  Dynamic mode decomposition (DMD), which the family of singular-value
decompositions (SVD), is a popular tool of data-driven regression. While
multiple numerical tests demonstrated the power and efficiency of DMD in
representing data (i.e., in the interpolation mode), applications of DMD as a
predictive tool (i.e., in the extrapolation mode) are scarce. This is due, in
part, to the lack of rigorous error estimators for DMD-based predictions. We
provide a theoretical error estimator for DMD extrapolation of numerical
solutions to linear and nonlinear parabolic equations. This error analysis
allows one to monitor and controls the errors associated with DMD-based
temporal extrapolation of numerical solutions to parabolic differential
equations. We use several computational experiments to verify the robustness of
our error estimators and to compare the predictive ability of DMD with that of
proper orthogonal decomposition (POD), another member of the SVD family. Our
analysis demonstrates the importance of a proper selection of observables, as
predicted by the Koopman operator theory. In all the tests considered, DMD
outperformed POD in terms of efficiency due to its iteration-free feature. In
some of these experiments, POD proved to be more accurate than DMD. This
suggests that DMD is preferable for obtaining a fast prediction with slightly
lower accuracy, while POD should be used if the accuracy is paramount.

Numerical Analysis