Deep Learning Approaches for Predicting Strain Energy in Heterogeneous Materials

Junesh Gautam ¹, Praneel Acharya ², Zhong Hu ^3,*

1. McComish Department of Electrical Engineering & Computer Science, Department of Mechanical Engineering, J. J. Lohr College of Engineering, South Dakota State University, Brookings, SD57007, USA

2. Department of Aviation, College of Education, Minnesota State University at Mankato, Mankato, MN56001, USA

3. Department of Mechanical Engineering, J. J. Lohr College of Engineering, South Dakota State University, Brookings, SD57007, USA

* Correspondence: Zhong Hu

Academic Editor: Sanjay Nimbalkar

Special Issue: Multi-Scale Modeling and Characterization of Functional Materials

Received: July 15, 2025 | Accepted: October 21, 2025 | Published: October 24, 2025

Recent Progress in Materials 2025, Volume 7, Issue 4, doi:10.21926/rpm.2504016

Recommended citation: Gautam J, Acharya P, Hu Z. Deep Learning Approaches for Predicting Strain Energy in Heterogeneous Materials. Recent Progress in Materials 2025; 7(4): 016; doi:10.21926/rpm.2504016.

© 2025 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

The need to continuously design advanced materials for highly specialized engineering applications requires robust and efficient methods for predicting mechanical properties. Traditional simulation-based techniques provide insight into stress distributions, strain fields, and strain energy stored in materials under various loading conditions. The finite element method (FEM) or analysis (FEA) is a commonly used method for numerically solving differential equations that arise in engineering and mathematical modeling. Although it is difficult to trace the invention date of FEM/FEA, its development can be traced back to the early 1940s [1,2,3]. Typical research areas include the traditional structural analysis heat transfer, fluid flow, mass transport, and electromagnetic fields. Computers are often used to perform the required calculations. With the help of high-speed supercomputers, better solutions can be obtained and are often used to solve the largest and most complex problems. However, these conventional computational methods become prohibitively expensive when exploring large design spaces, performing optimization cycles, or achieving near-real-time predictions for interactive applications [4,5].

Studying heterogeneous materials is generally more difficult than studying homogeneous materials. Examples of heterogeneous materials include composite laminates, functionally graded materials, biological tissues, and polymeric foams or rubbers. The main difficulties in studying heterogeneous materials are their inhomogeneity in composition and properties, complex microstructures, and the need for advanced modeling and characterization techniques, which results in higher computational and experimental costs than homogenous materials. Furthermor, predicting the mechanical response of heterogeneous materials is computationally intensive because it depends on spatially varying properties and nonlinear constitutive behavior. FEA remains the gold standard, but it can become prohibitively expensive for large design spaces, mesh-dependent microstructures, and repeated evaluations during optimization.

In many engineering applications, the relationship between stress and strain in the elastic region is not strictly linear, which makes the classical Hooke’s law inapplicable for large deformations, Instead, hyper-elastic material models (one of simplest of which is the Neo-Hookean model) describe stress-strain behavior through a strain energy function. For a two-dimensional Neo-Hookean solid, the strain energy density function is usually expressed as [6,7]:

\[ \psi=\frac{\mu}{2}(\mathrm{I}_1-2)-\mu\ln(J)+\frac{\lambda}{2}(\ln(J))^2 \tag{1} \]

where µ and λ are the Lamé parameters (related to Young’s modulus and Poisson’s ratio), I₁ is the first invariant of the right Cauchy-Green deformation tensor, and J is the determinant of the deformation gradient. This form can capture large deformation and is often used to simulate rubbers or soft biological tissues.

Despite their accuracy and reliability, computational methods such as FEA face fundamental challenges. For example, for heterogeneous materials with complex microstructures, the computational cost of nonlinear problems with n elements is as high as O (n^1.5-2), making real-time prediction impossible [8]. The mesh dependency and convergence requirements of non-uniform stiffness distributions limits the feasibility of materials with sharp property gradients (e.g. patches of the Mechanical MNIST image dataset used in this study) [9]. The manual expertise needed for constitutive model selection and solver tuning hinders automated design optimization efforts, which is the core process of deep learning (DL) approaches [10]. The high computational cost mainly comes from the need for dense meshes to resolve detailed spatial variability and the iterative solvers required for nonlinear constitutive relationships such as the Neo-Hookean model. These requirements severely hinder fast iterative design and optimization processes or real-time simulation applications.

