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度。效率在优化问题中尤为关键，在优化问题中，计算过程确定复杂系统的最佳工作条件。这些要求导致了多次求解具有数百万甚至数十亿未知数的模型。这一过程可能需要在高性能计算设施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，允许在本地计算设施上在几分钟到几个小时内运行模型。降阶模型是一种低维近似，可以将当前计算模型的计算代价降低几个数量级。考虑到生物医学和风能工程的应用，该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，以获得达到前所未有的效率和精度的ROM。新的数据驱动的只读存储器框架将使在笔记本电脑上进行主动脉夹层、儿科手术或风电场优化的数值模拟在几分钟内成为可能，并旨在成为决策过程中的关键和值得信赖的工具。数据同化(DA)、不确定性量化(UQ)和形状优化(SO)是为重要的生物医学和工程应用开发计算模型的核心。由于这些应用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(ROM)可以将FOM的计算成本降低数量级。因此，对于上述临床和工程应用，相对于在高性能计算设施上运行的昂贵的FOM，只读存储器似乎是一种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，或者为了处理风电场优化中的大雷诺数，成百上千的只读模式是必要的。对于这些应用，这些相对高维的ROM仍然不能有效地执行DA、UQ等。需要的是不仅低维、高效，而且准确的只读存储器。为了开发出在现实的、未得到充分解决的状态下精确的只读存储器，需要解决只读存储器的闭合问题，即需要对被丢弃的只读存储器模式对只读存储器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只读存储器范式，该范例以变分多尺度(VMS)方法的层次结构为中心，利用现代机器学习以及数值和观测数据来开发结构化的只读存储器闭包，该闭包可以以适度的计算代价显著地提高只读存储器的精度。这种新颖的数据驱动的VMS-ROM范例保持了当前只读存储器的低计算成本，但显著提高了只读存储器的精度。生物医学在胸外科和儿科手术中的应用(主动脉夹层和Fontan手术--患者的命运在很大程度上取决于血管的形状)以及风力工程应用是特别有针对性的。数据驱动的VMS-ROM框架最终将使这些领域以及可能依赖于数学和计算建模的其他领域的高效DA、UQ等成为可能。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 models with millions or even billions of unknowns many times. 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. Reduced order models (ROMs) are 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 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. This project will support one graduate student each year at each of the three institutions.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, that is, 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 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 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.