Multi-fidelity design and optimization framework of damage-tolerant fiber-reinforced composite structures via CNN-assisted homogenization

Lightweight fiber-reinforced materials (FRM) have emerged as essential materials in various engineering domains due to their exceptional strength-to-weight ratios and customizable behaviors (Talreja and Waas, 2022; Duan et al., 2025; Dou et al., 2025), including aerospace, automotive, and civil engineering. Their structural composition heavily influences their mechanical performance, particularly the arrangement of fibers and the type of filler materials used within the matrix (Nguyen-Thanh et al., 2022). By integrating multiple representative units, each featuring distinct fiber trajectories and filler compositions, engineers can design advanced composite structures with properties precisely tailored to meet the demands of specific applications (Zhu et al., 2025; Niu et al., 2024; Hyde et al., 2020).

However, identifying the optimal combination and spatial arrangement of these unit cells can become computationally prohibitive as more unit cells are introduced, limiting design innovation and efficiency. To overcome this, various advanced computational optimization techniques have been explored, including evolutionary algorithms (Wang and Sobey, 2020; Cai and Aref, 2015), reinforcement learning (Shen et al., 2025; Li et al., 2024d), Bayesian optimization (Coelho et al., 2025), swarm intelligence methods (Martínez-Muñoz et al., 2023), and neural network-based methods (Li et al., 2024d; Zhan et al., 2025). These approaches can explore the extensive design space within an affordable computational cost by strategically selecting and evaluating promising unit cell configurations. Nevertheless, full simulations on the mechanical behaviors of massive combinations are required for each of these methods (Shih et al., 2023; Barbero and Barbero, 2023). When dealing with large amounts of design variations, these methods show less efficiency. Current advancements on minimizing the number of full simulations while still identifying high-performance designs are neural network (NN) models (Li et al., 2024c; Kim et al., 2023), multi-fidelity simulations (Ahn et al., 2022; Pan et al., 2025), transfer learning from small simulations (Kim et al., 2021; Wang et al., 2025), and active learning coupled with Bayesian optimization (Zhao et al., 2024), etc.

Among these advanced methodologies, the multi-fidelity (MF) simulation method has recently gained considerable attention due to its ability to balance computational efficiency with predictive accuracy (Wang et al., 2024; An et al., 2023; Liu et al., 2024b). Traditional MF simulations run fast, low-fidelity simulations (homogenized unit cells) first to search for high-performance candidates (Li et al., 2024b, 2025). Then, high-fidelity finite element analyses (FEA) will be conducted on these high-performance candidates to identify the optimal combination further. Although homogenized models can save excessive computational resources, they often sacrifice accuracy (interactions between different local units not thoroughly considered), potentially posing threats of error that propagate into optimization outcomes. Moreover, the process of calibrating and validating low-fidelity models against high-fidelity simulations can be computationally expensive and labor-intensive. These considerations are the focus of this paper.

Introducing machine learning into the MF workflow can improve the predictive accuracy. Recent machine learning (ML)-based homogenization methods learn a mapping from micro-structural descriptors to global effective properties of a single representative volume element (RVE), and most MF pipelines use such predictions to screen candidates prior to high-fidelity analyses (Lee et al., 2023). Deep Material Networks approximate RVEs via hierarchical building blocks with guaranteed kinematic/traction compatibility, offering strong accuracy but nontrivial training cost and limited handling of spatial context across interacting units (Su et al., 2022). U-Net-style NN learn pixel-to-property mappings directly from image RVEs (Croom et al., 2022); variants include encoder–decoder NNs for elasticity/thermal fields and attention/U-Net hybrids that improve spatial feature reuse. Graph neural networks (GNNs) (Storm et al., 2024) and message-passing models operate on phase/voxel graphs, capturing long-range dependencies and irregular topologies; extensions include graph U-Nets and spectral GNNs for field prediction. Earlier surrogate-based homogenization combined handcrafted descriptors (e.g. 2-point correlations) with regressors (Bishara et al., 2023). These methods typically return global RVE properties for an isolated cell and do not model unit-level property shifts induced by local neighbors in an assembled structure.

In contrast, this study introduces a novel convolutional neural network (CNN)-based optimization approach into the multi-fidelity simulation framework to improve accuracy and efficiently identify the optimal configurations of RVE, each characterized by unique fiber arrangements and fillers (numerically encoded). CNNs were chosen because the spatial configuration of local RVEs in the composite structure can be naturally represented as two-dimensional matrices, analogous to image data. This makes CNNs particularly effective for capturing RVE interactions and spatial correlations that directly influence the effective material properties. We conducted FEAs separately on RVEs with all phases explicitly modeled to complete homogenization. Spatial arrangements of RVEs were encoded through a series of n × m layout matrices. High fidelity simulations on the RVE assemblies were conducted to identify the property deviations of the same RVE under different surrounding geometries. For each RVE in the assemblies, the local influence is quantified and stored as Pytorch tensors for subsequent training of convolutional neural networks (CNN) (Liu et al., 2024a). The trained models achieve a significant improvement in error compensation and reduction in computational cost without repeated high-fidelity simulation. By systematically removing some layers, a lightweight CNN model that shows comparable ability in capturing the nonlinear local influences with standard CNN models is proposed.