Alternative approaches, especially those that utilize data-driven machine learning (ML) methods, offer a promising alternative to rapidly predict mechanical responses based on relationships learned from pre-computed simulation data or experimental data. Such approaches significantly reduce computational costs and improve design efficiency. Recent developments in DL neural networks, particularly convolutional neural networks (CNNs) and residual neural networks (ResNets), show great potential because they can automatically learn hierarchical representations directly from raw image data without the need for complex manual feature extraction.

DL is revolutionizing materials science by enabling the prediction of material properties, simulation of material behavior, and the design of new materials with specific characteristics. It excels at analyzing complex unstructured data and automating identifying features, especially as large material databases become increasingly available. DL methods are applied to atomistic simulations, materials imaging, spectral analysis, and even natural language processing of material-related textual data [11]. Zhang et al. [12] proposed a hierarchical DL neural network (HiDeNN) that integrates adaptive FEM meshing with DL and uses CNN-predicted stress concentrations for dynamic mesh refinement, achieving a 70% reduction in degrees-of-freedom (DOF) while maintaining 99% strain energy accuracy and combined r-adaptivity (node movement) and rh-adaptivity (remeshing). Li et al. [13] established an implicit mapping between the effective mechanical properties of heterogeneous material part and their mesoscopic structure. Image processing techniques were employed to convert sample images into FEA models, and FEA was used to evaluate the effective mechanical properties of the sample. A CNN model was trained based on the sample images and their effective moduli. The trained network was validated to be able to accurately and efficiently predict the effective moduli of the sample. In addition, recent studies have demonstrated the potential of ML to overcome the limitations of FEA. For example, Guo et al. [14] calculated composite materials and achieved 380 times faster than commercial FEA software with an error of less than 3%. Henkes et al. [15] developed a hybrid PINN-FEA framework for nonlinear elasto-plasticity. Wang et al. [16] applied a vision transformer to predict stress concentrations in perforated plates. These developments have created technical precedents and performance benchmarks for our research framework.

The Mechanical MNIST image dataset developed by Lejeune et al. [17] provides a robust benchmark consisting of 60,000 28 × 28-pixel material stiffness image maps with precomputed FEA results of heterogeneous materials based on the Neo-Hookean constitutive law that match the nonlinear material and are subjected to large deformations (multiple loading scenarios: uniaxial tension, equi-biaxial tension, confined compression, and shear), allowing for a comprehensive evaluation of both classical and advanced ML methods. Exploring these advanced neural networks that learn directly from pixel intensity mapping represents a significant advance over traditional feature-engineered modeling approaches. The Mechanical MNIST dataset was simulated using a compressible Neo-Hookean material model with a strain energy density function expressed as:

\[ \psi=\frac{\mu}{2}[\mathbf{F}:\mathbf{F}-3-2\ln(\det\mathbf{F})]+\frac{\lambda}{2}\left[\frac{1}{2}(\det\mathbf{F})^2-1)-\ln{(\det\mathbf{F})}\right] \tag{2} \]

where F is the deformation gradient and detF is the determinant of deformation gradient. To convert the MNIST bitmap images to material properties, the material domain is partitioned to correspond to the grayscale bitmap [17]. Young’s modulus (E) is mapped from the MNIST grayscale bitmap intensity as follows:

\[ E=\frac{b}{255}(100-1)+1 \tag{3} \]

where b is simply the grayscale bitmap intensity of a single MNIST pixel, the raw MNIST pixel brightness at that location.

This study won’t replace FEA but rather explored how DL can complement it for heterogeneous materials. DL has shown its value in many fields, including agriculture [18]. While such models require intensive computation and fine tuning, this study will demonstrate that they could provide practical benefits even under these constraints. The main contributions of this will aim to:

• Constructing formulations of direct image-to–strain-energy mapping from 28 × 28 stiffness-field images under large deformation conditions.
• Developing and evaluating end-to-end CNN and ResNet surrogates for rapid strain-energy prediction.
• Performing a systematic and fair comparison with classical regression models and deep learning to predict strain energy in heterogeneous elastic materials.

This study will also outline the path to experimental validation by referring to recent studies that combine acoustic emission (AE) sensing with deep learning for failure analysis and early warning, thus concretizing real-world deployments.

