Generative Adversarial Networks in Business and Social Science

Citation: Ruiz-Gándara, A.; Gonzalez-Abril, L. Generative Adversarial Networks in Business and Social Science. Appl. Sci. 2024,14, 7438. https://doi.org/10.3390/ app14177438 Academic Editors: Adnane Cabani and Kévin Bouchard Received: 24 July 2024 Revised: 16 August 2024 Accepted: 20 August 2024 Published: 23 August 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Review Generative Adversarial Networks in Business and Social Science Africa Ruiz-Gándara and Luis Gonzalez-Abril * Department Applied Economic I, Faculty of Sciences of Economics and Business, University of Seville, Avda. Ramón y Cajal, 1, E-41018 Sevilla, Spain; [email protected] *Correspondence: [email protected]; Tel.: +34-954-557539 Featured Application: The importance of generative adversarial networks (GANs) in economics is growing and is driven by successes in other fields. Many economic problems could benefit from GANs, although few studies exist and progress is needed. This paper argues for the use of GANs as a novel and effective tool in economics. An important issue is the need for large datasets, where traditional techniques fall short due to a multiplicity of problems, making the use of GANs very useful in this task. Abstract: Generative adversarial networks (GANs) have become a recent and rapidly developing research topic in machine learning. Since their inception in 2014, a significant number of variants have been proposed to address various topics across many fields, and they have particularly excelled not only in image and language processing but also in the medical and data science domains. In this paper, we aim to highlight the significance of and advancements that these GAN models can introduce in the field of Business Economics, where they have yet to be fully developed. To this end, a review of the literature of GANs is presented in general together with a more specific review in the field of Business Economics, for which only a few papers can be found. Furthermore, the most relevant papers are analysed in order to provide approaches for the opportunity to research GANs in the field of Business Economics. Keywords: GANs; multidisciplinary application; business economics; artificial intelligence; machine learning 1. Introduction Currently, we are facing a continuous technological advance that necessitates significant societal changes. In light of this new social structure, massive data holds particular interest, since access to this resource signifies major privileges. Traders, manufacturers, suppliers, insurers, and entrepreneurs all require access to and the ability to utilise extensive databases that enable them to study and uncover the necessary information to establish commercial strategies. This access also allows them to understand customer profiles and preferences, thereby enabling informed decision-making on pricing, promotions, risk assessment, competition, business models, and more. However, these highly coveted volumes of data have surpassed human capacity for collection, storage, and analysis. Consequently, to undertake these tasks effectively, humans require suitable tools, such as statistics intelligence and machine learning. Major advances are being made in machine learning, which has been utilised to recognise patterns and identify objects. However, recently there has been a leap towards generating entirely new objects and individuals. Generative models are capable of creating all kinds of content from images and text. Among them, we highlight: generative adversarial networks (GANs), transformers, variational autoencoders (VAEs), diffusion models, and neural radiance fields (NeRFs). It is worth noting that when choosing one model or another, one should consider choosing a model that works well for the problem to be tackled. Thus, for example, one would use transformers for translation tasks, NeRFs Appl. Sci. 2024,14, 7438. https://doi.org/10.3390/app14177438 https://www.mdpi.com/journal/applsci Appl. Sci. 2024,14, 7438 2 of 23 for 3D scenes, and GANs for the generation of synthetic datasets. It is important that a sufficient quantity and quality of data be available; thus, for example, transformers need a large volume of data, but GANs work better with less data. Another aspect to consider is the quality of the results: GANs are better at obtaining clear and detailed images, while VAE obtains more uniform results. This paper is focused on GANs. Thus, by using two antagonistic neural networks, ref. [ 1 ] proposes a tool called “Generative Adversarial Networks” (GANs), which are capable of creating faces of people that are not real, and it is difficult to differentiate these from real faces. While this is one of the most popular applications of GANs, their true importance lies in their capability to generate synthetic data that appear indistinguishable from real data. In this paper, a bibliographic review of GANs is conducted that focuses on contributions made in the areas of Business and Social Science. Presently, within the literature, various reviews on GANs in general can be found [ 2 – 7 ]. However, none of these specifically concentrates on the potential applications within the field of Business Economics. The objective is to showcase the potential of this tool within these two areas. The rest of this paper is structured as follows: Section 2briefly defines a GAN and how it functions. Section 3presents a statistical study of publications on GANs gathered from the Web of Science. In Section 4, the focus is placed on GAN publications within the scope of this work: Business Economics. Section 5details the relevance of this tool in the field of Business Economics. Section 6outlines the conclusions drawn. 