To be more specific, this work contributes: (1) a unit-level, local-neighbor correction that predicts geometry-dependent ratios deviations for each embedded unit cell inside an assembly, rather than only estimating bulk properties of an isolated RVE; (2) a two-channel spatial encoding that preserves both macro-scale layout (cell identifiers) and meso-scale phase content (feature matrices), enabling the CNN to learn short-range interactions; (3) an NN-informed MF selector that applies these corrections to re-rank candidates at low cost, recovering HF quality while promoting a smaller fraction of designs; and (4) a lightweight CNN with block-wise down-sampling that achieves accuracy comparable to deep benchmarks with substantially lower training time and inference latency. We further provide a computational efficiency analysis demonstrating HF savings under identical hardware and solver settings.

The principle of homogenization (Li et al., 2025; Jain, 2019) is replacing the original elements within a specific volume/area with homogenized global material properties, ensuring the homogenized elements store approximately the same amount of strain energy as the original elements. In numerical homogenization, the properties of FRMs are commonly determined by modeling RVEs, which capture the key characteristics of the underlying structure (Jain, 2019).

To form the fundamental RVEs, five 2-dimensional, square unit cells with a length of L, each featuring unique internal fiber paths and filler material distributions, were modeled using the Abaqus FEA package as illustrated in Figure 1.

These unit cells serve as the basic components for assembling larger composite structures with tailored mechanical performance. Each unit consists of three different phases: a Polypropylene (PP) matrix (Hossain et al., 2024), which performs as a stable base; Several embedded carbon fibers, which offer strength; and fillers, strategically positioned to boost energy absorption and vibration control (Ramesh et al., 2022). The specific arrangement and interaction of these three phases within each unit cell are critical in determining the overall mechanical performance of the formed structure.

The start and end points of inner fibers are set at every 1/3L points on the edges of each square block to ensure the continuity of fibers in adjacent blocks when assembled. The PP matrix was treated as an isotropic elastic-plastic material defined by its Young's modulus and Poisson's ratio. Plasticity of PP was specified in Abaqus via a table of yield stress and plastic strain pairs (no kinematic hardening or rate dependence was included). Fillers were treated as a linear elastic material. For the carbon fibers, an orthotropic elastic model was used, implemented in Abaqus via engineering constants. Detailed material properties are listed in Table 1.

Mesh convergence was first established at the RVE level by refining matrix/filler and fiber seeds until changes in E₁₁, E₂₂, ν₁₂ were less than 1%. Specifically, three refinement levels were considered: (1) Matrix/filler seed 0.0060 mm, fiber seed 0.0020 mm. (2) Matrix/filler seed 0.0045 mm, fiber seed 0.0015 mm. (3) Matrix/filler seed 0.0030 mm, fiber seed 0.0010 mm.

For each unit cell, the effective stiffness and Poisson's ratio were computed at each refinement level under the same macroscopic loading. Across all cases, differences between level 2 and level 3 satisfied the precision criteria (less than 1%). Eventually, the meshing seed size of the PP matrix was set as 0.0045 mm, while the carbon fibers, which experience higher stress gradients and require higher accuracy, are refined to a seed size of 0.0015 mm.

Applying periodic boundary conditions (Tian and Qi, 2023) (PBC) to each unit cell is critical to accurately simulate the unit cells' mechanical response and ensure continuity across adjacent units. This approach allows the models to mimic infinite composite material by ensuring that the displacements and tractions on opposing boundaries remain consistent, thereby preserving the representativeness of the micro mechanics in larger-scale assemblies. Figure 2 shows the principle of PBC application on the boundaries of each unit cell.

To apply the PBCs rigorously, we begin with the deformation gradient tensor F:

where I is the identity tensor and ▽u is the displacement gradient tensor. The general periodicity condition for a unit cell of length L in each direction is:

where ɛ is the sympmetric macroscopic strain tensor under small deformation assumptions. Expressed in component form for square unit cells:

These relations enforce a displacement field that is periodic while allowing a uniform average strain to develop across the RVE.

Following the implementation of PBCs, multiple reference points (RP) were introduced and kinematically coupled to the boundary nodes of the unit cells with the multi-point constraints (MPC) method (Tian and Qi, 2023). The periodic constraints were implemented in Abaqus using the *EQUATION command. Nodes on opposite boundaries were paired according to their coordinates, and their displacements were coupled through linear constraint equations (Wang et al., 2025; Qi et al., 2015). The macroscopic strain components were represented by reference points (RPs) whose prescribed displacements correspond to ɛ₁₁L_x, ɛ₂₂L_y, and ɛ₁₂L_x, ɛ₁₂L_y. For example, the constraint between the left and right boundaries was defined as.

