Active-Learning-Driven Surrogate Modeling for Efficient Simulation of
  Parametric Nonlinear Systems
Harshit Kapadia Lihong Feng Peter Benner
When repeated evaluations for varying parameter configurations of a
high-fidelity physical model are required, surrogate modeling techniques based
on model order reduction are desired. In absence of the governing equations
describing the dynamics, we need to construct the parametric reduced-order
surrogate model in a non-intrusive fashion. In this setting, the usual
residual-based error estimate for optimal parameter sampling associated with
the reduced basis method is not directly available. Our work provides a
non-intrusive optimality criterion to efficiently populate the parameter
snapshots, thereby, enabling us to effectively construct a parametric surrogate
model. We consider separate parameter-specific proper orthogonal decomposition
(POD) subspaces and propose an active-learning-driven surrogate model using
kernel-based shallow neural networks, abbreviated as ActLearn-POD-KSNN
surrogate model. To demonstrate the validity of our proposed ideas, we present
numerical experiments using two physical models, namely Burgers' equation and
shallow water equations. Both the models have mixed -- convective and diffusive
-- effects within their respective parameter domains, with each of them
dominating in certain regions. The proposed ActLearn-POD-KSNN surrogate model
efficiently predicts the solution at new parameter locations, even for a
setting with multiple interacting shock profiles.