CAREER: A Nonlinear Model Reduction Framework for Oscillatory Systems and Associated Data-Driven Inference Strategies

该奖项由2021年美国救援计划法案提供全部或部分资金（公法117-2）。该教师早期职业发展计划（CAREER）拨款将资助能够改善对集体神经节律的理解和控制的研究，例如导致帕金森病运动症状的神经元的病理同步，从而促进科学的进步，促进国民健康。深部脑刺激术将电脉冲注入帕金森氏症患者的大脑，以减轻肌肉震颤和僵硬。由于需要大量的输入，用于预测和分析神经元反应的标准理论工具是不够的，因为它们假设与同步行为的微小偏差。该项目将通过开发一种新的理论方法来克服这些限制，该方法适用于具有振荡动力学的复杂高维系统，甚至适用于由系统非线性主导的大偏差。通过将这种方法与机器学习技术相结合，将有可能完全从测量结果中导出相关的动力学模型，并可能应用于治疗时差反应或控制车辆和飞机周围的气流。通过研究和教育的紧密结合，该项目将有助于工程和科学课程的探究为基础，动手学习经验的高中学生参加外展活动或专门设计的，多日沉浸式节目在孤独橡树农场，干教育中心在田纳西州西部，服务于大量人口的代表性不足的学生从资源不足的当地社区。本项目的完成还将产生一系列的辅导课程，一套在线学习模块，和一个计算工具箱，每一个促进使用强大的数学技术的动力系统分析的成员，更大的研究社区。本研究旨在作出基本贡献的理论模型降阶技术的振荡高维系统的动力学是由系统的非线性，特别强调精度、分析易处理性和控制设计的适用性。它实现了这一目标，通过增强传统的基于相位的减少方法与横向动力学的描述等稳坐标，其特征在于最慢的衰减模式的系统Koopman运营商。然后引入模型参数的自适应更新来约束等稳坐标的时间演化，并确保模型简化中使用的渐近展开的收敛性。概括到非周期性的动态，重要的是，在没有已知的基本动力学方程的数据驱动的模型识别将在理论模型和昼夜节律，神经脑节律和流体流动system.This奖项的应用进行了探索反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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 improved understanding and control of collective neurological rhythms, for example pathological synchronization of neurons contributing to the motor symptoms of Parkinson’s disease, thereby promoting the progress of science, and advancing the national health. Deep brain stimulation injects electrical pulses into the brains of patients suffering from Parkinson’s to alleviate muscle tremors and rigidity. Because of the large inputs required, standard theoretical tools for predicting and analyzing the neuronal response are inadequate, since they assume small deviations from the synchronized behavior. This project will overcome such limitations by developing a new theoretical approach, suitable for complex and high-dimensional systems with oscillatory dynamics even for large deviations dominated by system nonlinearities. By combining this approach with machine-learning techniques, it will be possible to derive relevant dynamical models entirely from measurements, with potential applications to treating recovery from jet lag or controlling the air flow around vehicles and aircraft. Through close integration of research and education, this project will contribute to an engineering and science curriculum of inquiry-based, hands-on learning experiences for high school students attending outreach activities or specially designed, multi-day immersive programs at Lone Oaks Farm, a STEM education center in west Tennessee that serves large populations of underrepresented students from under-resourced local communities. Completion of this project will also yield a series of tutorial sessions, a set of online learning modules, and a computational toolbox, each promoting the use of powerful mathematical techniques for dynamical systems analysis to members of the larger research community.This research aims to make fundamental contributions to a theory of model reduction techniques for oscillatory high-dimensional systems whose dynamics are dominated by system nonlinearities, with particular emphasis on accuracy, analytical tractability, and suitability for control design. It achieves this aim by augmenting traditional phase-based reduction methods with a description of transversal dynamics in terms of isostable coordinates, which characterize the slowest decaying modes of the system Koopman operator. Adaptive updates to model parameters are then introduced to bound the time evolution of the isostable coordinates and ensure convergence of asymptotic expansions used in the model reduction. Generalizations to non-periodic dynamics and, importantly, to data-driven model identification in the absence of known underlying dynamical equations will be explored in theoretical models and in applications to circadian cycles, neural brain rhythms, and fluid flow systems.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.