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Self-interference (SI) is considered as a main challenge in full-duplex (FD) systems. Therefore, efficient SI cancelers are required for the influential deployment of FD systems in beyond fifth-generation wireless networks. Existing methods for SI cancellation have mostly considered the polynomial representation of the SI signal at the receiver. These methods are shown to operate well in practice while requiring high computational complexity. Alternatively, neural networks (NNs) are envisioned as promising candidates for modeling the SI signal with reduced computational complexity. Consequently, in this paper, two novel low complexity NN structures, referred to as the ladder-wise grid structure (LWGS) and moving-window grid structure (MWGS), are proposed. The core idea of these two structures is to mimic the non-linearity and memory effect introduced to the SI signal in order to achieve proper SI cancellation while exhibiting low computational complexity. The simulation results reveal that the LWGS and MWGS NN-based cancelers attain the same cancellation performance of the polynomial-based canceler while providing 49.87% and 34.19% complexity reduction, respectively.