Fluid mechanics often involves complex systems characterized by a large number of physical parameters, which are usually described by experimental and numerical sparse data (temporal or spatial). The difficulty of obtaining complete spatio-temporal datasets is a common issue with conventional approaches, such as computational fluid dynamics (CFDs) and various experimental methods, particularly when evaluating and modeling turbulent flows. This review paper focuses on the integration of machine learning (ML), specifically physics-informed neural networks (PINNs), as a means to address this challenge. By directly incorporating governing physical equations into neural network training, PINNs present a novel method that allows for the reconstruction of flow from sparse and noisy data. This review examines various applications in fluid mechanics where sparse data is a common problem and evaluates the effectiveness of PINNs in enhancing flow prediction accuracy. An overview of diverse PINNs methods, their applications, and outcomes is discussed, demonstrating their flexibility and effectiveness in addressing challenges related to sparse data and illustrating that the future of fluid mechanics lies in the synergy between data-driven approaches and established physical theories.

Established reduced‐order modeling (ROM) methods, for instance, Galerkin‐projection, approximate the solution by linearly projecting high‐dimensional spaces to a lower‐dimensional space spanned by the reduced basis. However, the accuracy of these methods may be insufficient for complex and multiscale simulations due to the restriction to a linear space. Alternatively, autoencoders (AEs) can be used for nonlinear dimensionality reduction. We combine nonlinear dimensionality reduction techniques with time series prediction to build data‐driven ROMs. The presented framework consists of two‐level neural networks. The high‐dimensional space is nonlinearly compressed in the first level using the encoder function of the AE. Subsequently, the temporal evolution of the latent vector is predicted in the second level. The original solution can be easily reconstructed using the decoder function of the AE. In comparison with the projection‐based ROMs, for example, proper orthogonal decomposition (POD) with Galerkin projection, this framework allows naturally to include parameters in the prediction without nonlinear interpolation of the linear basis. We demonstrate the framework on a two‐dimensional flow field simulation around circular bodies parameterized with the inlet fluid velocity.

Modeling the complex flow behavior of impingement jets is a problem of great importance in many industrial applications. Traditional modeling methods often fail to accurately predict these flows due to their nonlinear nature. This paper presents a neural network-based reduced-order model for experimental data of a circular impinging jet and compares several data assimilation frameworks for incorporating wall shear stress measurements obtained from different radial positions. The high-dimensional velocity field and the corresponding wall shear stress measurements are obtained using time-resolved particle image velocimetry and polarographic measurements, respectively. The developed reduced-order model results from a proper orthogonal decomposition (POD) step for dimensionality reduction with a neural ordinary differential equation (NODE) for temporal modeling. The performance of the POD-NODE framework is compared with dynamic mode decomposition and nonlinear temporal modeling using long short-term memory. Assessments are based on root mean squared error and spectral proper orthogonal decomposition of the reconstructed predicted solution. It is found that the POD-NODE framework provides the most accurate dynamical model. Furthermore, it is evident that incorporating wall shear stress measurements in the NODE model as additional states significantly improves the prediction accuracy, outperforming traditional filtering techniques such as extended Kalman filters.