Of course, one of the core focuses of this work is to study heterogeneous materials, whose properties vary spatially, resulting in mechanical behaviors that are far more complex than those of homogeneous materials. By bridging the gap between computational mechanics and DL, this work aims to explore the capability, efficiency, practicality, and real-time applicability of DL approaches (e.g., CNN and ResNet) for strain energy prediction in mechanical design of heterogeneous materials. The Mechanical MNIST image dataset will be used for DL study. This advanced approach will completely bypass mesh generation and refinement by developing an end-to-end image-to-property mappings – a key innovative step for heterogeneous materials where mesh dependencies plague conventional FEA. The performance of the DL algorithms will be systematically compared with the classical statistical regression methods (e.g., simple linear regression, random forest, gradient boosting).

2. Methodology

2.1 Data Acquisition and Preprocessing

In this work, each pixel intensity in the 28 × 28 image represents the local stiffness distribution in a uniaxial tensile specimen of heterogeneous material. The finite element meshes with spatially varying stiffness were created from these intensities. The boundary conditions were imposed (e.g., fixed bottom, tension at the top), and then an FEA solver was used to compute the deformation of the specimens under load, thereby storing elastic energy in each element, as shown in Figure 1. Mathematically, the total strain energy can be expressed as:

\[ U=\int_V\psi dV \tag{4} \]

where V is the volume of the specimen.

Figure 1 Schematic representation of a heterogeneous material specimen under uniaxial tension. (a) vertical displacement distribution and (b) strain energy vs. applied displacement.

The study begins with 28 × 28-pixel grayscale images stored in an external file (e.g., minst_img_train.txt). Each row corresponds to a flattened 28 × 28-pixel sample. In practice, each row can be reshaped into a 2D array of shape (28, 28) so that the (28, 28, 1) features can be extracted and fed into the neural network [17]. Each image was reshaped into a 28 × 28 grid for learning. Input images were normalized. To preserve the physical coordinate system and load direction of the uniaxial-tension setup, geometric augmentation involving rotations was avoided, and the images were not smoothed to maintain the integrity of the stiffness contrast. The data was partitioned into training, validation, and testing sets, using a fixed seed to ensure reproducibility. 60% of the data was used for training, 20% for validation, and the remaining 20% for testing. The validation set supported model selection and early stopping, while the testing set was used for final evaluation. The supervised target was the scalar total strain energy at the final load step in the reference simulation. All DL and classical models were trained and evaluated on a single NVIDIA A100 GPU to ensure a fair and direct comparison of their predictive performance. This setup allows us to distinguish methodological differences between models, as their performance is not affected by hardware differences. While a thorough analysis of training time, inference speed, and memory usage is key for real-world deployments, studying the efficiency of such hardware normalization is beyond the scope of this paper; the focus here is on demonstrating the feasibility of this approach.

The strain energy values were extracted from an accompanying text file. Each row contains multiple columns corresponding to different load steps, with the strain energy for the last step stored in the last column. The Reaction Force Data were used for multi-task models that predict the reaction forces. This allows us to train a network with two outputs: strain energy and reaction force.

If a simpler, classical ML pipeline is chosen, the features such as Average Intensity, Contrast, and Edge Features for each 28 × 28-pixel image can be extracted.

Once these features are extracted, the correlation analysis on how strongly these features related to each other and with the final-step strain energy can be investigated. This helps identify if a single feature (e.g., Average intensity) dominates the variance, or if multiple features work together to explain the strain energy.

Reproducible experiments require a consistent method of splitting the dataset (e.g., first separate out a training set of 60% of the entire dataset, then split the remaining (40%) dataset into validation and test dataset (20% each)). This ensures that each subset is mutually exclusive. The random-state parameter guarantees that future reruns will produce random seeds with exactly the same partitions. By carefully separating the training, validation, and test data subsets, the models can be reliably tuned on the validation set while leaving the test set untouched for final performance evaluation.

2.2 Classical Regression Models

In this work, an initial attempt was made to predict the strain energy of the heterogeneous material samples using simpler models, classical models based on hand-crafted features such as Average Intensity, Contrast, Edge features. To ensure a fair and controlled comparison with our deep learning models, we intentionally restricted classical ML baselines to a minimal set of physically perceptual features (e.g., mean intensity, contrast, edge signatures). This approach was taken to isolate the effects of deep neural networks’ representational learning capabilities. By restricting classical models to a smaller “information budget” comparable to the pixel-level inputs used by CNNs and ResNets, the confounding effects of extensive manual feature engineering processes was avoided.

