Numerically studying the degeneracy problem in extreme finite-source microlensing events

Most transit microlensing events due to very low-mass lens objects suffer from extreme finite-source effects. While modeling their light curves, there is a known continuous degeneracy between their relevant lensing parameters, i.e., the source angular radius normalized to the angular Einstein radius $\rho_{\star}$, the Einstein crossing time $t_{\rm E}$, the lens impact parameter $u_{0}$, the blending parameter, and the stellar apparent magnitude. In this work, I numerically study the origin of this degeneracy. I find that these light curves have 5 observational parameters (i.e., the baseline magnitude, the maximum deviation in the magnification factor, the Full Width at Half Maximum $\rm{FWHM}=2 t_{\rm{HM}}$, the deviation from top-hat model, the time of the maximum time-derivative of microlensing light curves $T_{\rm{max}}=t_{\rm E}\sqrt{\rho_{\star}^{2}-u_{0}^{2}}$). For extreme finite-source microlensing events due to uniform source stars we get $t_{\rm{HM}}\simeq T_{\rm{max}}$, and the deviation from the top-hat model tends to zero which both cause the known continuous degeneracy. When either $\rho_{\star}\lesssim10$ or the limb-darkening effect is considerable $t_{\rm{HM}}$, and $T_{\rm{max}}$ are two independent observational parameters. I use a numerical approach, i.e., Random Forests containing $100$-$120$ Decision Trees, to study how these observational parameters are efficient in yielding the lensing parameters. These machine learning models find the mentioned 5 lensing parameters for finite-source microlensing events from uniform, and limb-darkened source stars with the average $R^{2}$-scores of $0.87$, and $0.84$, respectively. $R^{2}$-score for evaluating the lens impact parameter gets worse on adding limb darkening, and for extracting the limb-darkening coefficient itself this score falls as low as $0.67$.