arXiv 2010.034152020-10nlin.CDcs.LG

基于知识的非线性动力学与混沌学习

Knowledge-Based Learning of Nonlinear Dynamics and Chaos

University of Pennsylvania（宾夕法尼亚大学）；Montclair State University（蒙特克莱尔州立大学）

Tom Z. Jiahao, M. Ani Hsieh, Eric Forgoston

备注：: 8 pages, 12 figures in main text, 6 figures and 8 tables in supplement
发表信息：: Chaos 31, 111101 (2021)
DOI：: 10.1063/5.0065617

摘要：

从非线性系统中提取预测模型是科学机器学习的核心任务。一个关键问题是现代数据驱动方法和首要原则之间的协调。尽管机器学习技术发展迅速，但将领域知识嵌入到数据驱动的模型中仍然是一个挑战。在这项工作中，我们提出了一个通用的学习框架，用于从基于观测的非线性系统中提取预测模型。我们的框架可以很容易地结合第一原理知识，因为它自然地将非线性系统建模为连续时间系统。这既提高了提取模型的外推能力，又减少了训练所需的数据量。此外，我们的框架具有鲁棒性的观测噪声和不规则采样数据的适用性的优点。我们证明了我们的计划的有效性，通过学习各种各样的系统，包括刚性范德波尔振荡器，洛伦兹系统，和Kuramoto-Sivashinsky方程的预测模型。对于Lorenz系统，不同类型的领域知识被纳入到数据驱动的系统识别中，以证明知识嵌入的强度。

英文摘要：

Extracting predictive models from nonlinear systems is a central task in scientific machine learning. One key problem is the reconciliation between modern data-driven approaches and first principles. Despite rapid advances in machine learning techniques, embedding domain knowledge into data-driven models remains a challenge. In this work, we present a universal learning framework for extracting predictive models from nonlinear systems based on observations. Our framework can readily incorporate first principle knowledge because it naturally models nonlinear systems as continuous-time systems. This both improves the extracted models' extrapolation power and reduces the amount of data needed for training. In addition, our framework has the advantages of robustness to observational noise and applicability to irregularly sampled data. We demonstrate the effectiveness of our scheme by learning predictive models for a wide variety of systems including a stiff Van der Pol oscillator, the Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz system, different types of domain knowledge are incorporated to demonstrate the strength of knowledge embedding in data-driven system identification.

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