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

通常在科学和工程中，系统需要使用多个不同的参数配置，输入和系数进行数值模拟，以更深入地了解潜在的物理现象。模型简化（复杂性降低）已成为一种无处不在的工具，使这些重复的数值模拟计算易于处理，利用内在的低秩结构的系统动力学。然而，大多数今天的模型简化方法利用线性结构，只存在于有限的一类问题，这排除了，例如，重要的问题，描述波和其他移动相干结构的传输现象。这严重限制了当今模型简化方法在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模式形成等广泛应用中的使用。本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简化方法，利用非线性结构对传输现象进行易于处理的重复模拟，以支持科学家和工程师。本项目提出的非线性模型简化技术将在流形上导出作为低秩近似的合成的简化解。这意味着放弃了当今模型简化方法的传统线性向量空间结构，以提高表达能力（“打破Kolmogorov障碍”）。工作包如下：（一）将开发新的近似理论概念，以表明原型运输主导偏微分方程，拟议的简化模型打破了当今线性模型简化方法的Kolmogorov障碍，这意味着拟议的非线性模型的最佳近似误差比线性模型的最佳近似误差以指数方式衰减得更快。(ii)数值方法将提出，利用非线性低秩的建议的非线性简化模型的组成近似，以实现加速比传统的模型。非线性解将通过对控制方程进行时间积分来进行数值计算，以确保非线性简化模型能够很好地推广到新的输入/参数，而不是纯粹依赖于数据拟合图而不返回控制方程。(iii)与风暴潮预测和增材制造领域科学家合作的两个现实应用将证明，所提出的技术有可能影响广泛的应用，而这些应用目前受到运输现象模拟成本过高的限制。(iv)集成的教育组件，如通过翻转教室的互动在线课程，工程师教程，和推广活动，建立在两个真实世界的应用程序在风暴-该奖项反映了NSF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进行评估，被认为值得支持的搜索.

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.