Artificial neural networks (ANN) from the package NeuroShell 2 are applied to the modeling of the dayside magnetopause. The model data set is based on magnetopause crossings and on the corresponding hour-averaged measurements of the solar wind plasma and the interplanetary magnetic field. The ANN model represents the three-dimensional shape and size of the dayside magnetopause as a function of the solar wind dynamic pressure and the Bz and By components of the interplanetary magnetic field. The model describes the dynamics of the magnetic cusp and global asymmetry of the dayside magnetopause under a wide range of external conditions. Small variations in the size of the magnetopause as a function of the absolute value of By are presented quantitatively in the form of an analytic expression. The ANN model shows three regimes of magnetopause dynamics that are controlled by the Bz component: a pressure balance regime (Bz>0 nT), a reconnection regime (0>Bz>-10 nT), and a regime of reconnection saturation (Bz<-10 nT). Also, three different regimes of magnetopause dynamics at the subsolar point are described by a modified pressure balance equation obtained from the ANN model.