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.
