Multi-objective Optimization-Oriented Generative Adversarial Design for Multi-principal Element Alloys

Multi-principal element alloys (MPEAs) or high-entropy alloys (HEAs) have recently garnered significant academic interest in the field of material science, primarily driven by their unique properties and vast potential in many demanding applications [1‐5]. The term “high entropy” encapsulates the concept of substantial configurational entropy stemming from a nearly balanced molar ratio of constituent elements [1]. Multi-principal element alloys, unlike conventional (single-principal element) alloys, often consist of a blend of five or more elements in roughly equal atomic ratios [2]. Their distinct compositions can often result in superior mechanical properties such as remarkable hardness, yield strength, and durability to corrosion or wear [6, 7]. However, the inherent complexity of MPEAs poses significant challenges to alloy exploration using traditional methodologies. As a result, leveraging computational techniques is a compelling approach to facilitate and enhance the exploration of such complex alloys.

Launched by the US Federal government as multi-stakeholder initiative in 2011, the Material Genome Initiative (MGI) ushered in the concept of the materials genome, which has since become an integral facet of how one may approach computational advances in materials science [8]. Drawing inspiration from the systematic characteristics of biological genomes, the materials genome approach envisages a data-centric methodology, detailing the intrinsic properties, structures, and processes of materials, while also elucidating the inherent attributes that govern material performance. This methodology underscores the systematic generation and organization of data, which enables the application of advanced computational tools for predictive modeling, allowing researchers to more rapidly identify and optimize materials to manifest specific properties [9]. Complementing this evolution, computational methodologies like density functional theory (DFT) and high-throughput screening (HTS) have emerged as compelling approaches [10‐12]. These facilitate swift evaluation of myriad compositions and structures, significantly accelerating new material discoveries. Further enriching this landscape is the integration of machine learning (ML) and artificial intelligence (AI) [13‐17]. Drawing from comprehensive databases, these technologies can predict and optimize material properties with efficiency (often surpassing mechanistic models like DFT [18, 19]). The application of ML techniques has gained notable traction in recent years in the discovery and design of MPEAs. In 2019, Wen et al. introduced a method that integrates ML with experimental designs to identify high-hardness HEAs in the Al–Co–Cr–Cu–Fe–Ni system [20]. More recently, Bundela et al. developed an ML framework for predicting HEA mechanical properties [21]. Building upon these advancements, the present authors’ previous study delved deeper into the complexities of MPEAs by employing generative design techniques [22]. This approach merged data science with generative design principles, propelling alloy design into a more automated and predictive paradigm. It is noted that the utilization of ML tools extends beyond some of the original concepts envisioned by the MGI, whereby inherent mechanistic-based materials properties are embodied in the relationships that are embedded within the large datasets used—and do not require explicit calculation. This does not mean that fundamental mechanisms that govern materials properties are neglected, but instead means that the ML tools themselves play a role in providing mechanistic insights that aid in “reasoning” (for materials properties) that can also assist materials designers in understanding complex alloy materials.

In the present research, a novel generative design framework was employed, termed the non-dominant sorting optimization-based generative adversarial networks (NSGAN), originally introduced in Ref. [23]. The approach was developed for single-principal element aluminum (Al) alloys—where a more mature database of alloy properties was utilized in developing the model framework. Herein, the framework has been applied to the study of MPEAs, in order to assess its general applicability, and to explore its use on a complex system which can benefit from computational approaches. In this paper, we will provide a brief introduction to the generative framework employed. For a full treatise of the approach, and the associated algorithms, readers are referred to Ref. [23].

The NSGAN framework integrates the multi-objective optimization methodology with generative ML models (in this case, GANs), offering a data-centric metaheuristic that holds significant potential for applications in materials science. By leveraging GANs to learn data distributions from existing alloys, the design process operates across two spaces: a design space and a latent space. The design space encompasses detailed specifications for each alloy, like elemental composition and processing conditions. These intricate, high-dimensional data points from the design space are mapped through GANs into a simplified, lower-dimensional latent space. This mechanism enables the multi-objective optimization and exploration of generated candidate solutions within the latent space, adeptly circumventing the challenges posed by the so-called curse of dimensionality and thereby markedly enhancing the search efficiency of the optimization algorithm [23]. Furthermore, the incorporation of multi-objective optimization genetic algorithms empowers the GAN model to generate novel samples with specified properties tailored to target demands.

