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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公共法律117-2)。这项教师早期职业发展计划(职业生涯)赠款将资助能够改善对集体神经节律的理解和控制的研究，例如导致帕金森氏症运动症状的神经元的病理同步，从而促进科学进步，促进国民健康。脑深部刺激将电脉冲注入帕金森氏症患者的大脑，以缓解肌肉震颤和僵硬。由于所需的大量输入，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。本项目将通过发展一种新的理论方法来克服这种局限性，该方法适用于具有振荡动力学的复杂和高维系统，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中得出相关的动力学模型，并可能应用于治疗时差恢复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，该项目将为参加在Lone Oaks Farm参加外展活动或特别设计的为期数天的沉浸式课程的高中生提供工程和科学课程，该课程以探究为基础，实践学习体验。Lone Oaks Farm是田纳西州西部的STEM教育中心，服务于来自资源不足的当地社区的大量代表不足的学生。该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，每个模块都向更大的研究社区成员推广使用强大的数学技术进行动态系统分析。这项研究旨在为振荡高维系统的模型简化技术理论做出基础性贡献，其动力学由系统非线性主导，特别强调精确度、分析可控性和控制设计的适宜性。它通过用等稳坐标描述系统最慢衰减模式的横向动力学来增强传统的基于相位的约化方法，从而达到这一目的。然后引入模型参数的自适应更新，以限制等稳坐标的时间演化，并确保模型降阶中使用的渐近展开的收敛。我们将在理论模型和昼夜节律、神经脑节律和流体流动系统的应用中探索对非周期动力学的概括，更重要的是，在缺乏已知基本动力学方程的情况下对数据驱动的模型识别进行探索。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。

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.