2. Generative Adversarial Networks The main aim of GANs is the automatic generation of data. Its main difference from other generative models is that it does not directly use the distribution of real data; instead, it operates through a classifier. The generative model is random and adjusts itself step-by-step based on the classifier’s response, continually refining the output until the discriminator is unable to differentiate between real and synthetic data. In order to achieve this objective, two neural networks—namely, the generator and the discriminator—are employed, which engage in a competitive process. The generator produces synthetic data resembling real data, while the discriminator endeavours to distinguish between real data and the data supplied by the generator. This competition between the generator and discriminator can be viewed through game theory as a zero-sum game since the two networks have conflicting objectives; as one network improves, the other deteriorates. Consequently, the minimax algorithm can be employed, which is a decision-making algorithm that minimises the maximum loss in the game [ 8 ]. Thus, the training objective of GANs is to find the Nash equilibrium. Ref. [ 1 ] demonstrated that the Nash equilibrium is reached when the generator produces samples indistinguishable from the training dataset and the discriminator can only randomly guess whether a sample is real or fake, which means the generated samples are indistinguishable to the discriminator. When the Nash equilibrium is achieved, it is said that the network converges. In practice, GAN training is conducted using techniques that are not explicitly designed to find the Nash equilibrium, which may potentially lead to non-convergence [ 9 ]. At times, achieving convergence might necessitate the imposition of stringent conditions, which can be challenging to meet in practice [10–12]. The goal of the generator is to fool the discriminator, and the goal of the discriminator is to not be fooled. This confrontation leads to the generator being increasingly capable of providing synthetic data that is more similar to real data. The ideal solution in the GAN model is that the percentage of success of the discriminator for the real data and synthetic data is 50% for both cases. The structure of a GAN model for a database can be observed in Figure 1. Appl. Sci. 2024,14, 7438 3 of 23 Figure 1. Structure of a generative adversarial network model [13]. 2.1. Mathematical Framework Let us give a technical introduction of the GAN model: Given a database with m real samples ( x ) (training data) and a random noise vector ( z ), the following terms are considered: •G(z)is the output of the generator from the noise z: that is, it is the synthetic data. •D(x)is the output of the discriminator when a real sample xis processed. •D(G(z)) is the prediction of the discriminator on the synthetic data. •Pxand Pzare the distributions of real and noise data, respectively. •Ex and EG(z) are the expected log likelihoods from the different outputs of real and generated data, respectively. •θDand θGare the weights of the discriminator and generator model, respectively. The expression to be considered for the complete network comprising the discriminator and generator is denoted by Vand is as follows: V(θD,θG) = Ex∼Px[log D(x)] + Ez∼Pz[log(1−D(G(z)))]. (1) This value function is submitted to a min–max strategy with the goal of maximising the discriminator loss and minimising the generator loss: min θGmax θDV(θD,θG). (2) Furthermore, a suitable setup for each artificial neural network employed and its training is required for the implementation of a GAN. 2.2. Variants of GANs The training of GANs presents some major challenges: for example, mode collapse [ 14 ], non-convergence [ 11 , 15 , 16 ], and instability [ 17 ], among others [ 18 , 19 ]. Furthermore, a GAN can be structured to address a specific problem. Thus, some variants of GANs have been proposed in order to overcome these challenges by redesigning the generator and discriminator in the network architecture, changing the form of loss functions, and altering optimization algorithms. Hence, the following variants of GANs are worthy of note: Conditional GAN (CGAN): This is similar to the classic GAN but allows for the generation of data from a specific class defined within the real dataset, such as the generation of MNIST digits conditioned on class labels [20]. Vanilla GAN: This is a simple type of GAN where the discriminator and the generator are simpler, multilayer perceptrons [1]. Appl. Sci. 2024,14, 7438 4 of 23 Deeper