• *EQUATION

• 3, Rnode,1,1.0, Lnode,1,-1.0, RP_EXX,1,-1.0

• *EQUATION

• 3, Rnode,2,1.0, Lnode,2,-1.0, RP_GXYx,1,-1.0

Analogous equations were applied for the top and bottom boundaries.

For each unit cell, three independent loading conditions were applied: uniaxial tension in the x and y direction, and shear in the x-y plane. The effective stiffness components were calculated using the volume-averaged form of Hooke's law (Jain, 2019):

Where $σ¯ij$ is the average stress tensor components over the unit, $ε¯kl$ is the applied average strain tensor components, $Eijkleff$ is the effective stiffness tensor of the unit cell. For 2D plane stress or plane strain, the fourth-order stiffness tensor (Bauer and Böhlke, 2022) reduces to a symmetric 3 × 3 matrix using Voigt notation:

where E₁₁ and E₂₂ are the effective axial stiffness in x and y direction, E₁₂ is the coupling between ɛ₁₁ and σ₂₂, E₆₆ is in-plane shear modulus. These were computed by solving the system of equations generated from the three loading cases. The macroscopic stress $σ¯ij$ can be calculated by volume/area averaging:

This integration is implemented in Abaqus using field output averaging over the entire unit cell area. Through defining ɛ₁₁ ≠ 0, ɛ₂₂ = ɛ₁₂ = 0, E₁₁ and E₁₂ can be extracted. Similarly, E₂₂ and E₁₂ can be extracted through defining ɛ₂₂ ≠ 0, ɛ₁₁ = ɛ₁₂ = 0. The corresponding effective properties were derived as:

All simulations were run in Abaqus (standard) under a 2-D plane-stress assumption with nominal thickness 1.0 mm and all models were built using CPS4R elements with reduced integration and hourglass control enabled. Frictionless contact was assumed for the elastic simulations, while continuity across unit boundaries and interfaces was enforced using tie constraints.

From the resulting stiffness tensors, the effective Young's modulus E and Poisson's ratios ν were extracted for each unit cell. These homogenized properties were later used as key parameters to construct representative, low-fidelity models and served as baseline properties (denoted as E^eff and ν^eff) for comparison in the multi-fidelity framework.

Before assembly, the unit cells were encoded using a unique integer identifier ranging from 1 to 5. Through randomly positioning the integers into a series of two-dimensional arrangement matrices of size n × m, diverse spatial configurations of the larger composite structures with size n ⋅ L × m ⋅ L were generated. The arrangement matrices serve as a spatial encoding of specific composite structures, where the position of each integer reflects the placement of a particular unit cell within the global structure (Figure 3). Such encoding methods allow for efficient and scalable input generation for both high-fidelity simulations and NN models. Each matrix encodes not only the constituent types but also the geometry information of unit cells, which is essential for capturing local-neighbor influences.

After translating the unit cells into the grid, dedicated node sets were created for every unit. These node sets include all relevant phases within each unit and serve as spatial references for post-processing the stress and strain fields at the unit-cell level using Python scripts. Figure 4 shows a representative of the composite structures built with 30 randomly generated 20 × 20 arrangement matrices. Only simulation results of structures 1 to 25 were collected for NN training.

Unlike the typical homogenization process that extracts bulk properties of the entire structure (Li et al., 2024c), the focus here is on the local homogenized properties of individual unit cells within the structure. The effective properties of a specific unit vary not only depending on its composition but also on the geometrical arrangements of its neighboring units (Ramesh et al., 2022; De et al., 2024). For instance, unit 1 may exhibit different effective behavior when surrounded by different geometrically distributed unit cells, as shown in Figure 5.

By comparing these context-sensitive effective properties against the baseline properties obtained from single-unit-cell simulations, the influence of spatial interaction can be quantified.

Parallel simulations were conducted on the assembled structures. The ground truth responses of the structures were captured through high-fidelity simulation, where the fibers, fillers, and matrix phases within the assembled structures remain explicitly resolved (Cheung and Mirkhalaf, 2024; Li et al., 2024a). Meanwhile, the homogenized surrogate behaviors of the structures were captured through standard low-fidelity simulation, where the three phases within the units were replaced with equivalent properties.

Global loading conditions were applied using the same RP-driven approach to simulate the mechanical behavior of the structures. Specifically, PBC was applied to ensure continuity and compatibility with the behavior of the unit cells, thereby capturing the effective macroscopic response. To ensure mechanical continuity throughout the structure, the adjacent boundaries of units are connected through tie constraints. Similarly, fibers were also tied across unit boundaries to preserve the intended load transfer pathways and maintain microstructural realism.

Prescribed vertical displacements corresponding to a 1% global strain in the y direction were then applied to the RPs, inducing a uniform tensile load across the height of the structure. This setup ensures consistent global deformation, enabling stress responses to be computed across all embedded units.

