Integrating 3D CNN and phase-field simulation for TiAlN coating property prediction via 3D microstructure

Framework for fusing numerical simulation and experimental information towards materials property prediction

A chain model for alloy property prediction can be constructed based on limited experimental data:

where X denotes alloys with different compositions and processes, $$\tilde{\mathbf{M}}$$ is the simulated microstructure, $$\tilde{\mathbf{m}}$$ is the extracted microstructure information, and $$\tilde{\mathbf{P}}$$ is the predicted alloy property. By aligning phase-field simulation results with scarce experimental microstructures, high-fidelity 3D microstructure simulation results of materials under different conditions can be obtained (corresponding to $$\tilde{\mathbf{M}}$$ = f(X)). Following, a low-dimensional latent vector representing 3D microstructural features is extracted using a 3D-CNN-based encoder model (corresponding to $$\tilde{\mathbf{m}}$$ = EC($$\tilde{\mathbf{M}}$$)). Finally, an ANN model for material property prediction is constructed using microstructural latent vectors as input features (noted as $$\tilde{\mathbf{P}}$$ = ANN($$\tilde{\mathbf{m}}$$)). With this framework, we can overcome the constraints of heavy calculation burden for 3D microstructures and obtain an accurate quantitative composition-process-structure-property relation by training on a small amount of experimental data.

Phase-field model of spinodal decomposition

To simulate the microstructure evolution of TiAlN coatings during spinodal decomposition, the Cahn-Hilliard model was applied. The total energy functional in the TiAlN coating system is expressed as follows:

where V_m represents the molar volume, G denotes the free energy, and κ is the gradient energy coefficient. 1/2κ(∇c_AIN)² is the contribution of the component gradient to the energy of the system. Based on the free energy functional equation, the governing equation can be derived in

where M refers to the chemical mobility. κ represents the gradient energy which can be calculated based on thermodynamic parameters, as given by

where the interatomic distance, denoted as b, is typically chosen as a value ranging between 1/3 and 1/4 of the lattice constant a. x_A and x_B stand for the molar fractions of components A and B respectively, and L_i represents the interaction parameter, which can be determined by the CALPHAD thermodynamic database^[37].

E_el in Equation (1) is elastic energy density given as:

where λ_ijkl is the inhomogeneous elastic modulus tensor; the elastic strain $$\varepsilon_{ij}^{e l}$$ is the difference between the total strain and the composition heterogeneity introduced eigenstrain. According to Hook’s law, the local elastic stress can be written as $$\sigma_{i j}^{e l}=\lambda_{i j k l} \varepsilon_{k l}^{e l}$$. Eventually, the equilibrium elastic field of elastically heterogeneous system can be determined by solving the mechanical equilibrium $$\frac{\partial \sigma_{i j}^{e l}}{\partial r_{j}}=0$$. The phase-field simulations were performed using the home-made PFCSU code (https://pfcsu.ppmgroupcn.com/) and visualized with Paraview software (version 5.10.1).

Structure of feature extraction model and property prediction model

The raw data for each simulated microstructure represent a numerical array of Al concentration values (a floating-point number between 0 and 1 for each grid point). However, directly using such large scale 3D phase-field simulation numerical array data as training input may lead to unnecessarily large network sizes and poor convergence during the model training process. With the lower-dimensional representation of data, the relation towards prediction for the microstructure-determined property would also be beneficial. Therefore, aiming to extract the low-dimensional latent vectors of microstructural features, a 3D-CNN-based autoencoder model was constructed. Figure 2 illustrates a schematic with the underlying network architectures for encoder, decoder, and regressor networks used in this study. This autoencoder model consists of two parts: an encoder-decoder model for extracting and reconstructing 3D microstructure information, and an encoder-regressor model for predicting the hardness of TiAlN coatings. All models are implemented in Python 3.12.3 within the PyTorch framework (version: 2.7.1+cu118).

Figure 2. Schematic diagram of 3D-CNN-based reconstructible microstructure feature extraction model (Encoder-Decoder) and property prediction model (Encoder-Regressor).

