  This work proposes a machine-learning framework for constructing statistical
models of errors incurred by approximate solutions to parameterized systems of
nonlinear equations. These approximate solutions may arise from early
termination of an iterative method, a lower-fidelity model, or a
projection-based reduced-order model, for example. The proposed statistical
model comprises the sum of a deterministic regression-function model and a
stochastic noise model. The method constructs the regression-function model by
applying regression techniques from machine learning (e.g., support vector
regression, artificial neural networks) to map features (i.e., error indicators
such as sampled elements of the residual) to a prediction of the
approximate-solution error. The method constructs the noise model as a
mean-zero Gaussian random variable whose variance is computed as the sample
variance of the approximate-solution error on a test set; this variance can be
interpreted as the epistemic uncertainty introduced by the approximate
solution. This work considers a wide range of feature-engineering methods,
data-set-construction techniques, and regression techniques that aim to ensure
that (1) the features are cheaply computable, (2) the noise model exhibits low
variance (i.e., low epistemic uncertainty introduced), and (3) the regression
model generalizes to independent test data. Numerical experiments performed on
several computational-mechanics problems and types of approximate solutions
demonstrate the ability of the method to generate statistical models of the
error that satisfy these criteria and significantly outperform more commonly
adopted approaches for error modeling.
