Sindy on attracting manifolds

Abstract

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 dynamical systems with attracting invariant manifolds, the regression problem becomes simultaneously computationally intractable and ill-conditioned. Although, in principle, modeling only the dynamics evolving on the underlying manifold addresses both of these challenges, the truncated 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 extension that is able to robustly and efficiently identify dynamics restricted to an attracting manifold in two steps: (i) identify the manifold, that is, an algebraic equation for the driven (slaved) variables as functions of the driving ones, and (ii) learn a model for the dynamics of the driving variables on 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 nonlinearities 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 reduces both the condition number and the size of the SINDy library, thus enabling accurate identification of the dynamics on attracting manifolds.