标题： SINDy 在慢流形上 显示英文标题

标题： SINDy on slow manifolds

摘要： 稀疏非线性动力学识别（SINDy）已被确立为一种有效的方法，可以从数据中学习可解释的动力系统模型。 然而，对于高维的慢-快动力系统，回归问题变得同时计算上不可行且病态。 尽管原则上仅对底层慢流形上演变的动力学进行建模可以解决这两个挑战，但必须通过包含更高阶的非线性项作为模型的候选项来补偿截断的快变量，从而导致SINDy库的大小呈指数增长。 在本工作中，我们开发了一种SINDy变体，能够在两个步骤中稳健且高效地识别慢-快动力学：(i) 识别慢流形，即快变量作为慢变量函数的代数方程，以及(ii) 学习限制在流形上的慢变量动力学模型。 关键的是，在(i)中学习的方程被用来构建(ii)的流形信息函数库，该库仅包含必要的高阶非线性项作为候选项。 与包含所有至一定次数的单项式的库不同，所产生的定制库是后者的一个稀疏子集，专门针对当前的具体问题。 该方法在snap-through屈曲梁和NACA 0012机翼上方流动的数值例子中得到验证。 我们发现，我们的方法显著降低了SINDy库的条件数和大小，从而实现了对慢流形上动力学的准确识别。 显示英文摘要

摘要： The sparse identification of nonlinear dynamics (SINDy) has been established as an effective method to learn interpretable models of dynamical systems from data. However, for high-dimensional slow-fast dynamical systems, the regression problem becomes simultaneously computationally intractable and ill-conditioned. Although, in principle, modeling only the dynamics evolving on the underlying slow manifold addresses both of these challenges, the truncated fast variables have to be compensated by including higher-order nonlinearities as candidate terms for the model, leading to an explosive growth in the size of the SINDy library. In this work, we develop a SINDy variant that is able to robustly and efficiently identify slow-fast dynamics in two steps: (i) identify the slow manifold, that is, an algebraic equation for the fast variables as functions of the slow ones, and (ii) learn a model for the dynamics of the slow variables restricted to the manifold. Critically, the equation learned in (i) is leveraged to build a manifold-informed function library for (ii) that contains only essential higher-order nonlinearites as candidate terms. Rather than containing all monomials of up to a certain degree, the resulting custom library is a sparse subset of the latter that is tailored to the specific problem at hand. The approach is demonstrated on numerical examples of a snap-through buckling beam and the flow over a NACA 0012 airfoil. We find that our method significantly reduces both the condition number and the size of the SINDy library, thus enabling accurate identification of the dynamics on slow manifolds.