The reconstructible microstructure feature extraction model is constructed by a series of 3D convolutional layers (Conv3d), batch normalization, flattened, and dense layers, and each adjacent Conv3d layer is connected by a Leaky Rectified Linear Unit (LeakyReLU) as activation function. The detailed hyperparameters regarding each network can be found in Supplementary Table 1. The microstructure results of TiAlN coatings, obtained from high-fidelity phase-field model simulations with dimensions of 48 × 48 × 48 grids, were gradually compressed and transformed into a latent vector with a latent space size (Ls). Compared with feature dimensionality reduction methods such as principal component analysis and multidimensional scaling, the encoder can handle much more complex features using deeper Conv3d layers. The decoder then maps the latent vector back to the 3D microstructure image of 48 × 48 × 48 grids through a series of 3D transpose convolutional layers (ConvTranspose3d), gradually reconstructing the original microstructures and features. By comparing the reconstruction results with the original microstructure, we can further analyze the reliability of the feature extraction model and evaluate the applicability of the extracted features. The regressor unit is an ANN consisting of two hidden layers. The low-dimensional microstructure latent space from the encoder module was further processed by the regressor. Finally, the structure-property relation was established by introducing fully connected layers, which were used for the hardness prediction of TiAlN coatings.

Model training and evaluation metrics

During the training process, the mean absolute error, L1 loss function, was used to evaluate the differences between the original and reconstructed microstructures. The model parameters are inversely optimized by minimizing the L1 loss:

where y_i is the true value and f(x_i) is predicted value. n is the number of samples.

To get a qualitative analysis of the prediction accuracy, a structural similarity index measure (SSIM) was applied to calculate the 3D similarity between reconstructed and simulated structures, which is defined as:

where μ_φ, μ_ϕ, σ_φ, and σ_ϕ denote the average value and variance of original image φ and reconstructed ϕ images, respectively. σ_φφ indicates the covariance between the two images. C₁ and C₂ are constants.

Construction of high-fidelity 3D microstructure dataset

High-fidelity microstructure datasets are essential for establishing a linkage between composition/temperature and hardness. For c-TiAlN coatings, the Cahn-Hilliard model is a crucial approach for simulating the microstructure evolution during spinodal decomposition. However, precise model parameters are generally not readily obtainable, which leads to subtle distinctions between the simulated results and those of high fidelity, e.g., regarding the wavelength and corresponding distribution. Zhang et al. propose a quantitative simulation framework so as to accurately quantify model parameters of high confidence^[38]. Specifically, multi-objective optimization is employed for aligning the phase-field model and related model parameters to precisely demonstrate multiple microstructure features, i.e., wavelength and the corresponding distribution, simultaneously. Using this framework, we optimized the model parameters for phase-field simulations of TiAlN coatings in our previous work^[38]. By comparing the average wavelength and wavelength distribution density of spinodal decomposition between simulated structures and experimental measurements, the accuracy and validity of the simulation results were confirmed. The same calibrated model and parameters were consistently applied to simulate microstructural evaluation of TiAlN coatings over a large range of composition and temperature.

In this work, the microstructure evolution of TiAlN coatings with Al content of 0.42~0.66 and annealing temperatures in the range of 1,073~1,273 K was simulated, following the same free energy formulation, diffusion coefficients, and gradient energy coefficients as described in our previous work^[38]. Microstructure simulation data were extracted every 60 s for a total simulation duration of 2 h (partially 4 h). The specific microstructure information list is illustrated in Supplementary Table 2. An ultra-large dataset containing 4,962 high-fidelity simulation microstructure results was finally constructed for the autoencoder model training, where the simulation domain is 48 × 48 × 48 nm³ with a grid spacing of 1 nm. It is important to acknowledge the simplifications inherent in our phase-field model to contextualize the simulation results. The current model focuses on the bulk spinodal decomposition driven by chemical free energy and interfacial energy, neglecting other potential factors such as residual stresses from deposition, the presence of grain boundaries, and the influence of the substrate. These factors can, in reality, influence the kinetics and morphology of decomposition. However, for the primary objective of this study, to establish a generalized, high-throughput framework for linking 3D microstructure to properties, the chosen model is both appropriate and effective. It successfully captures the fundamental physical process (uphill diffusion and phase separation) that governs the age-hardening behavior in TiAlN coatings. The high fidelity of the simulation microstructure provides a robust foundation for the subsequent property prediction.

