Iterative gradient methods such as Levenberg-Marquardt (LM) are in widespread use for source localization from electroencephalographic (EEG) and magnetoencephalographic (MEG) signals. Unfortunately, LM depends sensitively on the initial guess, necessitating repeated runs. This, combined with LM's high per-step cost, makes its computational burden quite high. To reduce this burden, we trained a multilayer perceptron (MLP) as a real-time localizer. We used an analytical model of quasistatic electromagnetic propagation through a spherical head to map randomly chosen dipoles to sensor activities according to the sensor geometry of a 4D Neuroimaging Neuromag-122 MEG system, and trained a MLP to invert this mapping in the absence of noise or in the presence of various sorts of noise such as white Gaussian noise, correlated noise, or real brain noise. A MLP structure was chosen to trade off computation and accuracy. This MLP was trained four times, with each type of noise. We measured the effects of initial guesses on LM performance, which motivated a hybrid MLP-start-LM method, in which the trained MLP initializes LM. We also compared the localization performance of LM, MLPs, and hybrid MLP-start-LMs for realistic brain signals. Trained MLPs are much faster than other methods, while the hybrid MLP-start-LMs are faster and more accurate than fixed-4-start-LM. In particular, the hybrid MLP-start-LM initialized by a MLP trained with the real brain noise dataset is 60 times faster and is comparable in accuracy to random-20-start-LM, and this hybrid system (localization error: 0.28 cm, computation time: 36 ms) shows almost as good performance as optimal-1-start-LM (localization error: 0.23 cm, computation time: 22 ms), which initializes LM with the correct dipole location. MLPs trained with noise perform better than the MLP trained without noise, and the MLP trained with real brain noise is almost as good an initial guesser for LM as the correct dipole location.