Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

This work derives a residual-based posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations control inputs. It is shown that quantities are necessary the can be either obtained exactly as solutions least-squares problems in way such initial conditions, inputs, and solution trajectories or bounded probabilistic sense. The computational procedure follows an offline/online decomposition. In offline (training) phase, system judiciously solved black-box fashion to generate set up estimator. online used bound reduced-model predictions new conditions inputs without recourse system. Numerical results demonstrate workflow proposed approach certified predictions.