  This work proposes a machine-learning framework for modeling the error
incurred by approximate solutions to parameterized dynamical systems. In
particular, we extend the machine-learning error models (MLEM) framework
proposed in Ref. 15 to dynamical systems. The proposed Time-Series
Machine-Learning Error Modeling (T-MLEM) method constructs a regression model
that maps features--which comprise error indicators that are derived from
standard a posteriori error-quantification techniques--to a random variable for
the approximate-solution error at each time instance. The proposed framework
considers a wide range of candidate features, regression methods, and additive
noise models. We consider primarily recursive regression techniques developed
for time-series modeling, including both classical time-series models (e.g.,
autoregressive models) and recurrent neural networks (RNNs), but also analyze
standard non-recursive regression techniques (e.g., feed-forward neural
networks) for comparative purposes. Numerical experiments conducted on multiple
benchmark problems illustrate that the long short-term memory (LSTM) neural
network, which is a type of RNN, outperforms other methods and yields
substantial improvements in error predictions over traditional approaches.
