Learning Nonlinear Dynamics from Data Using Sparse Optimization and Compressed Sensing

自科学革命开始以来，科学家们一直对从实验和观测数据中了解复杂的底层结构感兴趣。在过去，这个过程是由相关领域的专家手工完成的。由于数据量大，行为复杂，对设计高效的算法和分析该学习问题的理论方面提出了很高的要求。这一建议的重点是回答这些重要的问题，从时间相关的数据学习动态系统。具体来说，我们计划通过结合优化和压缩感知理论的先进技术，开发学习非线性动力学的创新数值方法。

Since the beginning of scientific revolution, scientists have always been interested in learning sophisticated underlying structures from experimental and observational data. In the past, this process was done manually by experts in related fields. Due to the huge amount of data as well as its complicated behaviours, there are great demands in designing efficient algorithms and analyzing theoretical aspects of that learning problem. This proposal is focused on answering those important questions for learning dynamical systems from time-dependent data. Specifically, we plan to develop innovative numerical methods for learning nonlinear dynamics by combining advanced techniques from optimization and compressed sensing theory. The motivation of the proposed method is based on two main observations. Firstly, the form of the governing equations is rarely known a priori; however, based on the sparsity-of-effect principle, one may assume that the number of potential functions needed to represent the dynamics is very small. In practice, sparsity is promoted through the addition of an L1 term (or related quantity) as a constraint or penalty in the optimization model. While sparse optimization techniques have demonstrated their success in image and signal processing, information sciences, and others, their applications in dynamical systems is still limited. On the other hand, compressed sensing theory has provided solid theoretical results in reconstruction guarantees for general data. For time-dependent data coming from dynamical flows with complicated behaviours and additional restrictions, current theory need to be extended and studied carefully. Using sparse-inducing methods and results from random sampling theory, this proposal aims to develop sparse models and sampling strategies to recover the governing equations of nonlinear dynamics from time-dependent data as well as understanding the reconstruction guarantees for the related minimization problems. Preliminary results by the PI and collaborators show that in physical spaces of dimension three, it is possible to identify the underlying equations exactly from possibly highly corrupted data as the solution of an L1 minimization problem, provided that the flow is sufficiently ergodic. Based on those initial results, the PI will investigate further the effective combination of sparse learning for dynamical systems and reconstruction guarantees from compressed sensing in studying the dynamics from a wide range of data such as high-dimensional data, noisy data, and data from bifurcation diagram. This research will provide new perspectives from sparse optimization and compressed sensing in learning data structures. It can be applied to problems in weather predictions and atmospheric models, controls for fluid flows, aircraft development, and disease control models.