Training of reconstructible microstructure feature extraction model

For an accurate autoencoder model, over 4,962 phase-field simulation structures were used to enrich the high-quality dataset and train the model. Databases constructed from microstructural results of phase-field simulations provide more detailed and varied data information, which in turn leads to more complex and variable data patterns that need to be handled. Exact training on such large datasets is a time-consuming task. Thus, as shown in Figure 3, a transfer learning-based model training strategy was applied in this work. A pretraining process was first implemented, and only a small data sample was used for multi-step model training and optimization. After that, based on the trained pretraining model, the model parameters are retrained and fine-tuned in a few epochs using the 4,962 large dataset. The final encoder-decoder model with high-fidelity feature extraction ability is obtained. Supplementary Figure 1 shows the loss curve of 2,000 epochs of pretraining and 500 epochs of retraining. Each autoencoder training took only 1 h (approximately 17 min for pretraining and 43 min for retraining) on one NVIDIA GeForce RTX 4070Ti GPU. Since a pretrained base model was already provided, it can converge to the optimal solution faster when training on large sample datasets.

Figure 3. A transfer learning-based model training strategy. Start by training the model parameters with small data samples to obtain a base model, and then continue to train on large data samples in small batches to obtain a high-performance model.

To obtain low-dimensional and exhaustive microstructure information, the impact of the latent space size of autoencoder, Ls, on the model performance was investigated. To this end, we trained four autoencoder models with Ls = 512, 256, 128, and 64, respectively. The L1 loss values of train and test datasets for different models are reported in Table 1. From this survey, we observe that reducing the size of latent space does not influence the accuracy of models pretrained on small datasets, but significantly worsens the performance of retrained models with large datasets. Adhering to the goal of describing large-scale microstructural information with as few features as possible, we finally chose the model with Ls = 128. The loss value of this model is less than 0.015 on both the training and testing sets. These results demonstrate not only the ability of our framework to accurately provide a compact representation of the complex 3D microstructure data in a low-dimensional latent space but also illustrate the robustness of the training of this framework.

Accuracy of 3D microstructure reconstruction

A comparison of the reconstruction microstructure from the decoder model with that from high-fidelity phase-field simulations is demonstrated in Figures 4 and 5. The microstructure evolution of Ti_0.44Al_0.56N coatings during annealing at 1,073.15 K is illustrated in Figure 4A. During the initial time steps (t < 480 s), small compositional undulations occur in the coatings, and form randomly distributed Al-rich micro-regions. The microstructure is rich with multiple features and evolves rapidly with respect to time. As the annealing process continues (t = 960 s), a typical spinodal decomposition microstructure with c-TiN-rich and c-AlN-rich phases was formed by the uphill diffusion of Al atoms and reached an equilibrium state. When the annealing time is greater than 1,000 s, the coating enters the coarsening stage, the reticular c-TiN-rich and c-AlN-rich phases grow up, and the wavelength of the spinodal structure increases gradually. Increasing the annealing temperature has the effect of accelerating the spinodal decomposition process. As shown in Figure 4A and B, when the annealing temperature was up to 1,173.15 K, the coating formed a two-phase structure with c-TiN and c-AlN phases at 120 s, and the spinodal decomposition structure began to be coarsened at 360 s. By comparison, it is clear that the reconstructed coating microstructure from the encoder-decoder model is similar to the phase-field simulation results, and most micro-features can be captured. The distribution of Al content is well reproduced even in the initial phase, with only a slightly higher average Al content than that in the simulation results [Figure 4A]. Our 3D-CNN-based encoder-decoder can successfully predict the enrichment of Al elements, and the overall morphology of the network structure.

Figure 4. Comparison of decoder reconstruction and phase-field simulation microstructure. (A) Result of Ti_0.44Al_0.56N coating annealed at 1,073.15 K. (B) Result of Ti_0.44Al_0.56N coating annealed at 1,173.15 K.

Figure 5. Result of accuracy analysis between all the reconstructed and simulated microstructures. (A) L1 loss value. (B) Structural similarity index. (C) Kernel density contour fill map for L1 loss-SSIM values.

