Mechanical metamaterials derive their effective properties primarily from their microstructural geometry rather than their constituent material alone. Designing such microstructures to achieve desired mechanical properties requires systematic and efficient optimization approaches. In this work, a neural network-based topology optimization framework is proposed for designing two-dimensional mechanical metamaterial unit cells with tailored elastic properties. The optimization problems are formulated for different objectives: maximizing the bulk modulus, shear modulus, and effective elastic modulus, and minimizing Poisson’s ratio. In the proposed methodology, a neural network predicts the pseudo-density distribution over the unit-cell domain. The predicted density field is subsequently used to perform finite element analysis (FEA) to evaluate the effective mechanical properties through numerical homogenization. A loss function is constructed based on the target elastic property. The gradients of the loss function are backpropagated through the finite element analysis pipeline to update the neural network parameters via gradient-based optimization. This framework allows the generation of optimized metamaterial unit-cell layouts without relying on conventional iterative density-update schemes. Numerical results demonstrate that the proposed framework can discover mechanically efficient unit-cell topologies with improved stiffness characteristics and reduced Poisson’s ratio. The results suggest that neural network-assisted topology optimization offers a promising alternative to conventional density-based optimization methods for computational metamaterial design. The proposed methodology shows potential for the discovery of microstructures with tailored mechanical properties for lightweight and multifunctional engineering applications.