2.2.1 Simple Linear Regression Model

A linear regression model is a statistical method that examines the relationship between one or more independent variables (predictors) and a dependent variable (outcome) by fitting a linear equation to the observed data. It aims to find the “best-fit” line or multidimensional hyperplane that minimizes the difference between the predicted and actual values. A Simple Linear Regression model involves one independent variable and one dependent variable. The relationship between the two variables is represented by a straight line [19].

2.2.2 Random Forest Regression Model

An ML model that uses an ensemble of decision trees to perform predictions on regression tasks. It works by training multiple decision trees on different sub-datasets and then averaging their predictions to produce the final output. This approach helps reduce overfitting and improves the overall stability and accuracy of the model [20].

2.2.3 Gradient Boosting Model

An ensemble learning algorithm that generates accurate predictions by combining multiple decision trees into a single model. This predictive modeling algorithm leverages the strengths of the base models, corrects errors, and improves predictions. Gradient Boosting excels in a variety of predictive modeling tasks by capturing complex patterns in the data [21].

2.3 Deep Learning Models

Compared to classical models that rely on simple, hand-crafted features, DL models can learn hierarchical feature representations directly from raw images (the 28 × 28-pixel images in this study). This may reveal spatial patterns that are critical for accurate prediction of strain energy.

2.3.1 Convolutional Neural Networks

Convolutional Neural Networks (CNNs): are a class of artificial neural networks that are particularly well suited for processing grid-like data, especially for image-related tasks, because they apply convolutional filters to local patches to more fully capture edges, shapes, and textures [22]. The CNN architecture consists of an input layer, hidden layers, and an output layer. The hidden layers contain one or more layers that perform convolution, as shown in Figure 2.

Figure 2 Schematic representation of a CNN architecture for regression.

2.3.2 Residual Neural Network

Residual Neural Network (also referred to as Residual Network or ResNet): is one of the most widely used neural networks in recent years. ResNet is based on a deep residual learning framework that effectively overcomes the degradation problem and enhances the ability to extract information from the raw (input) data. It does this by utilizing an identity mapping module that incorporates skipping connections or shortcuts to skip layers in the network. This makes ResNet as deep as 152 layers. ResNet was developed in 2015 for image recognition [23]. As shown in Figure 3, the representative ResNet module has a double layer skipping connection that performs nonlinearity and batch normalization, and ReLu represents the rectified linear unit activation function.

Figure 3 Schematic representation of a Residual Network.

3. Results and Discussion

First, the simpler models (i.e., Simple Linear Regression model, Random Forest Regression model, and Gradient Boosting model) were investigated to predict strain energy of the MNIST dataset, and the models were trained on hand-crafted features (e.g., Average Intensity, Contrast, and Edge features).

For a Simple Linear Regression model, the performance achieved mean square errors (MSE) of 50.02 and 46.21 on the validation and test sets, and coefficients of determination (R²) of 0.79 and 0.80 on the validation and test sets, respectively. Figure 4 shows the predicted strain energy versus the actual strain energy for the validation data, indicating that most points are fairly close to the identity line. The red dashed line represents the ideal relationship between the variables.

Figure 4 The validation data of actual strain energy vs. predicted strain energy by a Simple Linear Regression model.

The same training/validation/test datasets were used for the Random Forest Regression model. After training, the model achieved promising results with MSE values of 34.18 and 34.69 on validation and test sets, and R² values of 0.86 and 0.85 on validation and test sets, respectively. Figure 5 shows the predicted strain energy versus the actual strain energy for the validation data. The red dashed line again represents the ideal relationship between the variables.

Figure 5 The validation data of actual strain energy vs. predicted strain energy by a Random Forest Regression model.

Next, the Gradient Boosting model was explored and tuned into the hyperparameters such as the number of estimators, learning rate, and maximum tree depth. Two configurations were created, i.e., for configuration 1, the number of estimators and the maximum tree depth are 200 and 10, respectively; for configuration 2, the number of estimators and maximum tree depth are 220 and 11, respectively. For configuration 1, the MSE values on the validation and test sets are 15.56 and 15.84, respectively, and the R² values for both the validation and test sets are 0.93. For configuration 2, the MSE values on validation and test sets are 18.44 and 18.69, respectively, and the R² values for both the validation and test sets are 0.92. Figure 6 shows the predicted strain energy vs. the actual strain energy for the validation data for both configurations. The red dashed line indicates the ideal relationship between the variables.

