Simultaneous Identification and Denoising of Dynamical Systems
Jeffrey M. Hokanson, Gianluca Iaccarino, Alireza Doostan
Abstract
.In recent years there has been a push to discover the governing equations of dynamical systems directly from measurements of the state, often motivated by systems that are too complex to directly model. Although there has been substantial work put into such a discovery, doing so in the case of large noise has proved challenging. Here we develop an algorithm for the simultaneous identification and denoising of a dynamical system (SIDDS). We infer the noise in the state measurements by requiring that the denoised state satisfies the dynamical system with an equality constraint. This contrasts to existing work where the mismatch in the dynamics is added as a penalty in the objective. Assuming the nonlinear differential equation is represented in a predefined basis, we develop a sequential quadratic programming approach to solve the SIDDS problem featuring a direct solution of the KKT system with a specialized preconditioner. We also show how to add a sparsity promotion regularization into SIDDS using an iteratively reweighted least squares approach. Our resulting algorithm obtains estimates of the dynamical system that achieve the Cramér–Rao lower bound up to discretization error. This enables SIDDS to provide substantial improvements compared to existing techniques: SIDDS substantially decreases the data burden for accurate identification, recovers optimal estimates with lower sample rates, and the sparsity promoting variant discovers the correct sparsity pattern with larger noise.Keywordsdynamical systemsmodel discoveryinverse problemsparameter estimationsparse recoveryMSC codes34A5565L0990C5593B30