The homogenized properties of each unit cell within its local environment were calculated through the volume-averaging method. This enabled a direct comparison between the baseline effective property E^eff(obtained from isolated simulation) and the local-neighbor-influenced effective property E^ln,eff when the same unit cell is embedded in different spatial geometries.

To quantify the effect, E^eff to E^ln,eff ratio R_e and ν^eff to ν^ln,eff ratio R_ν were introduced. We analyzed the R_e and R_ν of all unit cells across a series of geometrical configurations. Figure 6 illustrates the R_e and R_ν scatter plots of each unit cell across 30 randomly selected geometries extracted from the 25 structures.

The results reveal that the stiffness component (particularly in the loading direction) exhibits the most significant variation, suggesting that the load transfer paths and constraint effects attributed to neighboring units play critical roles in modulating local behavior (Liu et al., 2022; Bauer and Böhlke, 2022). This observation supports the need to incorporate local-neighbor effects into the modeling framework. The use of an NN model to capture these nonlinear interactions becomes essential for boosting the predictive accuracy of low-fidelity models without incurring the computational cost of full high-fidelity evaluations.

For each arrangement matrix, a series of 5 × 5 submatrices was extracted to represent the local geometry surrounding a target unit cell, as illustrated in Figure 7. The center of a submatrix corresponds to the target unit x, whose homogenized property variations under local-neighbor interaction are to be predicted. The first surrounding layer, which directly contacts unit cell x, is assumed to impose local influence on unit x's mechanical behavior. The second surrounding layer is considered to introduce neighboring influences, affecting how stress is transferred and distributed through the structure.

By systematically sliding this window across an arrangement matrix (padding the edge units that lack a full neighborhood with zeros), a set of local geometry sub-matrices can be extracted from a single assembled structure. Specifically, from each n × m arrangement matrix, up to n ⋅ m valid sub-matrices of size 5 × 5 can be generated (duplicated sub-matrices were removed). Each extracted 5 × 5 sub-matrix is then transformed into a two-channel 500 × 500 input array for the CNN. For the first channel, each integer in the 5 × 5 matrix, representing unit cell types and padding, is expanded into a corresponding 100 × 100 block using nearest-neighbor upsampling. This creates a 500 × 500 matrix preserving the original spatial arrangement at an enhanced resolution suitable for CNN processing.

The second channel encodes internal structural information of each unit cell type through predefined feature matrices. Six 100 × 100 feature matrices (including the value 0 in the padding area) were generated based on finite-element analysis images of the internal microstructure, representing the spatial distribution of matrix, fiber, and filler phases, and numerically encoded with distinct mechanical property values. Initially, the internal phase information of each unit cell was identified by segmenting RGB images obtained from FE simulations. As shown in Figure 8, each color region corresponded uniquely to one of the internal phases (matrix, fiber, or filler). Subsequently, these RGB-segmented images were translated into numerical feature values representing the distinct mechanical properties assigned to each phase.

The resulting numerical arrays form feature matrices that explicitly encode the heterogeneous internal composition of each unit cell. Each element in the original 5 × 5 arrangement matrix is then mapped to its corresponding feature matrix according to the unit type number. The mapped 100 × 100 feature matrices are subsequently assembled into a second 500 × 500 matrix, spatially matching the first channel. Each two-channel input effectively captures both the macro-scale geometric configuration and the meso-scale physical composition, enabling the neural network to learn local neighbor influences accurately.

After assembling all two-channel 500 × 500 input arrays, the complete dataset was compiled into a single NumPy file for efficient loading and access during model training. Each input sample in this dataset corresponds to a unique local neighborhood configuration extracted from a larger composite structure, paired with its respective target output values R_e and R_ν. The generated input dataset has a shape of (N, 2, 500, 500), where N denotes the total number of samples (N = 10,000 in this study), 2 represents the two input channels, and 500 × 500 corresponds to the spatial resolution of each sample derived from assembling 5 × 5 unit cells, each mapped to a 100 × 100 feature region.

Based on previous observations, the stiffness component aligned with the primary loading direction exhibits the greatest sensitivity to local-neighbor influences (Ramesh et al., 2022; Wang et al., 2025). Therefore, only the stiffness component E₂₂ and Poisson's ratio ν₂₁ are considered for inclusion in the output vector during dataset preparation. For each of the 5 × 5 sub-matrices, the corresponding output vector is calculated as:

where:

Considering the models were loaded in y direction, each output sample is a 2-dimensional vector:

When generated for an entire n × m structure, these outputs form a 3D tensor of shape:

The output tensors are spatially aligned with the corresponding position matrix of the assembled structure, ensuring a direct mapping between the encoded geometry (input) and the predicted property variations (output). To facilitate neural network training, all input–output pairs are consolidated into structured datasets: each assembly is assigned to exactly one of the sets (train, validation, test), and no structure contributes patches to more than one split. The assignment is implemented with a group-aware splitter keyed by the structure ID (Novac et al., 2022). The input/output datasets were then stored as Pytorch tensors and then split into three parts randomly according to the assignment. 70% of the data sets are used as training sets, 15% for validation, and 15% for evaluation.

