CDS&E: Data-driven Discovery of Probabilistic Closures in Turbulent Flows

湍流的可靠预测模型一直是经典物理学面临的重大挑战之一。这是因为湍流是多尺度问题。使用湍流的直接数值模拟来捕获所有这些尺度对于大多数科学和工程应用来说仍然是成本高昂的。因此，在实际湍流的预测模型中，小尺度是不能分辨的，因此必须模拟这些未分辨尺度对分辨尺度的影响。然而，除了非常简单的规范流，这些模型的函数形式是未知的。这是湍流预测中不确定性的主要来源。随着可扩展的科学机器学习算法的最新发展以及高性能计算资源的增长，现在可以生成高保真数据并使用大量参数训练机器学习模型。这为发现新的湍流模型以解决某些实际工程问题提供了新的机会，并显著提高了湍流预测的可靠性。众所周知，发现湍流模型的最准确框架之一是概率框架。然而，概率模型并不像确定性模型那样常用，主要是由于它们的计算成本。这些模型是相对于概率密度函数（PDF）的演变来表达的，这可能是非常高维的。概率湍流模型的发现需要求解正向和逆PDF输运方程。经典的科学计算技术由于其高维性而无法解决这一问题。在该项目中，将开发一种新的物理学深度学习方法，并用于求解正向和反向PDF传输方程。如果成功的话，这个项目将导致不同湍流的概率湍流模型的发现。沿着技术工具的开发，这项工作还将包括与教育相结合，组织新的会议，与工业和政府研究实验室互动，改善性别、民族和种族多样性，通过K-12.这一奖项反映了国家科学基金会的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评价来提供支持。

Reliable predictive modeling of turbulent flows has remained one of the grand challenges of classical physics. This is because turbulent flows are multi-scale problems. Capturing all these scales using direct numerical simulations of turbulent flows remains cost prohibitive for the majority of the scientific and engineering applications. Therefore, in the predictive models for practical turbulent flows the small scales cannot be resolved and as a result the effect of these unresolved scales on the resolved scales must be modeled. However, except for very simple canonical flows, the functional form of these models are not known. This is the main source of uncertainty in turbulent flow predictions. With the recent development of scalable scientific machine learning algorithms as well as the growth of high performance computing resources, it is now possible to generate high-fidelity data and train machine learning models with a very large number of parameters. This opens up new opportunities to discover new turbulence models for some of the practical engineering problems and significantly improve the reliability of predictions in turbulent flows.It is widely known that one of the most accurate frameworks to discover turbulent models is a probabilistic one. However, probabilistic models are not as commonly used as the deterministic models mainly due to their computational costs. These models are expressed versus the evolution of probability density functions (PDFs), which can be very high-dimensional. Discovery of probabilistic turbulence models requires solving both the forward and inverse PDF transport equations. Classical scientific computing techniques are unable to solve this problem due to the their high dimensionality. In the project, a new physics-informed deep learning methodology will be developed and utilized to solve both forward and inverse PDF transport equations. If successful, this project will lead to the discovery of probabilistic turbulence models for diverse turbulent flows. Along with the development of technical tools, this work will also include integration with education, organization of new conferences, interaction with industry and governmental research labs, improvement of gender, ethnic and racial diversity, and expansion through K-12 outreach.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.