CDS&E: Collaborative Research: Surrogates and Reduced Order Modeling for High Dimensional Coupled Systems

复杂科学系统的数学建模在理解和预测这些系统方面发挥着越来越重要的作用。为了理解不同模型假设和参数值的影响，人们可能需要数百万次计算模拟来全面探测系统的行为。许多复杂系统不仅由一个计算模型或算法建模，而且还由需要彼此交互的子模型和代码集建模，这一事实使计算负担变得更加复杂。该项目旨在开发对这种耦合计算模型的高效和灵活的近似，以消除基于数学建模的科学发现过程的计算瓶颈。这项研究将集中在流体流动和机械变形的原型耦合模型上，该模型可以同时适用于模拟水力压裂和软骨生物力学。学生将参与并接受该项目跨学科方面的培训。对具有科学和实际意义的系统进行计算模拟通常需要耦合两个或多个物理现象的数学模型。这类模拟通常依赖于许多输入参数，而模型的验证受可用数据有限的限制。耦合数学模型的数值模拟器采用完全或松散耦合的方法。完全耦合涉及同时求解一组方程，但由于计算的复杂性，可行的运行时间通常需要简化的物理模型。相反，松散耦合将模拟不同物理过程的独立代码连接在一起；即使是松散耦合的模拟也常常无法运行执行不确定性量化所需的大量次数。该项目旨在开发高维耦合模拟器的高斯过程仿真方法，从而能够对具有挑战性的但普遍存在的多物理模型进行不确定性量化；该方法将涉及控制模型方程的代理或降阶模型，以允许在不同输入下快速逼近整个物理模拟器。这些努力将需要开发(I)降维的新想法，(Ii)模拟时空场的新方法，以及(Iii)因降维和模拟而引起的误差分析。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。

Increasingly, mathematical modeling of complex scientific systems plays a crucial role both in understanding and in making predictions about these systems. To understand the effects of different model assumptions and parameter values, one might need millions of computational simulations to fully probe a system's behavior. This computational burden is compounded by the fact that many complex systems are modeled not by just one computational model or algorithm, but rather by sets of sub-models and codes that need to interact with each other. This project aims to develop efficient and flexible approximations to such coupled computational models, to take the computational bottleneck out of the mathematical modeling-based scientific discovery process. The research will focus on a prototype coupled model of fluid flow and mechanical deformation that can be adapted to model both hydraulic fracturing and cartilage biomechanics. Students will be involved and trained in interdisciplinary aspects of the project. Computational simulation of systems of scientific and practical interest often requires coupling two or more mathematical models of physical phenomena. Such simulations typically depend on many input parameters, while validation of the models is constrained by limited available data. Numerical simulators of coupled mathematical models use approaches of full or loose coupling. Full coupling involves solving a single set of equations simultaneously, but due to computational complexity, feasible run times often necessitate simplified physics models. Conversely, loose coupling connects independent codes simulating distinct physical processes; it is often infeasible to run even loosely coupled simulations the large number of times required to perform uncertainty quantification. This project aims to develop methodology for Gaussian process emulation of high-dimensional coupled simulators and thus enable uncertainty quantification for challenging yet ubiquitous multi-physics models; the approaches will involve surrogate or reduced-order models of the governing model equations to allow quick approximation of the full physical simulator at different inputs. The efforts will entail developing (i) new ideas for dimension reduction, (ii) a novel method to emulate space-time fields, and (iii) an analysis of errors owing to both dimension reduction and emulation.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.