  When evaluating quantities of interest that depend on the solutions to
differential equations, we inevitably face the trade-off between accuracy and
efficiency. Especially for parametrized, time dependent problems in engineering
computations, it is often the case that acceptable computational budgets limit
the availability of high-fidelity, accurate simulation data. Multi-fidelity
surrogate modeling has emerged as an effective strategy to overcome this
difficulty. Its key idea is to leverage many low-fidelity simulation data, less
accurate but much faster to compute, to improve the approximations with limited
high-fidelity data. In this work, we introduce a novel data-driven framework of
multi-fidelity surrogate modeling for parametrized, time-dependent problems
using long short-term memory (LSTM) networks, to enhance output predictions
both for unseen parameter values and forward in time simultaneously - a task
known to be particularly challenging for data-driven models. We demonstrate the
wide applicability of the proposed approaches in a variety of engineering
problems with high- and low-fidelity data generated through fine versus coarse
meshes, small versus large time steps, or finite element full-order versus deep
learning reduced-order models. Numerical results show that the proposed
multi-fidelity LSTM networks not only improve single-fidelity regression
significantly, but also outperform the multi-fidelity models based on
feed-forward neural networks.