convolutional GAN (DCGAN): This is a GAN whereby the architecture of both the generator and discriminator is composed of ConvNets instead of multilayer perceptrons [21–23]. Laplacian pyramid GAN (LAPGAN): This combines the CGAN model with a Laplacian pyramid representation [24]. Other known networks include: SRGAN, which employs a deep neural network [ 25 ]; StackGAN [ 26 , 27 ]; CycleGAN, as proposed in [ 28 ]; PassGAN [ 29 ]; WGAN, as seen in [ 30 ]; spatio–temporal GAN, as studied in [ 31 ]; constrained GAN, which can be seen in [ 32 ]; H-GAN [ 33 ]; pix2pix, as proposed in [ 34 ]; Android-GAN [ 35 ]; UNIT, which was developed in [ 36 ]; RGAN and RaGAN, as introduced in [ 37 ]; and AnoGAN, as proposed in [ 38 ]. Ref. [ 39 ] introduces GANomaly and RCGAN, which are showcased in [ 40 ], and EGAN [ 41 ] and TimeGAN are seen in [42], among others. Another approach to addressing the limitations of GANs is through the proposal of hybrid models that combine GANs with other deep learning models such as transformers [ 43 – 45 ], diffusion models [ 46 – 48 ], and large language models [ 49 , 50 ]. This line of research on the hybridisation of GANs with other architectures represents a promising field of investigation. Despite the continuous advancements and rapid growth that GANs have experienced since their inception, several challenges remain to be addressed. These include the consistency, the convergence rate, improving training stability and diversity, enhancing resources and real-time scalability, and the need to establish robust ethical regulations. 2.3. Applications of GANs GANs have been employed to address various types of problems: • Generating synthetic data for training: In [ 13 , 51 ], GANs are employed to generate data related to lung cancer patients, and statistical tests are employed to validate such synthetic data. For the generation of time-series data, one can employ SeqGAN [ 52 ]. It is also possible to generate synthetic data in tabular datasets using CTGAN [53]. • Generating text and natural language: There are several notable examples: Seq- GAN [ 52 ], which generates text sequences; LeakGAN [ 54 ], which introduces a search policy to enhance text generation; and RankGAN [ 55 ]. Others, like textGAN [ 56 ], are used for natural language generation. CTRL, proposed by [ 57 ], facilitates controlled text generation and enables users to specify style and content. Additionally, GPT-3, one of the most influential studies in text generation using GAN, is based on a transformer architecture that produces high-level, coherent, and natural text [58]. • Generating realistic images: One of the initial applications of GANs for generating realistic images was introduced by [ 21 ], where the authors proposed a DCGAN. Subsequently, other authors suggested different GAN modalities for distinct objectives. For instance, [ 28 ] applied CycleGAN to generate images by learning domain correspondence without the need for labelled data pairs. Another example is given by [ 26 ], which used StackGAN to produce detailed, high-resolution images by introducing a cascaded generator architecture. Other GAN variants, such as those employed for generating human faces [ 59 ], include StyleGAN [ 60 – 62 ] and BigGAN [ 63 ]. These models are utilised to enhance the quality and resolution of generated images and employ a scalable architecture. Other DCGAN variants are employed for the generation of human poses from a photograph [ 64 ] and can even project the age progression of an individual [65]. • Enhancement and restoration of damaged or low-quality images: To address this issue, several alternatives to the classic GAN have been proposed, such as SRGAN [ 25 ], Pix2Pix [ 34 ], and CycleGAN [ 28 ]. The latter addresses inpainting tasks (by filling in missing or damaged regions of an image) using DeepFill [ 66 , 67 ]. To restore images that are corrupted by noise, RedNet was designed [ 68 ], and for noise filtering, nCNN and FFDNet were created [69,70]. Appl. Sci. 2024,14, 7438 5 of 23 • Speech synthesis: This constitutes one of the earliest application fields of GANs. For instance, WaveGAN was designed for waveform-based voice synthesis [ 71 ]. Similarly, MelGAN was developed for voice synthesis using high-quality Mel spectrograms, and it provided enhanced quality and naturalness to synthesised voices [ 72 ]. To expedite real-time voice waveform generation, ParallelWaveGAN was proposed [ 73 ]. Other generative models include MelGAN-VC, which enables conversation [ 74 ], and HiFi-GAN for high-fidelity voice synthesis, which provides greater quality and detail [75,76]. GANs have also been used for the creation of music and songs by blending the different styles and compositions of recognised musicians and composers [ 77 ]. However, new approaches to apply GANs are constantly emerging, e.g., in anomaly detection [ 78 ], data imbalance problems [ 79 ], probabilistic forecasting of financial time series [45], stock price prediction [80], etc. 3. Statistics on GAN Publications In this section, the evolution of scientific articles related to GANs is presented. To this end, a search was conducted on 1 July 2023 using the Web of Science (WOS). In the search field, the keyword used was “Generative Adversarial” or “generative adversarial” or “GENERATIVE ADVERSARIAL”. References to countries and research areas were extracted exactly as they appear in the WOS. The conducted search yielded a total of 18619 documents, categorised according to the type of publication, as shown in Table 1. Table 1. Frequency per type of document. Document Type # Document Type # Document Type # Articles 11,168 Meeting Abstracts 131 Letters 5 Proceedings 7027 Editorial Materials 21 Retracted Publications 3 Early Access 503 Corrections 17 Data Papers 2 Review Articles 335 Book Chapters 12 — In this table, we can observe that 60% of GAN publications are in the form of papers, and 38% are papers of proceedings. Furthermore, in Table 2, the countries with the highest number of publications on this topic are displayed (“Others” includes all countries with a percentage lower than 1%). Table 2. Publications per country. Country # % Country # % China 9123 48.998 USA 3837 20.608 South Korea 1165 6.259 England 958 5.145 India 920 4.941 Japan 835 4.485 Germany 704 3.781 Australia 670 3.563 Canada 630 3.384 Taiwan 415 2.229 France 398 2.138 Singapore 369 1.982 Italy 350 1.880 Spain 296 1.570 Switzerland 251 1.348 Saudi Arabia 203 1.090 Others (103 countries) 3267 17.546 — — — In this table, it can be observed that nearly half of all publications related to GANs originate from China, which showcases the country’s positioning in emerging technologies. With just over 20%, the USA ranks second and is significantly ahead of other countries. Appl. Sci. 2024,14, 7438 6 of 23 3.1. Evolution of GANs The evolution of GANs over time is depicted in Figure 2. In this figure, an estimation of publications for the year 2023 is made based on a linear fit derived from the information collected in the table for previous years. The years 2014 and 2015 were not considered for the fit since the concept of GANs was still emerging and due to the lack of data to accurately model the real growth (on 1 July 2023, there were 1825 publications in the preceding 12 months). Figure 2. Number of articles published per year. It can be observed in this figure how the growth experienced by the papers addressing GANs follows an increasing trend that has not yet reached its peak since there are still fields, such as Business and Social Sciences, where the potential provided by this GAN tool to the scientific world remains to be fully explored. 3.2. Knowledge Areas and GANs Table 3has been obtained by taking into account the research areas collected by the WOS and reveals the most relevant research areas (areas that comprise at least 1% of all publications). Regarding the areas that feature a significant number of scientific publications for these GANs, the field of Computer Science stands out and encompasses a wide variety of applications in image processing, deep learning, and synthetic content generation [21,28,34,63,81]. The field of Engineering presents multiple applications, ranging from the optimisation of renewable energies to manufacturing and monitoring of structural health, which showcases its versatility and utility in solving real-world problems [ 82 – 85 ]. Of particular interest is the application to cognitive cars [86–88] and electronic home appliances [89]. Another field where numerous applications are found is in Imaging Science and Photographic Technology. This field is undergoing continuous evolution, for which GANs are contributing significant advancements [ 25 , 90 – 95 ]. It is a major challenge to enumerate the multitude of applications and problems addressed in this field that have been successfully resolved using GANs. Work in Telecommunications also stands out [96–101]. In the field of Radiology, Nuclear Medicine and Medical Imaging, GANs appear particularly relevant regarding the processing and analysis of medical images by improving their quality and usefulness, aiding in the diagnosis and treatment of diseases, and facilitating advancements in research [102–107]. Appl. Sci. 2024,14, 7438 7 of 23 Table 3. Publications according to research areas. Research Area # % Computer Science 10,974 58.940 Engineering 8418 45.212 Imaging Science/Photographic Technology 2368 12.718 Telecommunications 1983 10.650 Radiology/Nuclear Medicine/Medical Imaging 1182 6.348 Remote Sensing 897 4.818 Optics 868 4.662 Physics 854 4.587 Chemistry 667 3.582 Instruments/Instrumentation 674 3.620 Automation/Control Systems 631 3.389 Materials Science 507 2.723 Geology 444 2.385 Mathematics 404 2.170 Science/Technology/Other Topics 404 2.170 Mathematical/Computational Biology 389 2.089 Environmental Sciences/Ecology 361 1.939 Acoustics 351 1.885 Neurosciences/Neurology 329 1.767 Geochemistry/Geophysics 336 1.805 Robotics 283 1.520 Medical Informatics 245 1.316 Transportation 235 1.262 Energy Fuels 212 1.139 Operations Research/Management Science 189 1.015 Others (90 areas) 2283 12.261 Of great significance is the field of Remote Sensing, where GANs have contributed towards enhancing the quality and quantity of data, thereby enabling the interpretation and analysis of images, and introducing new applications and