The article "Turbulence Modeling in the Age of Data" by Karthik Duraisamy, Gianluca Iaccarino, and Heng Xiao reviews recent advancements in turbulence modeling, particularly focusing on the use of data-driven approaches to improve the accuracy and reliability of turbulence models. The authors highlight the historical reliance on experimental calibration and direct numerical simulations (DNS) to calibrate engineering models like those based on the Reynolds-averaged Navier-Stokes (RANS) equations. However, with the advent of large and diverse datasets, researchers are now exploring methods to systematically inform RANS models with data, aiming to quantify and reduce model uncertainties. The review covers key principles, achievements, and challenges in bounding uncertainties in RANS models using physical constraints, statistical inference to characterize model coefficients, and machine learning to enhance turbulence models. The authors advocate that by leveraging foundational knowledge in turbulence modeling and physical constraints, data-driven approaches can yield useful predictive models. The article also discusses the elements of a data-driven model, including the model itself, the data, the output, and the discrepancy, which describes the model's ability to represent the data. It explores the four layers of simplification required to formulate a RANS closure, from ensemble averaging to the selection of functional forms and the calibration of coefficients. Additionally, the review covers uncertainty quantification techniques, such as statistical inversion and the propagation of uncertainty through the model. It highlights methods for quantifying uncertainties in the Reynolds stress tensor and model parameters, including bounding approaches and probabilistic descriptions. The article further delves into predictive modeling using data-driven techniques, including the embedding of inference-based discrepancy and the generalization of these discrepancies. It discusses the use of machine learning, particularly neural networks, to construct mappings between large datasets and quantities of interest. The authors emphasize the importance of ensuring objectivity and rotational invariance in learned Reynolds stress models. Finally, the review explores the combination of statistical inference and machine learning to improve the predictive capabilities of turbulence models, addressing the challenges and potential of hybrid PDE/neural networks settings.The article "Turbulence Modeling in the Age of Data" by Karthik Duraisamy, Gianluca Iaccarino, and Heng Xiao reviews recent advancements in turbulence modeling, particularly focusing on the use of data-driven approaches to improve the accuracy and reliability of turbulence models. The authors highlight the historical reliance on experimental calibration and direct numerical simulations (DNS) to calibrate engineering models like those based on the Reynolds-averaged Navier-Stokes (RANS) equations. However, with the advent of large and diverse datasets, researchers are now exploring methods to systematically inform RANS models with data, aiming to quantify and reduce model uncertainties. The review covers key principles, achievements, and challenges in bounding uncertainties in RANS models using physical constraints, statistical inference to characterize model coefficients, and machine learning to enhance turbulence models. The authors advocate that by leveraging foundational knowledge in turbulence modeling and physical constraints, data-driven approaches can yield useful predictive models. The article also discusses the elements of a data-driven model, including the model itself, the data, the output, and the discrepancy, which describes the model's ability to represent the data. It explores the four layers of simplification required to formulate a RANS closure, from ensemble averaging to the selection of functional forms and the calibration of coefficients. Additionally, the review covers uncertainty quantification techniques, such as statistical inversion and the propagation of uncertainty through the model. It highlights methods for quantifying uncertainties in the Reynolds stress tensor and model parameters, including bounding approaches and probabilistic descriptions. The article further delves into predictive modeling using data-driven techniques, including the embedding of inference-based discrepancy and the generalization of these discrepancies. It discusses the use of machine learning, particularly neural networks, to construct mappings between large datasets and quantities of interest. The authors emphasize the importance of ensuring objectivity and rotational invariance in learned Reynolds stress models. Finally, the review explores the combination of statistical inference and machine learning to improve the predictive capabilities of turbulence models, addressing the challenges and potential of hybrid PDE/neural networks settings.