Figure 6 The validation data of actual strain energy vs. predicted strain energy by a Gradient Boosting model: (a) configuration 1 (number of estimators of 200 and maximum tree depth of 10) and (b) configuration 2 (number of estimators of 220 and maximum tree depth of 11).

Compared to classical regression models that rely on simple, hand-crafted features, DL models can learn hierarchical feature representations directly from raw images. This may reveal spatial patterns that are critical for more accurate prediction of strain energy.

For the CNN model, a series of searches were performed to systematically identify the best set of hyperparameters, and ultimately an optimized CNN architecture for regression was achieved and used for this study. Figure 7 shows the optimized CNN model, which contains two convolution-pooling blocks (64 and 128 filters, respectively) and two dense layers (128 and 64 units, respectively).

Figure 7 Optimized CNN architecture in this study (Batch Size: 32, Learning rate: 1 × 10^-4, Epochs: 50, Test MSE: 4.21, R²: 0.982).

Table 1 highlights a few representative hyperparameter configurations that affect the performance of the model. Finally, the optimized model selected the combination of [64,128] convolution filters, [128,64] dense layers, batch size of 32, learning rate of 1 × 10^-3, and 50 epochs. Based on this optimal configuration, the MSE values on the validation and test sets are 4.54 and 4.21, respectively, and the R² values for both the validation and test sets are 0.98.

Table 1 Selected hyperparameter configurations and their performance.

In addition to the CNN model, a ResNet model was also studied. To determine the optimal ResNet configuration, a series of searches were performed and the hyperparameter sets were adjusted. Finally, an optimized ResNet model configuration was selected: batch size of 32, epochs of 20, and number of residual blocks of 3, as shown in Figure 8. Based on this model configuration, the MSE values on the validation and test sets are 3.06 and 4.33, respectively, and R² values for both validation and test sets are 0.98.

Figure 8 Optimized ResNet architecture in this study.

Table 2 summarizes the performance of all models adopted in this study for validation and test processes.

Table 2 Summary of the performance of validation/test processes by all models.

A small, controlled hyperparameter search was conducted to balance accuracy and reproducibility. For the CNN, the architecture was fixed a priori—three 3 × 3 convolutional layers (128, 64, 64 filters) with 2 × 2 max-pooling, followed by three fully connected layers with ReLU activations (1028, 512, 256) — and trained used Adam with mean-squared error (MSE) loss. Pilot runs showed stable validation performance with a batch size of 32 and a 30-epoch schedule, which were used for all CNN experiments. For the ResNet, an explicit grid search was performed across batch size {32, 64}, training epochs {10, 20}, and depth varying by the number of residual blocks {2, 3} (8 configurations total), using Adam with a learning rate of 1 × 10^-3 and MSE loss in each case. All models were trained on the training set and evaluated on the validation set at each epoch using the same 60/20/20 partition and fixed random seed. The primary selection criterion was the lowest MSE on the validation set at the end of training; in the event of a tie, the lower mean-absolute error (MAE) on the validation set was used, followed by model parsimony (fewer parameters) and actual training time. The best configuration for each architecture, based on these criteria, was then evaluated on the testing set without further tuning.

Based on the above results and discussion, although these simpler models can achieve moderate accuracy on validation and test datasets, they often have difficulty capturing the full complexity of the 28 × 28-pixel distribution. Therefore, the performance of the predicted new image is lower. The results clearly show that DL models significantly outperform classical regression models in predicting strain energy from image-based representations. The main advantages include (1) Learned representations – CNN and ResNet models can automatically identify and exploit complex spatial features (edges, patterns, local stiffness variations) directly from the raw image data, completely bypassing manual feature extraction; (2) Superior prediction accuracy – DL models greatly exceed the performance limitations exhibited by classical regression models (lower MSE and higher R² values), highlighting their ability to model complex nonlinear relationships in heterogeneous materials; (3) Potential multi-task learning benefits – jointly predicting correlated mechanical properties, such as reaction forces and strain energy, may enhance the learning process by incorporating shared underlying physical behaviors.

