  High-fidelity numerical simulations of partial differential equations (PDEs)
given a restricted computational budget can significantly limit the number of
parameter configurations considered and/or time window evaluated for modeling a
given system. Multi-fidelity surrogate modeling aims to leverage less accurate,
lower-fidelity models that are computationally inexpensive in order to enhance
predictive accuracy when high-fidelity data are limited or scarce. However,
low-fidelity models, while often displaying important qualitative
spatio-temporal features, fail to accurately capture the onset of instability
and critical transients observed in the high-fidelity models, making them
impractical as surrogate models. To address this shortcoming, we present a new
data-driven strategy that combines dimensionality reduction with multi-fidelity
neural network surrogates. The key idea is to generate a spatial basis by
applying the classical proper orthogonal decomposition (POD) to high-fidelity
solution snapshots, and approximate the dynamics of the reduced states -
time-parameter-dependent expansion coefficients of the POD basis - using a
multi-fidelity long-short term memory (LSTM) network. By mapping low-fidelity
reduced states to their high-fidelity counterpart, the proposed reduced-order
surrogate model enables the efficient recovery of full solution fields over
time and parameter variations in a non-intrusive manner. The generality and
robustness of this method is demonstrated by a collection of parametrized,
time-dependent PDE problems where the low-fidelity model can be defined by
coarser meshes and/or time stepping, as well as by misspecified physical
features. Importantly, the onset of instabilities and transients are well
captured by this surrogate modeling technique.
