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。******提出该方法的动机基于两个主要观察结果。首先，控制方程的形式很少是先验的；然而，基于效应稀疏性原理，人们可以假设表示动力学所需的势函数的数量非常少。在实践中，通过在优化模型中添加L1项（或相关数量）作为约束或惩罚来提高稀疏性。虽然稀疏优化技术在图像和信号处理、信息科学和其他领域取得了成功，但它们在动力系统中的应用仍然有限。另一方面，压缩感知理论为一般数据的重构保证提供了坚实的理论成果。对于来自具有复杂行为和附加限制的动态流的时变数据，需要对现有理论进行扩展和仔细研究。利用稀疏诱导方法和随机抽样理论的结果，本提案旨在建立稀疏模型和采样策略，从时变数据中恢复非线性动力学的控制方程，并了解相关最小化问题的重建保证。PI和合作者的初步结果表明，在三维物理空间中，只要流动足够遍历，就有可能从可能高度损坏的数据中准确地识别出底层方程作为L1最小化问题的解。基于这些初步结果，PI将进一步研究动态系统的稀疏学习和压缩感知的重构保证的有效结合，以研究来自高维数据、噪声数据和分岔图数据等广泛数据的动态。本研究将为稀疏优化和压缩感知在数据结构学习中的应用提供新的视角。它可以应用于天气预报和大气模型、流体流动控制、飞机开发和疾病控制模型中的问题。

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.