Uncertainty Quantification in Data-Driven Dynamical Models via Inverse Problem Solving

Data-driven model identification strategies can be used to obtain phenomenological models that capture the temporal evolution of observable data. While it is usually straightforward to obtain such a model from time series data, for instance with least-squares fitting, it is generally difficult to quantify the uncertainty associated with the prediction of the temporal evolution of the observables. This paper considers a general framework for uncertainty quantification in data-driven dynamical models by framing prediction error through the lens of inverse problem theory. Building on Koopman-inspired model identification strategies that are suited for nonlinear dynamical models, we consider a prediction as an approximate measurement from which the original input state can be faithfully recovered, and define the prediction error as the MSE of solving the inverse problem that would yield this prediction. We demonstrate the efficacy of this approach on both numerical models and experimental data showing that it provides a robust uncertainty measure of model performance.