All-dielectric metasurfaces exhibit exotic electromagnetic responses, similar to those obtained with metal-based metamaterials. Research in all-dielectric metasurfaces currently uses relatively simple unit-cell designs, but increased geometrical complexity may yield even greater scattering states. Although machine learning has recently been applied to the design of metasurfaces with impressive results, the much more challenging task of finding a geometry that yields a desired spectra remains largely unsolved. We propose and demonstrate a method capable of finding accurate solutions to ill-posed inverse problems, where the conditions of existence and uniqueness are violated. A specific example of finding the metasurface geometry which yields a radiant exitance matching the external quantum efficiency of gallium antimonide is demonstrated. We also show how the neural-adjoint method can intelligently grow the design search space to include designs that increasingly and accurately approximate the desired scattering response. The neural-adjoint method is not restricted to the case demonstrated and may be applied to plasmonics, photonic crystal, and other artificial electromagnetic materials.