Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition
This work addresses the challenge of accurately measuring model quality in data-driven dynamical systems modeling, offering a more reliable alternative to residual error for researchers in applied mathematics and control theory.
The authors tackled the problem of assessing prediction accuracy in Extended Dynamic Mode Decomposition (EDMD) by introducing a consistency index based on forward-backward temporal analysis, which provides a tight upper-bound for relative root mean square error across all function predictions in the dictionary's span.
Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.