For this study, the NSGAN framework was applied in the generative design of novel MPEAs. Furthermore, in comparison with the original NSGAN work, which was demonstrated upon Al alloys, the MPEA dataset incorporates a greater number of features and optimization objectives. Given the inherent complexity of MPEAs (i.e., widely varying composition space and embodied phases and microstructures that can develop), designing novel alloys with desired properties becomes even more intricate.

To facilitate the application of this model, an online web application was developed for the generative design of MPEAs using the trained NSGAN model. This user-friendly platform provides an interactive interface, allowing users to specify various optimization objectives and, in turn, obtain a range of alloy solutions tailored to their preferences.

In this study, an experimental dataset capturing the specifics of 1704 MPEAs was utilized, which includes the elemental composition of alloys, processing conditions, reported phases, and mechanical characteristics—compiled by the authors [24]. In addition to the alloy features, the training set also incorporates 15 empirical parameters that have been observed to exhibit correlations with the properties of MPEA. Examples of these include mixing entropy, mixing enthalpy, and valence electron concentration. In the training set, alloy samples are characterized by their “design features,” encoded as high-dimensional vectors. The encoding method varies based on the feature’s nature. For instance, the elemental composition is represented by a 32-dimensional vector \({\mathbf{c}} = \left[ {{\text{c}}_{1} , c_{2} , \ldots , c_{31} , c_{32} } \right]^{T}\), where \(c_{i}\) denotes the molar ratio of the ith element, subject to the constraint \(\Sigma_{i = 1}^{n} c_{i} = 1\). In contrast, for processing conditions, this information is encoded as categorical features into seven distinct categories, thereby employing a one-hot encoding approach. A thorough discussion on these alloy features and their correlation with alloy properties was previously provided [22].

In the generative design of alloys with desired properties, a pivotal aspect is the ability to predict the mechanical properties of the generated samples. In this study, we set out to evaluate the predictive performance of the following five ML models for the mechanical properties in the MPEA dataset, namely:

The training process for these ML models incorporated alloy descriptors, encompassing the elemental composition, reported phase, processing methods, and the empirical parameters. In total, 58 variables were considered to predict the mechanical properties. In a previous study [22], a comparative analysis of ML models revealed that the RF model revealed the best performance with an MPEA dataset—and this finding was revalidated in the present study. The R² score (coefficient of determination) [25] was used to evaluate the performance of each model, concurrently employing tenfold cross-validation. And z-score normalization (standardization) was performed on all numerical features prior to training to enhance the efficiency and accuracy of the learning algorithms.

As illustrated in Table 1, both tree-based ensemble models, RF and GBT, showcased reasonable performance. Notably, the RF model outperformed other models across all four properties, justifying its selection as the machine learning model of choice for predicting the mechanical properties of the alloys in subsequent experiments.

In this study, a WGAN-GP model was used to learn the data distribution of 1704 MPEAs, encompassing both the elemental composition of the alloys and their processing conditions. As an extension of the WGAN (Wasserstein GAN), WGAN-GP applies a gradient penalty mechanism to ensure stable training, circumventing the pitfalls of weight clipping in WGANs, and consequently offering better convergence and model flexibility [26].

The architecture of WGAN-GP, illustrated in Fig. 1, encompassing a generative neural network and a discriminative network (the critic) designed to approximate the Wasserstein distance between two data distributions. The generator receives a multivariate Gaussian distributed random noise variable z and is trained to map it to the data distribution of existing alloy samples. The critic is trained to estimate the Wasserstein distance between these two distributions. Distinct from the original WGAN, the loss function for the critic in WGAN-GP incorporates a gradient penalty value, which is calculated as follows:where \(D\left( x \right) \) denotes the output of the critic with input \(x\). The combined term \(E_{{\tilde{x}\sim P_{g} }} \left[ {D\left( {\tilde{x}} \right)} \right] - E_{{x\sim P_{r} }} \left[ {D\left( x \right)} \right]\) constitutes the original critic loss of WGAN, where \(P_{r}\) is the data distribution of the training samples and \(P_{g}\) is the model distribution. Additionally, the gradient penalty term \(\lambda E_{{\hat{x}\sim P_{{\hat{x}}} }} \left[ {\left( {\left\| {\nabla_{{\hat{x}}} D\left( {\hat{x}} \right)} \right\|_{2} - 1} \right)^{2} } \right]\) helps enforce the Lipschitz constraint on the critic. Here, \(P_{{\hat{x}}}\) represents a distribution defined by randomly interpolating between \(P_{r}\) and \(P_{g}\). In our experiments, we observed that the model performs optimally when the penalty coefficient λ is set to 0.01. An in-depth introduction and comparison of WGAN-GP with other popular generative machine learning models is available in Ref. [26].

