Classic circuit modeling for supercapacitors is limited in representing the strongly non-linear behavior of the hybrid supercapacitor technology. In this work, two novel modeling techniques suitable to represent the time-domain electrical behavior of a hybrid supercapacitor are presented. The first technique enhances a well-affirmed circuit model by introducing specific non-linearities. The second technique models the device through a black-box approach with a neural network. Both the modeling techniques are validated experimentally using a workbench to acquire data from a real hybrid supercapacitor. The proposed models, suitable for different supercapacitor technologies, achieve higher accuracy and generalization capabilities compared to those already presented in the literature. Both modeling techniques allow for an accurate representation of both short-time domain and steady-state simulations, providing a valuable asset in electrical designs featuring supercapacitors.