Accurate Data-Driven Surrogates of Dynamical Systems for Forward
  Propagation of Uncertainty
Saibal De Reese E. Jones Hemanth Kolla
Stochastic collocation (SC) is a well-known non-intrusive method of
constructing surrogate models for uncertainty quantification. In dynamical
systems, SC is especially suited for full-field uncertainty propagation that
characterizes the distributions of the high-dimensional primary solution fields
of a model with stochastic input parameters. However, due to the highly
nonlinear nature of the parameter-to-solution map in even the simplest
dynamical systems, the constructed SC surrogates are often inaccurate. This
work presents an alternative approach, where we apply the SC approximation over
the dynamics of the model, rather than the solution. By combining the
data-driven sparse identification of nonlinear dynamics (SINDy) framework with
SC, we construct dynamics surrogates and integrate them through time to
construct the surrogate solutions. We demonstrate that the SC-over-dynamics
framework leads to smaller errors, both in terms of the approximated system
trajectories as well as the model state distributions, when compared against
full-field SC applied to the solutions directly. We present numerical evidence
of this improvement using three test problems: a chaotic ordinary differential
equation, and two partial differential equations from solid mechanics.