50 offers rapid inference (8 ms), it suffers from extreme memory complexity (O(N³)), requiring 14.2 GB of VRAM to process a 100³ voxel grid. This cubic scaling severely limits batch sizes and precludes the analysis of higher-resolution volumes.

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Table 2. Computational Efficiency Analysis (single-sample, inference phase). Hardware: NVIDIA A100 40 GB SXM4, CUDA 12.1, PyTorch 2.1.0, PyG 2.4.0, Python 3.11. Inference times: medians over 100 runs (batch size = 1; 10 JIT warm-up runs discarded). Preprocessing: voxelization (ResNet), point subsampling (PointNet++, E3NN), Delaunay + eigendecomposition (graph methods).

https://doi.org/10.1371/journal.pone.0353692.t002

In contrast, our LSE-MGN operates on the sparse graph representation, reducing memory usage to 1.8 GB. This efficiency stems from the fact that graph networks only process the topologically relevant entities (pores and particles), ignoring the redundant voxel data that plagues grid-based methods. Although the inference time (12 ms) is slightly higher than the ResNet due to the irregular scatter-gather operations inherent to message passing, it remains negligible compared to the physics solver. When including preprocessing (graph construction + eigendecomposition = 190 ms), the total per-sample latency is 202 ms, yielding an approximately -fold acceleration over the physics oracle. The inference-only speedup is approximately -fold. This acceleration enables the active learning agent to query thousands of candidate designs per hour, a throughput rate that is essential for navigating the combinatorial design space and moving from localized optimization to global discovery.

Scalability analysis

To assess the practical applicability of LSE-MGN at higher resolutions, we conducted a scalability analysis varying the domain side length . The number of nodes scales as : from at L = 50 to at L = 200. LSE-MGN inference time scales approximately linearly with (12 ms at L = 100, 89 ms at L = 200), while VRAM usage remains within 14 GB even at L = 200. By contrast, 3D-ResNet-50 exceeds 40 GB VRAM at L = 150 (out of memory), rendering it infeasible at higher resolutions. The linear scaling of LSE-MGN is a direct consequence of the message-passing complexity per layer. Full scalability data including preprocessing times are reported in Table S6 (S4 Appendix) and Fig 6 (new panel).

Two scaling caveats raised in review warrant comment. First, the spectral preprocessing is the component that scales least favorably: the partial eigendecomposition (ARPACK, k = 16 modes) grows roughly as and reaches 1.82 s at L = 200 (). For industrial-resolution domains exceeding 500³ voxels, dense eigensolvers become impractical; the established remedy, which we have not yet implemented, is to replace global Laplacian eigenvectors with a hierarchical or locally-windowed spectral construction (e.g., multiscale/Nyström approximation of the eigenbasis), which preserves the long-range connectivity benefit at near-linear preprocessing cost. Because eigendecomposition is a one-time preprocessing step and is excluded from the inference-time speedup, it does not affect the reported -fold inference acceleration, but it does bound the practical domain size in the current implementation. Second, the present surrogate models the static-equilibrium permeability and processing-survival of a fixed topology; it does not evolve the graph as a crack propagates. Capturing the time-dependent diffusive-reactive healing process, in which the topology changes as cracks open and seal, would require a temporal or autoregressive extension in which edges are added/removed between message-passing steps, analogous to dynamic-graph MeshGraphNet variants. We treat this dynamic, time-resolved formulation as a distinct and substantial extension rather than a parameter change to the current model, and discuss it in Sec. 5.

Generalization to out-of-distribution topologies

A critical limitation of pure data-driven models is their tendency to memorize the training distribution, failing when exposed to unseen geometric regimes. To evaluate the robustness of the GTAL framework, we conducted an out-of-distribution (OOD) generalization test. The models were trained on microstructures with low-to-medium porosity (10% to 20%) and then evaluated on a held-out test set comprising high-porosity samples (25% to 40%). This shift is significant because high porosity often leads to percolation (the formation of connected pathways across the domain), which induces non-linear shifts in the permeability tensor that simple interpolation cannot capture.

Fig 6 depicts the degradation in prediction accuracy (NRMSE) as the test samples deviate from the training distribution. The Euclidean baseline (3D-ResNet) exhibits catastrophic failure in the high-porosity regime, with error rates spiking above 0.45. This suggests the CNN learned local texture heuristics, such as local pore density, rather than the global physics of flow connectivity. Conversely, the LSE-MGN demonstrates “graceful degradation,” maintaining an NRMSE below 0.12 even at 40% porosity. This robustness is attributed to the physics-informed loss functions () and the topological graph inductive bias. By enforcing Kirchhoff’s conservation laws at the node level, the LSE-MGN learns a generalizable physics rule that holds true regardless of the macroscopic porosity, allowing it to extrapolate effectively to denser, more complex pore networks.

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Fig 6. Out-of-Distribution Generalization.

The plot illustrates the Normalized RMSE for permeability prediction as the test set porosity increases beyond the training range (10% to 20%, shaded gray). The LSE-MGN (blue line) maintains stability in the unseen high-porosity regime, whereas the 3D-ResNet (red dashed line) and PointNet++ (green dotted line) fail to generalize, indicating overfitting to the training topology.

https://doi.org/10.1371/journal.pone.0353692.g006

As a preliminary validation on real data, we processed one experimental synchrotron CT scan from the dataset of Gallucci et al. [38] (Portland cement paste, w/c = 0.40, 28-day hydration, 0.70 m voxel resolution) through our standard pipeline without domain adaptation or fine-tuning. The LSE-MGN trained exclusively on synthetic data achieves on the three orthogonal permeability values, providing proof-of-concept that the model generalizes to real experimental microstructures. Details are provided in S6 Appendix. We deliberately refrain from interpreting this single-specimen result as evidence of broad real-data generalization: the R² = 0.88 is computed over only three orthogonal directional permeabilities from one specimen and is therefore a qualitative consistency check (correct anisotropy ordering and 10% directional error), not a statistically powered validation. Establishing generalization would require a multi-specimen campaign spanning several w/c ratios, curing ages, resolutions, and segmentation-noise levels, which we identify as the single most important experimental extension of this work (Sec. 5).