To get a qualitative analysis of the prediction accuracy, a SSIM was applied to calculate the 3D similarity between reconstructed and simulated structures. Since there is no available codebase that supports the direct calculation of 3D SSIM, we can use an approximation that slices the 3D image into 48 pieces of 2D pictures along the Z-axis and calculates the 2D SSIM for each slice, using the average value to express the similarity of 3D microstructures. Higher SSIM values refer to more realistic reconstructed microstructures. As shown in Figure 4, the SSIM value is low for the initial time steps, where features evolve rapidly with time. However, the similarity improves over time and maintains at 0.99 when the evolution process slows down and the microstructure coarsens. Increasing the annealing temperature leads to faster equilibrium of spinodal decomposition and improves the accuracy of prediction results. We further calculated the error and similarity of the microstructures for all 4,962 samples. Among these, 93.3% had a loss value below 0.04, and no more than 5% exhibited a large prediction error [Figure 5A]. Additionally, 94.3% of the predicted microstructures had a similarity greater than 0.96 to the simulation results [Figure 5B]. It is foreseeable that the dimensionality reduction from a 48 × 48 × 48 microstructure to a 128-dimensional latent vector inevitably leads to the loss of some details, especially in early-stage fine fluctuations. The complexity of the microstructure itself influences reconstruction fidelity. The initial decomposition stage features numerous, small-scale, and low-amplitude compositional fluctuations with high spatial frequency. These features are inherently more difficult to reconstruct compared to the larger, more stable and interconnected domains present in the coarsened stages. The model prioritizes capturing the dominant global morphology, which is most critical for property prediction. From the kernel density contour fill map [Figure 5C], we can see that nearly 94% of the sample points are local at the area with high SSIM and low loss value which confirms the feature extraction efficiency and detail microstructure preservation of the encoder-decoder model. While the 128-dimensional latent space is inherently complex, it encodes salient morphological characteristics critical to mechanical properties. These likely include quantitative measures of the interpenetrating network structure, such as the dominant wavelength of decomposition, the interfacial area density between the c-TiN-rich and c-AlN-rich phases, and the connectivity and curvature of the phases. By compressing the microstructure into a vector that preserves these key features, the encoder provides a highly efficient and information-rich descriptor that serves as a superior input for property prediction compared to traditional, hand-crafted metrics.

Training and validation for hardness prediction model

To further investigate the relationship between coating property changes and microstructure, a regression model for coating hardness prediction was established. The regressor unit is an ANN consisting of two hidden layers (For detailed hyperparameters, see Supplementary Table 1). All latent vectors of TiAlN coatings under different service conditions and time points are merged as input data. By training this regression model, the relationship between coating microstructure and properties can be established, enabling accurate prediction of coating hardness. Before training, the t-distributed stochastic neighbor embedding algorithm was used to perform dimensionality reduction on the microstructural feature vectors of the hardness dataset and compare them with all simulated microstructures. As shown in Supplementary Figure 2, the distribution of the hardness experimental data points is uniform, effectively reflecting the hardness information of the coating across the entire composition range. Through an exhaustive search, we found 52 hardness data for TiAlN coatings from 13 publications^[34,39-50] with hardness values ranging from 12.3 to 47 GPa. The hardness dataset is presented in Supplementary Table 3 and used for the microstructure-hardness relation construction. Figure 6 shows the training results of the encoder-regressor hardness prediction model. The hardness dataset is divided into training and testing sets according to the ratio of 0.8:0.2, and the variation of L1 error during the training process is recorded in Figure 6A. As training steps increase, the loss values for both the training and testing sets decrease rapidly and stabilize at a low value, proving that the model has converged and the training data performs well. To further verify the accuracy, the hardness prediction results obtained from the encoder-regressor model are compared with the actual hardness results and shown in Figure 6B. All sample points are close to the true values and distributed on the line y = x; the average error between the predicted and true values is only 1.6 GPa.

Figure 6. Result of training and evaluation for encoder-regressor model. (A) Loss curve of model training process. (B) Comparison of predicted hardness and actual hardness.

Based on the microstructure-hardness prediction model of TiAlN coatings, we further investigated the trends of hardness with coating composition, annealing time, and annealing temperature, and compared them with the experimentally measured hardness results to verify the accuracy of model prediction. As shown in Figure 7A, the coating hardness gradually increases with Al content and decreases after reaching the peak value. At the same annealing temperature of 1,073 K, extending the annealing time will reduce the hardness. For example, coatings with an Al content around 0.45 exhibit a hardness of ~35 GPa after 0.5 h of annealing, whereas the hardness drops to only ~10 GPa after 2 h of annealing (seen in Figure 7A). From Figure 7B, coatings with high Al content exhibit higher initial hardness in the as-deposited state. Annealing at appropriate temperatures can further enhance hardness, but higher temperatures accelerate the coarsening of the spinodal decomposition microstructure, which in turn reduces coating hardness. The model predictions are in good agreement with experimental measurements. Despite a slight performance gap between the training and testing sets [Figure 6B], this is typical when working with limited experimental data. The model’s primary success lies in its ability to accurately predict the overall trends of hardness evolution with composition and temperature, which align closely with experimental observations. This demonstrates that the model has learned the underlying physical structure-property relationships (i.e., hardening from coherent spinodal structures and softening from coarsening) rather than merely memorizing noise in the training data. Achieving a low prediction error of 1.6 GPa with only 52 data points underscores the effectiveness of the 3D-CNN autoencoder in extracting physically meaningful and predictive features from complex microstructure data.

