A New Framework for Nonlinear Dynamical Model Reduction

本项目的研究目标是将多变量和非线性时间序列分析应用于建立全面系统的动力学模型和数据约简框架。在这一框架下，光滑正交分解将被应用于确定性非线性动力系统的降维，并将其与其他备选方案--如真正交分解--进行对比。研究方法是在几个大型结构和分子动力学模型的短时间轨迹上测试框架，并评估得到的子空间对模型降阶的适宜性。基于这些子空间的降阶模型将使用频率响应函数、分叉图、相图和其他动力学特征(如分维和Liapunov指数)进行模拟并与全尺度模型进行比较。该框架将提供一种系统的方法来评估特定模型降阶过程的有效性及其相对于系统/强迫参数和初始条件的有界变化的稳健性。这个项目有望为研究复杂的工程和生物系统的慢时间动力学(如蛋白质构象变化或折叠的物理、蛋白质-蛋白质结合、大的多物理动力学等)提供新的工具。例如，对于蛋白质系统，这种方法可以用于下一代化合物拮抗剂的合理设计。该计划还支持扩展两门跨学科研究生课程，为低年级和高年级学生组织全系研究生研究讲习班，定期组织会议和小型专题讨论会，通过网络传播成果，个人交流，以及在顶级期刊上发表传统出版物。

The research objective of this project is to apply multivariate and nonlinear time series analyses in formulating comprehensive and systematic dynamical model and data reduction framework. In this framework, smooth orthogonal decomposition will be applied to the dimension reduction of deterministic nonlinear dynamical systems and will be contrasted with other alternatives - e.g., proper orthogonal decomposition. The research approach is to test the framework on several large-scale structural and molecular dynamics models' short time trajectories, and evaluate the resultant subspaces suitability for model reduction. Reduced-order models based on these subspaces will be simulated and compared with the full-scale models, using frequency response functions, bifurcation diagrams, phase portraits, and other dynamical characteristics such as fractal dimensions and Liapunov exponents.This framework will provide a systematic way of evaluating a particular model reduction procedure's effectiveness and its robustness with respect to the bounded variations in the system/forcing parameters, and initial conditions. This project is expected to provide new tools to study slow-time dynamics of complex engineered and biological systems (e.g., physics of protein conformational changes or folding, protein-protein association, large multi-physics dynamics, etc.). For a protein system, e.g., this methodology can be exploited in the rational design of next generation compound-antagonists. The plan also supports the curricular expansion of two interdisciplinary graduate courses, the organization of department-wide workshops on graduate research for junior and senior students, the regular organization of conferences and minisymposia, the dissemination of results through the web, personal communication, and traditional publications in top journals.