The authors/developers of the WGAN-GP noted in their work that the classic WGAN tends to generate overly simplistic data distributions when attempting to fit complex data distributions—which aligns with the observations from the implementation with our MPEA dataset. Figure 2 compares the t-SNE (t-distributed stochastic neighbor embedding) dimensionality-reduced data distribution plots of the data generated by both WGAN and WGAN-GP under identical training conditions, where t-SNE is a commonly used technique visualizing high-dimensional data distributions [27]. It is evident that (from Fig. 2), constrained by weight clipping in its critic network, the generator network trained by this method struggles to accurately capture the original data distribution. Noticeable “chunks” of the WGAN-generated data populates the regions between clusters of training data, and there is an evident underrepresentation of certain training samples in the generated dataset. In contrast, the data generated by WGAN-GP aligns exceptionally well with the original training data distribution. Therefore, it can be concluded that WGAN-GP demonstrates a significant improvement over WGAN in such generative design applications.

The NSGA-II algorithm was employed utilizing the pymoo library, which is versatile and comprehensive multi-objective optimization Python framework encompassing a collection of state-of-the-art algorithms [30]. The modular design capabilities of this approach facilitate straightforward customization and expansion, making it an appropriate option for the current application. For property prediction and the implementation of the WGAN-GP model, we utilized PyTorch, a powerful and widely used open-source machine learning library that enables both flexibility and efficiency in building complex models [31].

The associated repository for deploying the NSGAN model for MPEA design is available open-source at: https://​github.​com/​anucecszl/​generative_​design_​MPEA.

To assess the effectiveness and efficiency of the NSGAN model on the MPEA database, a comparative analysis was conducted between the NSGAN model and data-driven generative machine learning models, specifically utilizing the same WGAN-GP model. Both methods were employed to identify samples exhibiting superior tensile strength and elongation according to a pre-defined criterion: ultimate tensile strength (UTS) > 1500 MPa and elongation > 35%.

Generative adversarial networks (GANs) are recognized for their efficiency in producing synthetic samples by learning from existing datasets. Once trained, GAN models can rapidly generate a vast number of synthetic data points. However, samples generated by GANs often display a high degree of randomness and typically lack precise controllability.

Consequently, employing GAN models to explore innovative alloy designs frequently requires filtering through a large volume of randomly generated samples to find and assess those that show potential for superior performance and innovation. This presents a substantial challenge in leveraging GANs for efficient and targeted material design. In this study, ten WGAN-GP models, each with different initial parameters, were trained and utilized to evaluate the comparative effectiveness of direct GAN-generated alloy samples versus those produced through the NSGAN framework.

Using the direct GAN method, each of the ten WGAN-GP models was tasked with generating 40,000 alloy samples, resulting in an aggregate of 400,000 samples generated by all the models combined. On the other hand, the NSGAN strategy involved executing the NSGAN framework once for each WGAN-GP model, setting a population size of 200 for 500 generations, leading to a collective output of 2,000 alloy samples from all models.

The experimental results are displayed in Table 3, which reveals that due to the inherent randomness and lack of controllability in GAN-generated samples, only 0.035% of the large volume of samples produced by the WGAN-GP model met the pre-defined criteria.

In contrast, the NSGAN model, despite generating a significantly smaller total number of samples compared to the GAN model, succeeded in producing a greater number of samples that satisfied the pre-defined criteria.

Regarding computation time and efficiency, since both approaches necessitate pre-training of both the GAN models and the property prediction models, the comparison is focused solely on the time taken to generate and screen for the desired samples using the trained models. As indicated in the table, the total time consumed for generative design using the direct ten WGAN-GP models amounted to 363 s. Conversely, employing the NSGAN method required a total of 171 s, significantly less than the GAN approach. These experiments were performed on a computer with the specifications: CPU i9-13900 K 3.00 GHz and 32 GB of RAM.

In conclusion, the NSGAN model demonstrates a notable improvement in the efficiency of identifying and optimizing desired samples, highlighted by its reduced computation times and the ability to achieve a higher proportion of samples meeting specific mechanical properties with fewer generated samples. This underscores the model’s effectiveness in conducting a more targeted and efficient search for novel materials, illustrating its potential in streamlining material innovation processes.

