Astrophotonics offers a pathway towards compact and stable spectrographs for future large telescopes. The size of these spectrographs does not scale with the diameter of the telescope aperture; thus, they offer a cost-effective way to achieve astronomy science requirements compared to traditional optical spectrographs. A major limitation of current on-chip astrophotonic spectrographs is the significant loss incurred at interfaces between distinct components, particularly at the input and output fiber interfaces, limiting overall throughput. In this work, we investigate inverse-designed architectures to optimize transitions between these components to improve throughput. Inverse design formulates device design as an optimization problem, enabling non-intuitive geometries that can achieve high performance. We focus on optimizing coupling between an arrayed waveguide grating (AWG) spectrograph and an input single-mode fiber. Our approach explores two classes of fabrication-constrained inverse design. The first is shape optimization, which optimizes the geometry of the boundary of a coupling taper. The second is topology optimization, which optimizes the material distribution in a design region to gradually transform the optical mode between components. Both methods are implemented using the Tidy3D package from Flexcompute. We use germanium-doped silica for the core of the photonic components and silica as the cladding, and we use an SMF-28 fiber as input. This work demonstrates a pathway to improve astrophotonic spectrographs through computational inverse-design methods. Leveraging higher throughput, we can improve signal-to-noise ratios offered by photonic devices and position ourselves to design high-performance, cost-effective instrumentation for the next generation of telescopes, both on the ground and in space.