Data normalization (Fan et al., 2023) is crucial for achieving stable and efficient training of convolutional neural networks, particularly when input channels exhibit drastically different numerical ranges. Without appropriate normalization, gradient-based optimization methods may struggle to converge, due to numerical instability and uneven scaling between input channels.

In this study, the input data consists of two channels with distinctly different numerical characteristics. The first channel, representing the unit cell type (integers ranging from 0 to 5), was normalized using simple min–max scaling:

where x represents the original integer value, ensuring the channel's numerical range is constrained within [0, 1].

The second channel, containing the feature matrices with values of 30 (filler), 1700 (matrix), and 135,000 (fiber), exhibited significant numerical disparities. To address this, a logarithmic transformation followed by min–max scaling was applied to compress and standardize the data:

This normalization technique significantly reduced numerical variance, placing all values within a stable numerical interval of [0, 1]. The effectiveness of the normalization was confirmed by examining the resulting data ranges (Table 2), verifying that both input channels were now suitably scaled for efficient network training.

In addition to input normalization, min–max scaling was also applied to the output dataset, specifically targeting the effective property ratios R_e and R_ν. The normalization for output data followed a similar linear rescaling:

where y denotes each property difference or ratio in the original dataset. This step ensured that the network outputs also fall within a bounded range of [0, 1], significantly improving the stability and efficiency of gradient computations during network training.

Data loading was conducted with no parallel workers to maintain reproducibility and stability across platforms. Empirical results demonstrated significantly improved convergence speeds and predictive performance following normalization, underscoring its importance in the neural network training pipeline. After normalization, both inputs and outputs exhibited consistent numerical ranges, leading to enhanced training stability and predictive robustness.

In this study, we selected CNN for predicting R_e and R_ν due to their inherent capability to efficiently capture and leverage spatial dependencies and local geometric patterns embedded within image-based input data. Unlike fully connected or recurrent neural networks (Sathish et al., 2023; Ghane et al., 2024), CNNs excel at extracting hierarchical spatial features through convolutional kernels, which makes them particularly suitable for processing structured two-dimensional input arrays representing macro-scale geometry and meso-scale internal features of composite materials (Li et al., 2023; Novac et al., 2022). However, standard CNN typically requires extensive computational resources, substantial training durations, and large memory capacities to process high-resolution input data (Li et al., 2024c). These constraints can limit their practical utility, especially when rapid inference or resource-constrained deployment environments are required. Hence, a lightweight CNN architecture employing specialized strategies, such as block-wise down-sampling, was developed. The lightweight model significantly reduces computational complexity and memory footprint while maintaining comparable prediction accuracy to conventional CNNs.

We implemented two widely adopted, high-capacity CNN architectures as benchmarks: a baseline CNN model (model-1) and a ResNet-18 network(model-2). Each network was modified (first layer and output layer) to processes the full 2 × 500 × 500 input tensor through its respective stack of convolutional, pooling (or residual) layers and final fully-connected layers to predict the values of R_e and R_ν. Model-1 (approximately 11M parameters) uses cascades of 3 × 3 convolutions interleaved with max-pooling to compress spatial resolution, whereas Model-2 (randomly initialized, approximately 12.5 M parameters) employs eight residual blocks to ensure stable gradient propagation in a moderately deep topology (Azad et al., 2025). Specific architectures and tunable hyper-parameters for these two baseline models are summarized in Table 3.

The models were trained for 100 and 110 epochs on identical data-augmentation pipelines and optimizer settings. We systematically recorded key performance metrics—training duration, memory footprint, inference speed, and predictive accuracy to establish a rigorous baseline against which we compare our lightweight CNN variants.

The lightweight CNN model was developed by systematically reducing the complexity of model-1 and model-2, as shown in Figure 9. Unlike the standard models, this lightweight model utilizes a block-wise down-sampling strategy, compressing spatial dimensions into a manageable feature representation without sacrificing the critical spatial and structural information required for accurate prediction. Specifically, given the large input size (N, 2, 500, 500), the input is partitioned into smaller, non-overlapping blocks (block pooling layer). In this context, the CNN employs a convolutional layer with a kernel size and stride equal to the block size (100 × 100). Each block is processed individually by this convolutional kernel, effectively summarizing and compressing localized spatial and structural information into a smaller, lower-resolution feature map (in this case, from 500 × 500 down to 5 × 5). This block-wise pooling mechanism slashes both FLOPs and peak memory usage. Moreover, the entire model comprises approximately 3 million trainable parameters, roughly one-quarter of model-1, enabling training times on the order of 20–30 min and inference latencies below 5 ms per sample, all while matching the predictive accuracy of the larger architectures. This down-sampling strategy significantly reduces computational complexity and memory requirements, while retaining essential spatial patterns and features critical for accurately predicting local effective property ratios.

