Learning Nonlinear Dynamics from Data Using Sparse Optimization and Compressed Sensing

使用稀疏优化和压缩感知从数据中学习非线性动力学

• 批准号: RGPIN-2018-06135
• 负责人: Tran, Giang
• 金额: $ 1.68万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759275
• 关键词: Learning; Nonlinear; Dynamics; Data; Using

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.

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学习复杂的潜在结构感兴趣。过去，这一过程是由相关领域的专家手动完成的。由于数据的海量和行为的复杂性，设计高效的学习算法和对学习问题的理论分析提出了很高的要求。这一建议的重点是回答那些从时变数据中学习动力系统的重要问题。具体地说，我们计划通过结合最优化和压缩传感理论的先进技术来开发学习非线性动力学的创新的数值方法。提出的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知道的；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，稀疏性是通过在优化模型中添加L1项(或相关量)作为约束或惩罚来促进的。虽然稀疏优化技术已经在图像和信号处理、信息科学等领域取得了成功，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建提供了坚实的理论保证。对于来自具有复杂行为和附加限制的动态流动的时变数据，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，该建议旨在建立稀疏模型和抽样策略，以从随时间变化的数据中恢复非线性动力学的控制方程，并理解相关最小化问题的重构保证。PI和合作者的初步结果表明，在三维物理空间中，如果流具有足够的遍历性，则可以从可能高度受破坏的数据中准确地识别潜在方程作为L1最小化问题的解。基于这些初步结果，PI将进一步研究在从高维数据、噪声数据和分叉图数据等广泛的数据中研究动态系统的稀疏学习和压缩感知的重构保证的有效组合。本研究将从稀疏优化和压缩感知两个方面为学习数据结构提供新的视角。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。