Frequency-Domain Learning of Dynamical Systems From Time-Domain Data

Dynamical systems are useful tools for modeling many complex physical phenomena. In many situations, we do not have access to the governing equations to create these models. Instead, we have access to data in the form of input-output measurements. Data-driven approaches use these measurements to construct reduced order models (ROMs), a small scale model that well approximates the true system, directly from input/output data. Frequency domain data-driven methods, which require access to values (and in some cases to derivatives) of the transfer function, have been very successful in constructing high-fidelity ROMs from data. However, at times this frequency domain data can be difficult to obtain or one might have only access to time-domain data. Recently, Burohman et al. [2020] introduced a framework to approximate transfer function values using only time-domain data. We first discuss improvements to this method to allow a more efficient and more robust numerical implementation. Then, we develop an algorithm that performs optimal-H2 approximation using purely time-domain data; thus significantly extending the applicability of H2-optimal approximation without a need for frequency domain sampling. We also investigate how well other established frequency-based ROM techniques (such as the Loewner Framework, Adaptive Anderson-Antoulas Algorithm, and Vector Fitting) perform on this identified data, and compare them to the optimal-H2 model.