DL models face potential limitations when applied to heterogeneous material applications, largely due to the complexity of the microstructures and the lack of comprehensive data. These challenges impact on model accuracy, reliability, and interpretability. With advances in computing technology and algorithms, data-driven methods, particular ML and artificial intelligence (AI) are transforming the study of heterogeneous materials by accelerating discovery, optimizing design, and improving manufacturing processes. The performance of these materials (e.g., composites, alloys, concretes, ceramics with complex microstructures) is heavily dependent on their composition, structure, and interfaces, making traditional trial-and-error approaches inefficient. However, experimental validation of these methods is crucial to ensure accuracy and reliability and to bridge the gap between computational predictions and real-world performance. Recent studies combining Acoustic Emissions (AE), a non-destructive testing technique, with DL for heterogeneous material failure analysis have shown great potential in real-world applications, which demonstrated that AE time series features processed by neural models can identify precursor damage signals and enable online failure prediction in harsh environments, providing a practical path for experimental validation and field use of heterogeneous material surrogates, surpassing traditional analysis by addressing complexities such as noise and multiple co-existing damage mechanisms [24,25,26]. Combining AE with DL for failure analysis has great potential in real-world heterogeneous materials applications, such as damage identification, predictive modeling, and unsupervised learning in composite materials; damage assessment, damage classification, and damage stage identification in concrete structures; geotechnical and material testing, and experimental validation. The challenges and prospects include data quality and quantity, noise filtering, model transferability, and uncertainty quantification.

4. Conclusions

This study has systematically evaluated classical regression models (Simple Linear Regression, Random Forest, and Gradient Boosting) based on manually extracted image features and assessed their applications to strain energy prediction. The coefficient of determination (R²) of these models ranges from 0.79 to 0.93, highlighting inherent limitations due to their reliance on overall simplistic feature representations.

DL models (CNN and ResNet) show superior predictive capabilities by automatically identifying and exploiting complex spatial features in raw image data without the need for manual feature extraction. These models have R² of approximately 0.98, which is a significant improvement over classical regression models and have great potential for practical applications in computational mechanics, material property predictions, and material design.

This work establishes a foundational benchmark based on Mechanical MNIST, but we acknowledge several key limitations that will be addressed in future work. Our current surrogate model is evaluated under a single loading model (uniaxial tension) and does not yet encompass more complex scenarios, such as multiaxial states, multiple boundary conditions, or time-dependent phenomena. To extend the current approach, the deep learning network needs to be fine-tuned based on compact load descriptors and include multiaxial and shear cases in training and testing. For temporal behavior, smaller temporal heads or multi-output objectives need to be used to predict load-displacement or energy-cycle sequences. Furthermore, the current model employs a low-resolution stiffness field and lacks explicit physical constraints, which increases the risk of poor extrapolation; to mitigate this, more diverse data needs to be used. To achieve high R² values, explainable artificial intelligence (XAI) techniques would be considered, such as saliency maps and occlusion analysis, to gain deeper understanding of the physical properties of the learned features. Finally, a key future direction is to combine our deep learning model with real-time sensor data, such as acoustic emission monitoring, to validate predictions against experimental data and enable practical applications such as online fault prediction. Future work includes analyzing training/inference costs and exploring compression/quantization to enable real-time deployment.

Acknowledgments

This work was supported by the Department of Mechanical Engineering and the McComish Department of Electrical Engineering & Computer Science in J. J. Lohr College of Engineering, and the Department of Chemistry, Biochemistry and Physics in College of Natural Sciences at South Dakota State University, for which we gratefully acknowledge.

Author Contributions

Conceptualization, J.G., P.A., and Z.H.; methodology, J.G., P.A., and Z.H.; software, J.G., and P.A.; validation, J.G., and P.A.; formal analysis, J.G., P.A., and Z.H.; investigation, J.G., P.A., and Z.H.; resources, J.G., and P.A.; data curation, J.G., and P.A.; writing—original draft preparation, J.G., and P.A.; writing—review and editing, Z.H.; visualization, J.G., P.A., and Z.H.; supervision, Z.H.; project administration, Z.H. All authors have read and agreed to the published version of the manuscript.

Competing Interests

The authors have declared that no competing interests exist.