techniques for monitoring and understanding our planet from space [108–112]. GANs have overcome limitations in the field of Optics related to resolution and quality in the reconstruction of diffracted images, thereby enhancing the capacity for observation and analysis [113–115]. In the field of Physics, GANs have contributed towards significant advances in simulation techniques, data analysis, and modelling across various domains, including computational physics, particle physics, and quantum physics [116–121]. In Chemistry, GANs have contributed towards advances in molecular techniques, drug discovery, organic synthesis, and the prediction of chemical properties, thereby opening doors for chemical and pharmaceutical research [122–125]. In the field of Instruments and Instrumentation, one can find studies applying GANs in deepfake detection, synthetic data generation, instrument calibration, and improvements in resolution in microscopy techniques [126–129]. In Automatic Control Systems, GANs have contributed innovative solutions in areas such as anomaly detection, system modelling, and synthetic data generation for the design and evaluation of control algorithms [130–135]. In Materials Science, GANs provide innovative solutions in areas such as material discovery, molecular design, structure optimisation, and the generation of compounds with specific properties [136–139]. In Geology, several significant applications are related to facies classification techniques, geological pattern generation, image enhancement, and detection of geological features [140–143]. In Mathematics, innovative solutions have been achieved in areas such as robust optimisation, graph generation, geometric modelling, image compression, and solving Appl. Sci. 2024,14, 7438 8 of 23 differential equations, expanding the capabilities and applications of GANs in the mathematical domain [95,144–148]. In the field of Science, Technology and other Topics, advances have been made in image generation techniques, enhancements in the quality and stability of GANs, as well as in the manipulation and control of features in image synthesis. These advances have propelled the field of computer vision and machine learning [28,59,60,63,149]. In Mathematical/Computational Biology, advances have been made in unsupervised learning techniques, synthetic data generation, molecular modelling, and analysis of biological images, thereby opening new perspectives in mathematical and biological research [21,150,151]. The application of GANs in Environmental Sciences and Ecology has developed innovative solutions for species distribution modelling, land cover prediction, environmental data analysis, and water quality monitoring, thereby enhancing our understanding and management of ecosystems and the environment [152,153]. Also, in the field of Acoustics, GANs have contributed significant solutions in areas such as audio synthesis, speech enhancement, sound simulation, and speaker verification, thereby enhancing the quality and robustness of acoustic signal processing [71,154–158]. In the field of Neurosciences and Neurology, GANs have achieved advancements in diagnostic techniques, synthetic data generation, analysis of brain signals, and the study of neurological diseases, thereby providing new insights and tools for the field of neuroscience and neurology [159–161]. When studying GANs in Geochemistry and Geophysics, significant advancements have been achieved in techniques for the estimation of petrophysical properties, seismic data inversion, interpretation of geophysical data, and modelling of subsurface structures. These advances enhance our understanding and analytical capabilities in exploring and characterising natural resources and geological processes [162–166]. In the field of Robotics, GANs have been applied in areas such as synthetic data generation, reinforcement learning enhancement, robotic perception, terrain classification, and robotic grasp improvement, thereby enhancing the capabilities and applications of robots in various environments and tasks [167–171]. In the field of Medical Informatics, studies on GANs have developed innovative solutions in areas such as the generation of synthetic medical images, anomaly detection, incomplete image reconstruction, and data synthesis to enhance the analysis and interpretation of medical images, thereby significantly impacting disease diagnosis and treatment [38,172–175]. In the field of Transportation, advances have been made in areas such as traffic flow prediction, traffic simulation, traffic sign translation, vehicle re-identification, and estimation of public transport demand, thereby enhancing the efficiency, safety, and sustainability of transportation systems [176–180]. In the field of Energy Fuels, GANs have been applied in energy forecasting techniques, optimisation of energy systems, design of renewable energy systems, and assessment of climate-change mitigation strategies, thereby enhancing our ability to address current energy and environmental challenges [181–184]. In Operations Research and Management Science, GAN applications encompass fields such as synthetic data generation, revenue