  A machine-learning-based framework for modeling the error introduced by
surrogate models of parameterized dynamical systems is proposed. The framework
entails the use of high-dimensional regression techniques (e.g., random
forests, LASSO) to map a large set of inexpensively computed `error indicators'
(i.e., features) produced by the surrogate model at a given time instance to a
prediction of the surrogate-model error in a quantity of interest (QoI). This
eliminates the need for the user to hand-select a small number of informative
features. The methodology requires a training set of parameter instances at
which the time-dependent surrogate-model error is computed by simulating both
the high-fidelity and surrogate models. Using these training data, the method
first determines regression-model locality (via classification or clustering),
and subsequently constructs a `local' regression model to predict the
time-instantaneous error within each identified region of feature space. We
consider two uses for the resulting error model: (1) as a correction to the
surrogate-model QoI prediction at each time instance, and (2) as a way to
statistically model arbitrary functions of the time-dependent surrogate-model
error (e.g., time-integrated errors). We apply the proposed framework to model
errors in reduced-order models of nonlinear oil--water subsurface flow
simulations. The reduced-order models used in this work entail application of
trajectory piecewise linearization with proper orthogonal decomposition. When
the first use of the method is considered, numerical experiments demonstrate
consistent improvement in accuracy in the time-instantaneous QoI prediction
relative to the original surrogate model, across a large number of test cases.
When the second use is considered, results show that the proposed method
provides accurate statistical predictions of the time- and well-averaged
errors.
