约束稀疏Galerkin回归
Constrained Sparse Galerkin Regression
摘要：
虽然在过去的几十年里，线性系统的降阶和辨识已经取得了很大的进展，但非线性低阶模型的推导仍然是一个挑战性的任务。在这项工作中，我们开发了一个新的数据驱动的框架，通过结合降维技术（例如适当的正交分解）和稀疏回归技术从机器学习，识别非线性降阶模型的流体。特别是，我们扩展了稀疏识别的非线性动力学（SINDy）算法，以执行物理约束的回归，即能量保持二次非线性。所得到的模型（下文称为Galerkin回归模型）结合了Galerkin投影的许多有益方面，但不需要全阶或高保真解算器来投影纳维-斯托克斯方程。相反，最简约的非线性模型是确定的，是与观察到的测量数据一致，并满足必要的约束。Galerkin回归模型也容易推广到包括高阶非线性项，其对截断模式的影响进行建模。Galerkin回归的有效性证明了两种不同的流动配置：二维流动通过一个圆柱和剪切驱动的空腔流。对于这两种情况下，所确定的模型的准确性比较有利的降阶模型从一个标准的Galerkin投影程序。目前的研究结果突出了立方非线性的重要性，在建设中的准确的非线性低维近似的流动系统，这是不能很容易地获得使用标准的Galerkin投影的纳维-斯托克斯方程。最后，我们的约束稀疏Galerkin回归算法的整个代码库可以在线免费获得。
英文摘要：
Although major advances have been achieved over the past decades for the reduction and identification of linear systems, deriving nonlinear low-order models still is a chal- lenging task. In this work, we develop a new data-driven framework to identify nonlinear reduced-order models of a fluid by combining dimensionality reductions techniques (e.g. proper orthogonal decomposition) and sparse regression techniques from machine learn- ing. In particular, we extend the sparse identification of nonlinear dynamics (SINDy) algorithm to enforce physical constraints in the regression, namely energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a full-order or high-fidelity solver to project the Navier-Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed mea- surement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of Galerkin regression is demonstrated on two different flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favorably against reduced-order models obtained from a standard Galerkin projection procedure. Present results highlight the importance of cubic nonlinearities in the construction of accurate nonlinear low-dimensional approximations of the flow systems, something which cannot be readily obtained using a standard Galerkin projection of the Navier-Stokes equations. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.
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