Figure 7. Prediction of hardness evolutions. (A) Variation of coating hardness with composition under the same heat treatment conditions. (B) Variation of coating hardness with annealing temperature for the same composition. The predicted value is the numerical result obtained directly from the model. Experiment data with error bar represent the average value and standard deviation from at least three independent measurements.

Capturing the property-microstructure relation of TiAlN coating

Microstructural evolution in service significantly affects the performance of coatings; however, traditional experimental measures make it difficult to capture the microstructural features and the logistics between them and the property, especially the phase-to-phase positional relationships and interactions in three-dimensional space. By coupling numerical simulation and autoencoder, our proposed framework enables accurate simulation of the TiAlN coating microstructure under different conditions, and directly establishes a quantitative relationship between the microstructure and property by using the extracted key features, thus realizing the prediction of property evolution during varying service processes. Figure 8A demonstrates the evolution of coating hardness over time for Ti_0.46Al_0.54N coating annealed at 1,073 K. Under prolonged annealing, the coating gradually softens and the predicted hardness values show a decreasing trend in good agreement with the experimental observations.

Figure 8. Property and microstructure evolution of Ti_0.46Al_0.54N coating during annealing at 1,073 K. (A) Time-dependent variations of coating predicted and experimental hardness (B) 3D microstructure and wavelength. The predicted value is the numerical result obtained directly from the model. Experiment data with error bar represent the average value and standard deviation from at least three independent measurements.

Coating properties are closely related to microstructure. We extracted the 3D microstructures and spinodal decomposition wavelengths at different times using the phase-field simulation results [Figure 8B]. Throughout the annealing process, the spinodal decomposition structure of the coating grows, and the wavelength increases, which correspondingly leads to a gradual decrease in the hardness. It reveals that the encoder-regressor hardness model learns the microstructure evolution behavior well and reflects it to the coating hardness variation. Combining the results of hardness and microstructure evolution, an initial spinodal decomposition stage followed by a coarsening stage can be distinguished. The initial decomposition leads to the formation of a nanoscale compositionally modulated structure, which acts as a potent strengthener and results in a slight increase in hardness, reaching a peak of ~37 GPa at approximately 0.65 h. This age-hardening phenomenon is a well-documented characteristic of TiAlN coatings and other systems undergoing spinodal decomposition. As the c-TiN-rich and c-AlN-rich phases grow, the coating enters the roughening stage, and the hardening effect diminishes, leading to the observed gradual decrease in hardness. However, such subtle hardness variations are often difficult to detect experimentally because data points are typically sparse, resulting in an apparently monotonic decay. Overall, the momentary values and trends of hardness predicted by the 3D microstructure-augmented ML model are consistent with experimental results. It is worth noting that our feature extraction and hardness prediction model can capture microstructural evolution with higher temporal resolution. Predicted microstructures and hardness can be obtained for TiAlN coatings at any time point, including information that is often missed in experiments.

Overall, the analysis of the relationship between coating hardness and microstructure further demonstrates the reliability of the hardness prediction models. Using only a small amount of experimental data, we have successfully constructed a quantitative “composition-process-structure-property” relation and achieved dynamic prediction of coating hardness. This success is closely linked to quantitative phase-field simulations and an accurate structure feature extraction ML model. High-fidelity simulation results provide rich 3D microstructural information and high-quality datasets for model training that cannot be obtained through traditional experimental methods. The 3D-CNN-based autoencoder model enables the transformation of large amounts of actual 3D structural information into low-dimensional structural feature vectors, providing the necessary data for constructing a high-precision hardness model. Although this study focuses on hardness prediction of TiAlN coatings, the proposed framework is generalizable. The phase-field model can be adapted to different materials by incorporating appropriate thermodynamic and mobility databases, while the 3D-CNN autoencoder remains effective for extracting latent features from diverse microstructures. Moreover, with suitable experimental training data - such as fracture toughness, wear resistance, or thermal properties - the same framework could be used to predict other mechanical and functional properties. This flexibility underscores the potential of our approach to accelerate the design and optimization of a wide range of advanced materials.