management, fraud detection, supply-chain optimisation, and dynamic pricing. These applications improve decision-making and efficiency in operations and management processes [185–188]. However, there are still significant fields where the development of such models can offer major advances and advantages that are still emerging and hold great potential in their research. One of these fields is Business Economics, where GANs can provide solutions to major problems encountered in the economic world when conducting various studies of interest. This topic is addressed in the following section. Appl. Sci. 2024,14, 7438 9 of 23 4. Statistics on GAN Publications in Economics In the field of Economics, GANs constitute an emerging and highly promising tool, among other reasons due to their ability to generate synthetic data that can help to improve economic research and analysis. Researchers often face limitations in data availability, especially when dealing with economic data. It is worth noting that having access to synthetic data in these scenarios would help simulate and predict complex economic behaviours, thereby enabling the evaluation of various economic strategies and policies well in advance to facilitate appropriate decision-making tailored to each situation. In order to highlight these advances in the economic field and to reveal the extensive potential this tool holds, we filter and analyse the selected bibliography by limiting it to research directly related to this field. Hence, the number of obtained papers is reduced to a total of 42 by the WOS in the the research areas considered: Business Economics (29), Social Issues (3), Mathematics Methods in Social Science (7), and Social Sciences Other Topics (9). Out of these 42 publications, we select the publications that are categorised as research papers, narrowing down the number to 31. In order to analyse and present these research contributions more clearly, these papers are grouped into five specific fields identified by the researchers: Financial Economics (9), Management (4), Marketing and Publicity (8), Logistics Transport (5), and Others (5). These areas are analysed in the following sections. 4.1. Financial Economics The publications on GANs in the field of Financial Economics are displayed in Table 4. The first paper [ 189 ] proposes an innovative optimisation framework for the portfolio optimisation problem by leveraging GANs to develop a heuristic approach to the classic Markowitz model. This involves three steps: (i) using a GAN to select the initial set of assets for investment, (ii) solving the portfolio optimisation problem to determine the weights of the assets from the previous step, and (iii) enhancing the obtained solutions. In the year 2020, [ 190 ], proposed the use of CGAN for the calibration and aggregation of trading strategies. They designed an experiment involving multiple trading strategies across 579 assets and yielded successful results and clear advantages over other tools. In [ 191 ], a data-driven approach is proposed to calibrate local stochastic volatility models using GANs, and it demonstrates its feasibility and showcases several significant advantages over other methods. An approach using a GAN-based model, referred to as Quant GAN, is introduced in [ 192 ] for modelling discrete-time financial time series. Although the models of time series have historically been notably challenging to train, advances in GANs showcase their potential to provide competitive outcomes. Thus, as training procedures continue to evolve, GANs hold the promise of delivering even better performance in the future. In [ 193 ], a GAN model named WCGAN-GP is developed to address the challenges existing in actuarial domains, since actuaries often require a great amount of data. Furthermore, the datasets employed by actuaries frequently encounter issues with imbalanced classes, wherein events of interest related to risk are underrepresented. The authors provide several contributions to actuaries, with applications including the generation of new samples, data augmentation, enhancement of predictive models, anomaly detection, and imputation of missing data. Another paper from 2022 in the financial domain is [ 194 ], which addresses the problem of maximising utility under uncertain parameters through a robust optimisation process that can be interpreted as a stochastic differential game for two players with a zero-sum. This algorithm is tested in real markets and shows that robust portfolios tend to have higher expected utility and are more stable during market downturns. To solve the value function, the authors derive the analytical solution in the case of logarithmic utility and obtain numerical approximations using various methods, including GANs, and demonstrate a high level of accuracy in the numerical results. Appl. Sci. 2024,14, 7438 16 of 23 17. Zhang, Z.; Luo, C.; Yu, J. Towards the gradient vanishing, divergence mismatching and mode collapse of generative adversarial nets. In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, Beijing, China, 3–7 November 2019; pp. 2377–2380. 18. 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