CAREER: Formulations, Theory, and Algorithms for Nonlinear Model Reduction in Transport-Dominated Systems

通常在科学和工程中，系统需要用多个不同的参数配置、输入和系数进行数值模拟，以获得对潜在物理现象的更深层次的理解。通过利用系统动力学中固有的低阶结构，模型简化(复杂性降低)已经成为一种普遍存在的工具，使得这些重复的数值模拟在计算上变得容易处理。然而，今天的大多数模型降阶方法利用的是仅存在于有限类问题中的线性结构，这排除了例如描述波和其他运动相干结构的传输现象的重要问题。这严重限制了今天的模型简化方法在广泛的应用中的使用，从材料科学中的多相问题到工程中的燃烧，再到生物学中的图案形成。该项目的目标是在机器学习、数值分析和科学计算的交叉点上开发新型的模型降阶方法，利用非线性结构对运输现象进行易于处理的重复模拟，以支持科学家和工程师。该项目中提出的非线性模型降阶技术将得到流形上的降阶解作为低阶近似的组合。这意味着，今天的模型简化方法放弃了传统的线性向量空间结构，取而代之的是增加表现力(“打破科尔莫戈罗夫壁垒”)。工作包如下：(I)将发展新的逼近理论概念，以证明所提出的简化模型打破了现有线性模型降阶方法的Kolmogorov障碍，这意味着所提出的非线性模型的最佳逼近误差衰减的指数速度快于线性模型的最佳逼近误差。(Ii)将提出数值方法，利用所提出的非线性约化模型的合成近似的非线性低秩性来实现与传统模型相比的加速比。非线性解将通过及时积分控制方程来进行数值计算，以确保非线性简化模型很好地适用于新的输入/参数，而不是纯粹依赖于数据拟合的映射而不返回控制方程。(3)与领域科学家在风暴潮预报和添加剂制造方面合作的两个真实世界应用将表明，拟议的技术有可能影响目前受到令人望而却步的运输现象模拟的广泛应用。(Iv)综合教育部分，如通过翻转教室的互动在线课程、工程师教程和基于风暴潮预报和添加剂制造中的两个真实应用程序的外联活动，将教育广泛的受众并确保实践中的相关性。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。

Often in science and engineering, systems need to be numerically simulated with multiple different parameter configurations, inputs, and coefficients to gain a deeper understanding of the underlying physical phenomena. Model reduction (complexity reduction) has become a ubiquitous tool that makes these repeated numerical simulations computationally tractable by exploiting intrinsic low-rank structures in the system dynamics. However, most of today's model reduction methods exploit linear structures that are present only in a limited class of problems, which excludes, for example, important problems with transport phenomena that describe waves and other moving coherent structures. This severely limits the use of today's model reduction methods in a wide range of applications from multi-phase problems in materials science to combustion in engineering to pattern formation in biology. The goal of this project is to develop new types of model reduction methods at the intersection of machine learning, numerical analysis, and scientific computing that exploit nonlinear structures to make tractable repeated simulations of transport phenomena to support scientists and engineers.The nonlinear model reduction techniques proposed in this project will derive reduced solutions on manifolds as compositions of low-rank approximations. This means that the traditional linear vector-space structure of today's model reduction methods is given up in favor of increased expressiveness ("breaking the Kolmogorov barrier"). The work packages are as follows: (i) New approximation theory concepts will be developed to show for prototypical transport-dominated partial differential equations that the proposed reduced models break the Kolmogorov barrier of today's linear model reduction methods, meaning the best-approximation error of the proposed nonlinear models decays exponentially faster than the best-approximation error of linear models. (ii) Numerical methods will be proposed that exploit the nonlinear low rankness of the composed approximations of the proposed nonlinear reduced models to achieve speedups compared to traditional models. The nonlinear solutions will be numerically computed by integrating in time the governing equations to ensure that the nonlinear reduced models generalize well to new inputs/parameters, in contrast to purely relying on data-fitted maps without going back to the governing equations. (iii) Two real-world applications in collaborations with domain scientists in storm-surge forecasting and additive manufacturing will demonstrate that the proposed techniques have the potential to impact a wide range of applications that currently are limited by prohibitively expensive simulations of transport phenomena. (iv) Integrated education components such as interactive online courses via flipped classrooms, tutorials for engineers, and outreach activities building on the two real-world applications in storm-surge forecasting and additive manufacturing will educate a broad audience and ensure relevance in practice.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.