  The temporal dynamics of integrated silicon resonators has been modeled using
a set of equations coupling the internal energy, the temperature and the free
carrier population. Owing to its simplicity, Newton's law of cooling is the
traditional choice for describing the thermal evolution of such systems. In
this work, we theoretically and experimentally prove that this can be
inadequate in monolithic planar devices, leading to inaccurate predictions. A
new equation, that we train to reproduce the correct temperature behaviour, is
introduced to fix the discrepancies with the experimental results. We discuss
the limitations and the range of validity of our refined model, identifying
those cases where Netwon's law provides, nevertheless, accurate solutions. Our
modeling describes the phenomena underlying thermal and free carrier
instabilities, and is a valuable tool for the engineering of photonic systems
which relay on resonator dynamical states, such as all optical spiking neural
networks or reservoirs for neuromorphic computing.
