Collaborative Research: Data-Driven Variational Multiscale Reduced Order Models for Biomedical and Engineering Applications

数学模型是提高我们对自然和工业过程知识的基本工具。它们在实践中的使用取决于它们的可靠性和效率。可靠性要求对模型参数进行微调，并准确评估对噪声输入的敏感性。在优化问题中，效率是特别重要的，其中计算过程确定了一个复杂系统的最佳工作条件。这些需求导致求解具有数百万甚至数十亿未知数的多次模型。此过程可能需要在高性能计算设备上进行数天或数周的计算。为了降低这些成本，我们需要新的建模策略，允许模型在几分钟到几小时内在本地计算设备（比如笔记本电脑）上运行。降阶模型（ROMs）是一种极低维的近似，可以将当前计算模型的计算成本降低几个数量级。考虑到生物医学和风力工程的应用，这个项目提出了新的模型简化方法。来自昂贵（或高保真）模型的数据和数值结果与机器学习方法相结合，以获得在前所未有的水平上实现效率和准确性的rom。新的数据驱动ROM框架最终将在几分钟内在笔记本电脑上实现主动脉夹层、儿科手术或风力发电场优化的数值模拟，并旨在成为决策过程中关键且值得信赖的工具。数据同化（DA）、不确定性量化（UQ）和形状优化（SO）是重要生物医学和工程应用计算模型发展的核心。由于这些应用程序需要大量的模型模拟，因此运行昂贵的全订单模型（FOM）通常非常昂贵。对于显示主导结构的系统，降阶模型（ROMs）可以将FOM计算成本降低几个数量级。因此，对于上述临床和工程应用程序，rom似乎是在高性能计算设施上运行昂贵的fom的自然和实用的替代方案。不幸的是，为了捕捉主动脉夹层血流动力学中的所有几何尺度，或者为了应对风电场优化中的大雷诺数，需要成百上千的ROM模式。这些相对高维的rom仍然无法有效地为这些应用程序执行DA、UQ或SO。我们需要的是不仅低维、高效，而且精确的rom。为了开发在现实的、未解决的制度下准确的ROM，需要解决ROM关闭问题，即需要对丢弃的ROM模式对ROM动力学的影响进行建模。该研究提出了一种新的数据驱动的ROM范式，该范式以变分多尺度（VMS）方法的层次结构为中心，利用现代机器学习（ML）和数值和观测数据来开发结构ROM闭包，可以在适度的计算成本下显着提高ROM精度。新的数据驱动的VMS-ROM模式保持了当前ROM的低计算成本，但显著提高了ROM的精度。生物医学应用在胸外科和儿科手术（主动脉解剖和Fontan手术——病人的命运很大程度上取决于血管的形状）以及风力工程应用中是特别针对的。数据驱动的VMS-ROM框架最终将使这些以及可能依赖于数学和计算建模的其他领域的高效DA、UQ和SO成为可能。该项目每年将资助三所院校各一名研究生。该奖项反映了美国国家科学基金会的法定使命，并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估，被认为值得支持。

Mathematical models are a fundamental tool for improving our knowledge of natural and industrial processes. Their use in practice depends on their reliability and efficiency. Reliability requires a fine-tuning of the model parameters and an accurate assessment of the sensitivity to noisy inputs. Efficiency is particularly critical in optimization problems, where the computational procedure identifies the best working conditions of a complex system. These requirements lead to solving many times models with millions or even billions of unknowns. This process may require days or weeks of computations on high-performance computing facilities. To mitigate these costs, we need new modeling strategies that allow model-runs in minutes to hours on local computing facilities (such as a laptop). Reduced order models (ROMs) are extremely low-dimensional approximations that can decrease the computational cost of current computational models by orders of magnitude. Having in mind biomedical and wind-engineering applications, this project proposes novel methods of model reduction. Data and numerical results from the expensive (or high-fidelity) models are combined with machine learning approaches, to obtain ROMs that attain both efficiency and accuracy at an unprecedented level. The new data-driven ROM framework will finally make possible the numerical simulation of aortic dissections, pediatric surgery, or wind farm optimization on a laptop in minutes, and aims at becoming a critical and trustworthy tool in decision-making processes.Data assimilation (DA), uncertainty quantification (UQ), and shape optimization (SO) are central to the development of computational models for significant biomedical and engineering applications. Since these applications require a large number of model simulations, running an expensive full order model (FOM) is generally prohibitively expensive. For systems that display dominant structures, reduced order models (ROMs) can decrease the FOM computational cost by orders of magnitude. Thus, for the clinical and engineering applications above, ROMs appear as a natural and practical alternative to the prohibitively expensive FOMs running on high-performance computing facilities. Unfortunately, to capture all the geometric scales in the hemodynamics of aortic dissections or to cope with the large Reynolds number in the wind farm optimization, hundreds and thousands of ROM modes are necessary. These relatively high-dimensional ROMs are still not viable to effectively perform DA, UQ, or SO for these applications. What is needed is ROMs that are not only low-dimensional and efficient, but also accurate. To develop ROMs that are accurate in realistic, under-resolved regimes, the ROM closure problem needs to be solved, i.e., the effect of the discarded ROM modes on the ROM dynamics needs to be modeled. The proposed research puts forth a new data-driven ROM paradigm that centers around the hierarchical structure of variational multiscale (VMS) methodology and utilizes modern machine learning (ML) and numerical and observational data to develop structural ROM closures that can dramatically increase the ROM accuracy at a modest computational cost. The novel data-driven VMS-ROM paradigm maintains the low computational cost of current ROMs but dramatically increases the ROM accuracy. Biomedical applications in thoracic and pediatric surgery (aortic dissections and Fontan procedure – where the fate of the patient depends significantly on the shape of the vessels) as well as wind-engineering applications are specifically targeted. The data-driven VMS-ROM framework will finally make possible the efficient DA, UQ, and SO in these and, possibly, other fields relying on mathematical and computational modeling. This project will support one graduate student each year at each of the three institutions.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.