Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
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DOI：
10.1073/pnas.1517384113
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发表时间：
2016-04-12
影响因子：
11.1
通讯作者：
Kutz, J. Nathan
中科院分区：
文献类型：
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作者：
Brunton, Steven L.;Proctor, Joshua L.;Kutz, J. Nathan
在许多不同的科学和工程领域，从数据中提取管理方程是一个核心挑战。数据是丰富的，而模型通常仍然难以捉摸，例如在气候科学，神经科学，生态学，金融和流行病学中，仅举几个例子。在这项工作中，我们将促稀疏技术和机器学习与非线性动力学系统相结合，以从嘈杂的测量数据中发现管理方程。关于模型结构的唯一假设是，只有几个重要的术语来控制动力学，因此方程在可能的函数的空间中很少。此假设适用于许多物理系统。特别是，我们使用稀疏回归来确定准确表示数据所需的动态管理方程式中最少的项。这导致了简约的模型，这些模型与模型复杂性之间的精度平衡，以避免过度拟合。我们在各种问题上演示了算法，从简单的典型系统，包括线性和非线性振荡器以及混乱的洛伦兹系统到障碍物后面的流体涡流。流体示例说明了这种方法发现系统的基本动力学的能力，该系统使社区中的专家近30年才能解决。我们还表明，该方法概括为时变或具有外部强迫的参数化系统和系统。
Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.