In order to expand the utility and outreach of the proposed NSGAN framework, an online application has been developed. This software tool, utilizing cloud-based computing, enables users from various backgrounds to employ the proposed generative framework without the constraints of local software installations.

The development and deployment of this application were facilitated using Streamlit—an open-source Python library specifically designed for crafting intuitive and interactive web applications, and may be accessed at https://​generativedesign​mpea.​streamlit.​app/​. Integrated within this application are nine pre-trained machine learning models: a robustly trained WGAN-GP model, complemented by eight RF models, each dedicated to predicting specific phase and mechanical property of MPEAs.

The user interface of the application is designed to be intuitive and concise. It offers several parameters for customization, such as the population size and number of generations. Additionally, the platform presents selectable optimization objectives, allowing users to adjust and combine these targets to generate specific desired outcomes. The user interface is illustrated in Fig. 5, which also presents an example of an output (scatter plot) depicting the relationship between tensile strength and elongation of generated samples. This visualization is based on a population size of 50, spanning 200 generations, with optimization objectives factoring in tensile strength and elongation. All computationally generated samples can be batch downloaded directly from the user interface in Excel format, with screenshot of the generation provided in in Fig. 6. An entire generation process takes less than ~ 15 s to run. It should be noted that the final population not only hosts a considerable number of optimized non-dominant solutions, owing to the genetic operations performed in each generation, but it also includes dominated solutions present in the final generation. Consequently, it is important to acknowledge that not every solution within the final population can be considered optimal.

By combining different design objectives, the model may be customized to generate results fulfilling various design requirements. Taking phase structure as an example, with the training condition shown in Fig. 5, out of the 50 generated alloy samples, 12 were predicted to have an FCC structure, while the rest were predicted as either single-phase BCC or BCC with intermetallic compounds. Conversely, by setting the FCC phase as an additional optimization objective and keeping other parameters constant, the number of samples predicted to exhibit an FCC structure increased to 30. Moreover, beyond the eight properties mentioned, we integrated additional optimization objectives like density and aluminum content to demonstrate how this model can be tailored for custom generation. Using density as an example, under identical training conditions, when optimizing for tensile strength and elongation, the introduction of density as an additional optimization criterion led to a noticeable decrease in the average density of the final population, from 7.98 to 5.07 g/cm³.

While the study herein primarily focuses on results pertaining to strength–ductility, it is important to note that the model is versatile enough to accommodate any combination of design objectives, capable of effectively managing multiple objectives simultaneously. Through the strategic integration of various optimization objectives or the addition of new ones, users can deftly harness the proposed NSGAN framework to computationally tailor-design desired alloys and materials for a wide range of applications. Additionally, by training the model with diverse databases, this generative design method can be expanded to aid in the design of various materials across multiple domains.

In this study, the potential of the NSGAN (non-dominant sorting optimization-based generative adversarial network) framework was explored—as an innovative model to address the significant challenge of multi-principal element alloy (MPEA) design. The NSGAN is a blend of NSGA-II’s multi-objective optimization and the generative abilities of the WGAN-GP model. By utilizing the strengths of GANs to decipher the data distributions of the existing MPEA databases, NSGAN allows operations across two core dimensions: explicit design space and latent space. It was noted that the available (experimental) data for MPEAs is presently limited, and therefore the model performance as presented will continue to benefit from more experimental data from emerging literature or empirical testing as a result of predictions from the NSGAN framework presented. The approach will also benefit from the addition of properties that are relevant to some specific target applications (such as corrosion rate, fatigue performance, or high-temperature properties).

To allow open access to the NSGAN framework, the associated code is available, and we have also demonstrated the tangible applicability of NSGAN by developing an intuitive online application. This platform, powered by cloud computing, makes the NSGAN framework accessible to a global audience and provides an interactive environment where users (including non-computational specialists, alloy practitioners or students) can define optimization objectives; thereby generating a spectrum of alloy designs tailored to their unique requirements. The application of the NSGAN framework for the design and discovery of MPEAs showcases a convergence of material science and computational methodologies, aligning with the evolving ethos of the material genome initiative. The approach herein is tilted toward data science approaches as opposed to first principal approaches, however, it is the first (that the authors know of) alloy design tool that works as an “optimizer” for MPEAs—making it significant for the field. The framework offers a coherent theoretical base and a pragmatic approach to alloy design; opening avenues for further exploration and innovation in MPEAs.