CAREER: Goal-Oriented Variable Transformations for Efficient Reduced-Order and Data-Driven Modeling

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。这项教师早期职业发展计划（Career）赠款将资助能够对复杂的自然和工程过程（包括气候动力学和火箭燃烧）进行有效数据驱动建模的研究，从而促进科学进步，促进国家繁荣和福利。实时预测、控制干预或工程设计需要对这些过程进行快速、准确的计算机模拟。目前从测量中开发模拟模型的技术依赖于可能使分析和认证复杂化的近似值，而没有降低计算成本或保证基本物理定律得到尊重。该项目通过开发一种新的理论方法来系统地揭示系统动力学的最佳公式，这些公式在计算上易于处理和严格可验证，并保留了物理过程的关键属性，从而克服了这些挑战。这种公式可以使化学和热过程的计算效率和可靠的建模成为可能，或用于预测长期的海洋流动动力学，然后将其与耦合的气候模式结合起来。通过与工业界的合作，这项研究将推动空调系统的设计和控制，使他们能够使用更准确和更快的建筑物气流模型。通过研究和教育的紧密结合，该项目将通过外联、指导和本科研究，支持和吸引来自当地高中、社区学院和大学的第一代和低收入家庭的学生。面向大学生的免费教育材料将广泛传播，以促进培养具有较强计算技能的新一代工程师。本研究旨在发展一种新的理论和计算范式的基础，该范式利用变量变换来揭示非线性动力系统中的低维结构，并实现有效和准确的模型简化，从而可以在稳定性和结构保存方面得到认证。它通过利用符号计算算法来系统地识别转换和随后的降阶投影，从而在模型和数据驱动的设置中实现这一目标，从而产生最优的二次或多项式模型，这些模型也保留了哈密顿系统的辛结构。在数据驱动的情况下，需要寻找能够生成长期预测的降阶模型的转换，这些模型在物理上是可解释的，并且具有有利的数值特性。通过这一努力，化学反应动力学和增材制造的中等规模应用的新的低维物理模型将被发现。方法学的贡献将在反应流动和海洋动力学的大尺度模型上进行评估。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This Faculty Early Career Development Program (CAREER) grant will fund research that enables efficient data-driven modeling of complex natural and engineering processes, including climate dynamics and rocket combustion, thereby promoting the progress of science, and advancing the national prosperity and welfare. Fast and accurate computer simulation of such processes is required for real-time prediction, control intervention, or engineering design. Current techniques for developing simulation models from measurements rely on approximations that may complicate analysis and certification, without a reduction in computational cost or guarantees that underlying physical laws are respected. This project overcomes these challenges by developing a new theoretical approach for systematically uncovering optimal formulations of the system dynamics that are computationally tractable and rigorously certifiable, and that preserve key properties of the physical processes. Such formulations may enable computationally efficient and reliable modeling of chemical and thermal processes or be used to predict long-term ocean flow dynamics that can then be integrated with coupled climate models. In collaboration with industry, this research will advance the design and control of air-conditioning systems by allowing them to use more accurate and faster models of air flow in buildings. Through close integration of research and education, this project will support and engage with first-generation and low-income students from local high schools, community colleges, and universities through outreach, mentoring, and undergraduate research. Free educational material aimed at an undergraduate audience will be disseminated widely to promote training of new generations of engineers with strong computational skills.This research aims to develop the foundations of a new theoretical and computational paradigm that leverages variable transformations to uncover low-dimensional structures in nonlinear dynamical systems and achieve efficient and accurate model reduction that may be certified with respect to stability and structure-preservation. It accomplishes this aim in model- and data-driven settings by exploiting symbolic computing algorithms for systematically identifying transformations and subsequent order-reduction projections that result in optimal quadratic or polynomial models that also preserve symplectic structure for Hamiltonian systems. In the data-driven case, transformations are sought that lead to long-term predictive reduced-order models that are physically interpretable and have favorable numerical properties. Through this effort, new low-dimensional models of the physics of medium-scale applications of chemical reaction dynamics and additive manufacturing will be discovered. The methodological contributions will be assessed on large-scale models of reactive flows and ocean dynamics.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.