Linear response based parameter estimation in the presence of model error

Recently, we proposed a method to estimate parameters of stochastic dynamics based on the linear response statistics. The method rests upon a nonlinear least-squares problem that takes into account the response properties that stem from the Fluctuation-Dissipation Theory. In this article, we address an important issue that arises in the presence of model error. In particular, when the equilibrium density function is high dimensional and non-Gaussian, and in some cases, is unknown, the linear response statistics are inaccessible. We show that this issue can be resolved by fitting the imperfect model to appropriate marginal linear response statistics that can be approximated using the available data and parametric or nonparametric models. The effectiveness of the parameter estimation approach is demonstrated in the context of molecular dynamical models (Langevin dynamics) with a non-uniform temperature profile, where the modeling error is due to coarse-graining, and a PDE (non-Langevin dynamics) that exhibits spatiotemporal chaos, where the model error arises from a severe spectral truncation. In these examples, we show how the imperfect models, the Langevin equation with parameters estimated using the proposed scheme, can predict the nonlinear response statistics of the underlying dynamics under admissible external disturbances.