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This study explores the potential of physics-informed neural networks (PINNs) for the realization of digital twins (DT) from various perspectives. First, various adaptive sampling approaches for collocation points are investigated to verify their effectiveness in the mesh-free framework of PINNs, which allows automated construction of virtual representation without manual mesh generation. Then, the overall performance of the data-driven PINNs (DD-PINNs) framework is examined, which can utilize the acquired datasets in DT scenarios. Its scalability to more general physics is validated within parametric Navier-Stokes equations, where PINNs do not need to be retrained as the Reynolds number varies. In addition, since datasets can be often collected from different fidelity/sparsity in practice, multi-fidelity DD-PINNs are also proposed and evaluated. They show remarkable prediction performance even in the extrapolation tasks, with $42\sim62\%$ improvement over the single-fidelity approach. Finally, the uncertainty quantification performance of multi-fidelity DD-PINNs is investigated by the ensemble method to verify their potential in DT, where an accurate measure of predictive uncertainty is critical. The DD-PINN frameworks explored in this study are found to be more suitable for DT scenarios than traditional PINNs from the above perspectives, bringing engineers one step closer to seamless DT realization.

Airfoil shape optimization plays a critical role in the design of high-performance aircraft. However, the high-dimensional nature of airfoil representation causes the challenging problem known as the "curse of dimensionality". To overcome this problem, numerous airfoil parameterization methods have been developed, which can be broadly classified as polynomial-based and data-driven approaches. Each of these methods has desirable characteristics such as flexibility, parsimony, feasibility, and intuitiveness, but a single approach that encompasses all of these attributes has yet to be found. For example, polynomial-based methods struggle to balance parsimony and flexibility, while data-driven methods lack in feasibility and intuitiveness. In recent years, generative models, such as generative adversarial networks and variational autoencoders, have shown promising potential in airfoil parameterization. However, these models still face challenges related to intuitiveness due to their black-box nature. To address this issue, we developed a novel airfoil parameterization method using physics-aware variational autoencoder. The proposed method not only explicitly separates the generation of thickness and camber distributions to produce smooth and non-intersecting airfoils, thereby improving feasibility, but it also directly aligns its latent dimensions with geometric features of the airfoil, significantly enhancing intuitiveness. Finally, extensive comparative studies were performed to demonstrate the effectiveness of our approach., Comment: 33 pages, 19 figures, preprint for journal submission

This study aims to comprehensively investigate the deep ensemble approach, an approximate Bayesian inference, in the multi-output regression task for predicting the aerodynamic performance of a missile configuration. To this end, the effect of the number of neural networks used in the ensemble, which has been blindly adopted in previous studies, is scrutinized. As a result, an obvious trend towards underestimation of uncertainty as it increases is observed for the first time, and in this context, we propose the deep ensemble framework that applies the post-hoc calibration method to improve its uncertainty quantification performance. It is compared with Gaussian process regression and is shown to have superior performance in terms of regression accuracy ($\uparrow55\sim56\%$), reliability of estimated uncertainty ($\uparrow38\sim77\%$), and training efficiency ($\uparrow78\%$). Finally, the potential impact of the suggested framework on the Bayesian optimization is briefly examined, indicating that deep ensemble without calibration may lead to unintended exploratory behavior. This UQ framework can be seamlessly applied and extended to any regression task, as no special assumptions have been made for the specific problem used in this study.

Autoencoder-based reduced-order modeling (ROM) has recently attracted significant attention, owing to its ability to capture underlying nonlinear features. However, two critical drawbacks severely undermine its scalability to various physical applications: entangled and therefore uninterpretable latent variables (LVs) and the blindfold determination of latent space dimension. In this regard, this study proposes the physics-aware ROM using only interpretable and information-intensive LVs extracted by $\beta$-variational autoencoder, which are referred to as physics-aware LVs throughout this paper. To extract these LVs, their independence and information intensity are quantitatively scrutinized in a two-dimensional transonic flow benchmark problem. Then, the physical meanings of the physics-aware LVs are thoroughly investigated and we confirmed that with appropriate hyperparameter $\beta$, they actually correspond to the generating factors of the training dataset, Mach number and angle of attack. To the best of the authors' knowledge, our work is the first to practically confirm that $\beta$-variational autoencoder can automatically extract the physical generating factors in the field of applied physics. Finally, physics-aware ROM, which utilizes only physics-aware LVs, is compared with conventional ROMs, and its validity and efficiency are successfully verified.

The inverse approach is computationally efficient in aerodynamic design as the desired target performance distribution is prespecified. However, it has some significant limitations that prevent it from achieving full efficiency. First, the iterative procedure should be repeated whenever the specified target distribution changes. Target distribution optimization can be performed to clarify the ambiguity in specifying this distribution, but several additional problems arise in this process such as loss of the representation capacity due to parameterization of the distribution, excessive constraints for a realistic distribution, inaccuracy of quantities of interest due to theoretical/empirical predictions, and the impossibility of explicitly imposing geometric constraints. To deal with these issues, a novel inverse design optimization framework with a two-step deep learning approach is proposed. A variational autoencoder and multi-layer perceptron are used to generate a realistic target distribution and predict the quantities of interest and shape parameters from the generated distribution, respectively. Then, target distribution optimization is performed as the inverse design optimization. The proposed framework applies active learning and transfer learning techniques to improve accuracy and efficiency. Finally, the framework is validated through aerodynamic shape optimizations of the wind turbine airfoil. Their results show that this framework is accurate, efficient, and flexible to be applied to other inverse design engineering applications.