This paper presents new structure and adaptation criterion for equalization of two-dimensional magnetic recording channels, as opposed to typical linear equalizer with minimum mean square error (MMSE) as adaptation criterion. To compensate for the nonlinear channel noise, we propose a neural network based nonlinear equalizer and show it outperforms linear equalizer under the same criterion. To achieve minimum bit error rate (BER) at the detector output, we propose to adapt the equalizer with cross entropy between the true probability of the bit and detector's estimate of it. We show minimizing the cross entropy enables maximum likelihood adaptation, and results in lower detector BER than the MSE criterion. Several variations of nonlinear equalizer structures with cross entropy criterion are investigated. Compared to linear MMSE equalizer, the proposed scheme can provide up to 22.76% detector BER reduction with only 6x increase in complexity.