Multiscale topology optimization offers a powerful approach for designing lightweight structures with tailored microstructures. However, its practical implementation is often limited by the high computational cost of numerical homogenization and the complexity of multi-scale interactions. Concurrent multiscale structural optimization seeks to improve macroscale performance through the design of microscale architectures, requiring a design space that spans both scales. This work introduces a deep neural network (DNN)-assisted two-scale topology optimization framework that significantly accelerates multiscale design. A surrogate modeling approach using DNNs is developed to approximate the homogenized constitutive properties of parameterized microstructural cells. The DNN serves as a differentiable model for both microscale properties and material shape sensitivities, offering a key advantage: the ability to capture complex, multidimensional relationships in an explicit and computationally efficient manner. Compared to traditional finite element–based homogenization methods, the surrogate achieves sufficient accuracy and stability while reducing computational cost by orders of magnitude. The proposed framework trains DNNs to learn numerical homogenization for 2D and 3D microstructures, enabling the design of intricate architectures with increased complexity. Numerical examples demonstrate that DNN-based surrogates provide accurate shape sensitivity through backpropagation and yield optimized designs that closely match those obtained using interface-enriched finite element methods. To enhance structural robustness, the surrogate is integrated with a worst-case buckling load factor formulation, addressing both local (microscale) and global (macroscale) buckling constraints under two-scale volume and compliance limitations. The results highlight that the combination of machine learning and two-scale optimization accelerates the design process and produces multiscale structures with superior resistance to buckling, paving the way for practical, high-performance multiscale designs.