  In a recent preprint (arXiv:1211.4285v1) we addressed the problem of
constructing reduced models for time-dependent systems described by
differential equations which involve uncertain parameters. In the current work,
we focus on the construction of reduced models for systems of differential
equations when the initial condition is uncertain. While for both cases the
reduced models are constructed through the Mori-Zwanzig formalism, the
necessary estimation of the memory parameters is quite different. For the case
of uncertain initial conditions we present an algorithm which allows to
estimate on the fly the parameters appearing in the reduced model. The first
part of the algorithm evolves the full system until the estimation of the
parameters for the reduced model has converged. At the time instant that this
happens, the algorithm switches to the evolution of only the reduced model with
the estimated parameter values from the first part of the algorithm. The
viscous Burgers equation with uncertain initial condition is used to illustrate
the construction.
