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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667010
• 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，科学家一直对从实验和观测数据中了解复杂的基础结构感兴趣。过去，这个过程是由相关领域的专家手动完成的。由于数据量巨大及其复杂的行为，对设计有效的算法和分析学习问题的理论方面有很大的需求。该提案的重点是回答从瞬态数据学习动态系统的重要问题。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，开发用于学习非线性动力学的创新数值方法。 ******所提出方法的动机基于两个主要观察。首先，控制方程的形式很少是先验已知的；然而，基于效应稀疏原理，人们可以假设表示动态所需的潜在函数的数量非常小。实际上，通过在优化模型中添加 L1 项（或相关量）作为约束或惩罚来提高稀疏性。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，但它们在动态系统中的应用仍然有限。另一方面，压缩感知理论为通用数据的重构保证提供了扎实的理论成果。对于来自具有复杂行为和附加限制的动态流的瞬态数据，需要对当前理论进行扩展和仔细研究。该提案旨在使用稀疏诱导方法和随机采样理论的结果，开发稀疏模型和采样策略，以从瞬态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，只要流量足够遍历，就可以从可能高度损坏的数据中准确识别基础方程，作为 L1 最小化问题的解。基于这些初步结果，PI将进一步研究动力系统稀疏学习与压缩感知重构保证的有效结合，以研究高维数据、噪声数据和分叉数据等广泛数据的动力学。图表。这项研究将为学习数据结构中的稀疏优化和压缩感知提供新的视角。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等问题。