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.
翻译:从非线性系统提取预测模型是科学机器学习的一项核心任务。一个关键问题是现代数据驱动方法和第一原则之间的调和。尽管在机器学习技术方面进展迅速,但将域知识嵌入数据驱动模型仍然是一个挑战。在这项工作中,我们提出了一个从非线性系统中提取预测模型的普遍学习框架。我们的框架可以很容易地纳入第一原则知识,因为它自然地将非线性系统模拟为连续时间系统。这既改进了所提取模型的外推能力,也减少了培训所需的数据数量。此外,我们的框架具有对观测噪音的稳健性和对非常规抽样数据的可适用性的优势。我们通过学习各种系统的预测模型展示了我们的计划的有效性,这些系统包括坚硬的Van der Pol 振动器、Lorenz系统和Kuramoto-Sivashinsky等式。对于Lorenz系统来说,不同种类的域知识被整合到显示数据驱动系统中知识嵌入的强度。