Recent studies show promising performance gains achieved by non-linear equalization using neural networks (NNs) over traditional linear equalization in two-dimensional magnetic recording (TDMR) channels. But the examined neural network architectures entail much higher implementation complexities compared with the linear equalizer, which precludes practical implementation. For example, among the low complexity reported architectures, the multilayer perceptron (MLP) requires about 6.6 times increase in complexity over the linear equalizer. This paper investigates candidate reduced complexity neural network architectures for equalization over TDMR. We test the performance on readback signals measured over an actual hard disk drive with TDMR technology. Four variants of a reduced complexity MLP (RC-MLP) architecture are proposed. A proposed variant achieves the best balance between performance and complexity. This architecture consists of finite-impulse response filters, a nonlinear activation, and a hidden delay line. The complexity of the architecture is only 1.59 times the linear equalizer's complexity, while achieving most of the performance gains of the MLP.