The block pooling layer transforms the input of size (N, 2, 500, 500) into a representation, where 256 learnable filters extract abstract local spatial features across the downsampled grid. The number of filters in this block-pool layer was increased from the baseline to 256 to improve feature richness and capacity. The output from the block-wise layer is processed through a sequential stack of six convolutional layers with decreasing filter widths. Each convolution uses a kernel size of 3 × 3 with padding of 1, followed by batch normalization and ReLU activation (Song et al., 2023) to promote gradient stability and non-linearity. This deep stack enhances local pattern recognition and supports robust spatially distributed structural effects learning. Following convolutional processing, the 5 × 5 × 8 output is flattened and passed through a FC (Li et al., 2023) head consisting of three hidden layers with dimensions [512, 256, 64], each followed by ReLU activation and dropout (0.4) to reduce overfitting. The final FC layer maps the learned features to the two output targets, representing the normalized R_e and R_ν values, respectively. All parameters used in the training process of the lightweight model are listed in Table 4.

The CNN models are set up using the Pytorch framework (Novac et al., 2022) according to the above architectures and then trained with the Adam Optimizer (Li et al., 2023; Song et al., 2023; Ghane et al., 2024; Li et al., 2024c). Given the dataset size, the architecture is compact, and an early stopping criterion (Yang et al., 2019) based on validation loss is implemented to minimize the risk of overfitting while still maintaining enough expressive ability to mimic local interactions. At each epoch, training and validation losses, along with scores, were computed and visualized in real-time to monitor convergence behavior. The best model was selected based on validation performance and further evaluated on a held-out test set. All experiments were run on CUDA-enabled GPUs (Nvidia GeForce RTX 3070).

Figure 10 depicts the training and validation loss curves that reflect the training stability by indicating the convergence behavior of both training and validation losses. In the early stage, the training and validation losses of all models both decrease rapidly, indicating stable and effective model learning (Li et al., 2023). As the training epoch grows, the improvement rate (dropping velocity of the loss curves) gradually slows down. The validation losses of model 1 and model 2 plateau after approximately 90th and 70th epoch, respectively, while the training loss continues to decline slightly. Such divergence marks the point where the model begins overfitting. The training processes are halted at 90th and 70th epochs, respectively, to preserve the best generalization performance according to the preset early-stopping criterion (Yang et al., 2019). The stable gap between the training and validation loss curves, along with their smooth convergence behavior, indicates that the CNN models have achieved a good balance between underfitting and overfitting.

For the lightweight model, the dropping rate of its training loss slows down after the 85th epoch. Although it continues to decline modestly through to the 120th epoch, the rate of improvement falls below 0.1% per epoch, indicating diminishing returns. Meanwhile, validation loss reaches its minimum around the 90th epoch, after which it remains effectively flat, suggesting the network has captured the dominant feature mappings. Consequently, we adopt an early-stopping criterion at epoch 100, when validation improvement drops below our 0.05% threshold to prevent overfitting and minimize total training time to under 30 min. This strategy yields a final lightweight model whose test-set RMSE matches the standard models within 2%, while inference latency is reduced by more than 95%.

Figure 11 presents a set of overlaid scatter plots, each comparing the CNN-predicted values of the normalized effective stiffness ratio R_e (left panel) and the normalized Poisson's ratio R_ν (right panel) against the corresponding ground-truth data from the validation set. Points are color-coded by model type (model-1, model-2, and lightweight Model), with the solid black line denoting the ideal y = x relationship.

On both axes, values span the interval [0.83, 1.18], capturing the full range of local property perturbations induced by varying 5 × 5 neighborhoods (Hyde et al., 2020). All three models produce a tight cluster of points around the identity line, with negligible systematic bias or spread even in regions of extreme neighbor effects. Model-1 and Model-2 achieve root-mean-square errors of approximately 0.015, while the lightweight Model, despite its drastically reduced parameter count, attains an RMSE of just 0.017 for R_e and 0.013 for R_ν.

This visualization confirms that each network, and in particular our optimized lightweight architecture, robustly approximates the nonlinear mapping from local geometry to effective property corrections. The near-perfect alignment with y = x further demonstrates the models’ capacity to generalize across hundreds of assembly configurations, enabling rapid, high-fidelity microstructure predictions without resorting to costly full-scale finite-element simulations (Cheung and Mirkhalaf, 2024; Li et al., 2024a).

The homogenization quality of unit cells 1–5 was further validated by direct comparison with full-fidelity FE models of the same microstructures. Figure 12 displays the equivalent uniaxial stress–strain responses for each homogenized unit cell overlaid with the corresponding high-resolution simulation results. In every case, the two curves exhibit closely initial slopes (effective Young's moduli) and yield points, with maximum deviations below 5% in stress.

