SINDy on slow manifolds

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, such many unsteady fluid flows, 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 laminar unsteady flow over a NACA 0012 airfoil. We find that our method reduces both the condition number and the size of the SINDy library, thus enabling accurate identification of the dynamics on slow manifolds.