On the phenomenological modelling of physical phenomena

Mathematical modelling of physical phenomena is based on the laws of physics, but for complicated processes, phenomenological models could enhance the descriptive and prescriptive power of the analysis. This paper describes some hybrid models, where in addition to the physics-driven part, some phenomenological variables (based on observations) are added. The internal variables widely used in continuum mechanics for modelling dissipative processes and the phenomenological variables used in modelling neural impulses are described and compared. The appendices describe two models of neural impulses and test problems for two classical cases: the wave equation and the diffusion equation. These test problems demonstrate the usage of phenomenological variables for describing dissipation as well as amplification.