These comparisons confirm that our homogenized constitutive models faithfully reproduce the key mechanical behaviors—linear stiffness, onset of nonlinearity, and post-yield softening of the detailed micro-structural simulations (Wen et al., 2024; Barbero and Barbero, 2023). Such close alignment not only validates the baseline effective properties but also demonstrates that the reduced, homogenized representation captures the essential physics needed for accurate downstream predictions in both the standard and lightweight CNN frameworks.

Traditional homogenization treats each unit cell as possessing a fixed, context-independent set of effective properties, thereby overlooking the perturbations introduced by the neighboring environment (Li et al., 2024c). In contrast, our CNN-augmented frameworks explicitly account for these local interactions and predict the resulting property discrepancies. To evaluate its accuracy, we compare three sets of predictions: standard homogenization, NN corrections, and full-fidelity simulation results, using the deviation metrics as follows:

where $E22full$ and $ν21full$ stand for ground truth value of E₂₂ and ν₂₁ respectively. Similarly, the prediction errors of NN model can be expressed as:

where δ_NN and γ_NN are errors of NN predicted effective stiffness and Poisson's ratio, respectively. For a cell within its local geometry, a higher value of δ_hom reflects strong neighbor influence, whereas a lower value of δ_NN indicates the NN correction effectively compensates for the deviations. Conversely, a smaller δ_hom implies negligible neighbor influences.

Based on the quantified errors, two indicators, I_e and I_ν, are introduced to assess both the extent of deviations of homogenized properties from the ground truth value and the accuracy of NN correction within a cell's local geometrical context, as follows:

A higher value of I indicates higher homogenization error and greater error reduction through NN correction. In Figure 13, the heatmap shows the improvement ratios I of the three CNN models across unseen structure (specifically, structure No.27). The color gradations in the heatmap indicate how good the NN correction works. Regions with darker colors reflect lower improvements (lower values of I), while lighter areas indicate larger improvements.

Across all three models, a large majority of locations recover at least 80% of the high-fidelity correction. Furthermore, the models exhibit similar patterns, with the highest improvement ratios generally observed in regions where local influences are more pronounced. Indeed, the intrinsic error of the homogenization can substantially bias the initial candidate ranking. The designs with target performance that barely clear the threshold may be underestimated and discarded, creating false negatives. This spatial analysis supports the assertion that the incorporation of local geometry information significantly boosts prediction performance.

Furthermore, three simulation strategies were implemented on the unseen Structures: full-fidelity simulation, homogenized simulation, and homogenized simulation with NN corrections, enabling a rigorous assessment of the models' ability to extrapolate beyond known data. Results show that the model exhibited stable performance when predicting R_e and R_ν for unit cells embedded in unfamiliar environments.

The comparison between the full fidelity simulation and the homogenized simulation informed by the lightweight NN model, as shown in Figure 14, reveals the effectiveness of homogenization in simplifying complex composite behavior (Bauer and Böhlke, 2022). In the full model (left), stress and strain distributions show significant local variations due to fiber-matrix interactions and geometric complexities, reflecting microscale heterogeneity. In contrast, the homogenized model (right) exhibits smoother, more uniform fields, indicating that local effects have been averaged into effective material properties. The lightweight model informed result (center), however, exhibits a much closer match to the full-fidelity result. Notably, the red dashed circle highlights regions where the NN-informed model successfully recovers stress intensification patterns absent in the conventional homogenized model, thereby demonstrating its ability to reconstruct local stress effects without explicitly modeling the fine-scale geometry. The strain field shows similar trends. The lightweight model informed simulation captures both magnitude and spatial variation better than the standard model, which oversimplifies deformation. Overall, the lightweight CNN effectively reconstructs local effects, offering high-fidelity mechanical response with reduced computational cost, validating the proposed multi-fidelity framework.

Figure 15 depicts the equivalent stress–strain curves obtained from the simulation results. The standard homogenized model significantly underestimates the stress response, particularly beyond the elastic regime, due to its inability to account for local stiffness variations and stress intensification. In contrast, the CNN-informed and lightweight models provide stress–strain predictions that closely track the full-fidelity curve across the entire deformation range.

Importantly, despite a dramatic reduction in parameters and compute, the lightweight model maintains accuracy on par with the standard CNN models. By systematically pruning convolutional layers and adopting a block-wise, down-sampling strategy, the lightweight model achieves comparable R² and error statistics (within 5% of the standard model), while reducing training time to under 30 min and inference latency by over 10-fold. Such a level of predictive accuracy and robustness supports the model's suitability for accelerating exploration of targeted composite design without the computational burden of full-scale simulations (Cheung and Mirkhalaf, 2024).

An essential motivation for integrating NN corrections into the multi-fidelity framework is to reduce the overall computational cost with enhanced accuracy (Ghane et al., 2024). To quantify the NN models' efficiency, we compared three strategies on a representative set of 500 combinations using the three CNN models.

