  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.
