A major challenge in DNA nanotechnology is to extend the length scale of DNA structures from the nanoscale to the microscale for biomedical, sensing, optical, and soft robotics applications. Self-assembly of DNA origami building blocks provides a promising approach for fabricating such higher-order structures. Inspired by self-assembly of patchy colloidal particles, researchers have recently begun to introduce patches of mutually attractive moieties, including non-natural hydrophobic polynucleotide brushes, at designated sites on DNA origami to assemble them into complex higher-order architectures. However, the underlying relationship between the design of these DNA origami building blocks and the resulting assembly structure is complex. Machine learning is especially well suited for such inverse-design tasks. In this work, we developed a coarse-grained model of DNA origami nanocubes grafted with hydrophobic brushes and employed neural adjoint (NA) method to explore highly ordered target assemblies, including checkerboard, honeycomb, and Kagome lattices. We envision that such inverse design approaches can be generalized to more complex designs and used to tailor structural properties to expand the application space of DNA nanotechnology.