1. Full fidelity (FF) simulation for baseline: Structures are simulated by high fidelity FEA.

2. Standard multi-fidelity simulation (MF): all 500 structures undergo fast, low-fidelity homogenization; the top 10% by low-fidelity ranking are then re-evaluated with high-fidelity FEA.

3. NN informed multi-fidelity (NN-MF): as in the MF approach, but the NN correction is used to adjust low fidelity estimates and re-rank candidates. Only the top 4% are promoted to high fidelity FEA (NN-MF at 4% matched the top objectives achieved by MF at 10%).

All methods were applied to the same 500 assembled combinations, generated once with a fixed random seed. No candidates were added or removed between methods. MF ranks by the baseline homogenized objective; NN-MF ranks by the same objective after applying the CNN ratios to the effective properties. If multiple designs tie at the boundary score, selection is made deterministically by: (1) secondary key equals lower predicted variance, then (2) lexicographic index of the layout (row-major). No FF warm starts were used (to avoid bias). Geometry assembly, node or face sets, and meshing templates were cached and reused across methods; solver tolerances, element types, and hardware settings were identical for MF and NN-MF to ensure fair wall-clock comparisons. All running times were recorded on a 12-core AMD Ryzen9 5900X (3.7 GHz, 32 GB RAM) for CPU-bound homogenizations, and a single NVIDIA RTX 3070 GPU (8 GB) for NN inference. Detailed computational costs on training and inference per structure are listed in Table 5.

Table 6 shows the comparison of computational time cost for the 500 configurations. The standard MF strategy already cuts the simulation time by nearly 90% compared to FF evaluation. Incorporating the NN correction further halves this cost, yielding a total speed-up of over 22-fold relative to the baseline FF strategy, with negligible overhead of NN inference for all 500 samples.

The cost of low-fidelity simulation scales linearly with the number of candidates and remains small per sample (Cheung and Mirkhalaf, 2024); full-fidelity analyses are surely the dominant term in the entire optimization workflow. By dynamically adjusting that term with NN-based re-ranking, the NN-informed multi-fidelity workflow scales very favorably: doubling the design space induces only a proportional increase in low-fidelity cost and NN inference, while the number of full-fidelity simulations can be held roughly constant to maintain a given accuracy level. These results demonstrate that our Lightweight CNN strikes an optimal balance, maintaining parity in accuracy with deeper benchmarks while drastically reducing computational cost and resource demands.

This work has introduced and validated a neighbor-aware, NN correction within a multi-fidelity homogenization framework to predict effective property deviations of composite structures assembled from 2-dimensional RVE unit cells. The main findings include.

1. Local-neighbor influence captured by NN model. By encoding each unit's local context into a 5 × 5 matrix and training the NN model to predict the deviation, over 80% of the high-fidelity accuracy can be recovered while only requiring a small portion of full FE solves.

2. Models’ generalization to unseen assemblies. When evaluated on previously unseen structures, the NN predictions maintain sub-10% error on effective property deviations.

3. Substantial Computational Savings. By systematically pruning the architecture of conventional CNN models and implementing a down-sampling strategy, our lightweight model achieves a 92.42% reduction in total simulation cost while holding error below 10%.

Despite these promising outcomes, the approach has clear limitations. The training dataset is obtained from FEA on planar 2-dimensional unit cells with minor plastic behavior (Xu and An, 2022); extensions to 3-dimensional units or to inelastic and damage-evolution models would require retraining on higher-dimensional data (multi-channel inputs) and potentially more complex NN architectures. Likewise, while a fixed 5 × 5 local window proved sufficient for the materials and layouts studied here, highly anisotropic fiber trajectories or graded-volume architectures may demand adaptive neighborhood sizes or trainable attention weights. Given the localized, normalized targets and strong cross-structure heterogeneity, single scalars may obscure systematic, context-dependent effects. Accuracy is instead summarized via by-structure splits and steady-state validation R², future work may add complementary error metrics. Finally, the model has so far been demonstrated on a regular three-phase system (Tian and Qi, 2023; Cheung and Mirkhalaf, 2024); material systems with vastly different phase contrasts or interface behaviors will require further transfer-learning experiments to verify data efficiency and accuracy.

Future developments will extend the framework to include full plasticity and damage evolution to capture nonlinear and dissipative behaviors of composites. The approach will also be scaled to larger assemblies to examine size convergence of effective properties. Methodologically, adaptive neighborhood learning using attention or graph-based models, uncertainty-guided active sampling, and physics-informed loss functions enforcing mechanical symmetry and bounded corrections will be explored to enhance robustness and extrapolation to highly heterogeneous configurations (Ghane et al., 2024). Additionally, the methodology will be extended to three-dimensional simulations to capture out-of-plane interactions and validate its robustness for realistic composite architectures.