Accurate and efficient fluid flow models are essential for applications relating to many physical phenomena, including geophysical, aerodynamic, and biological systems. While these flows may exhibit rich and multiscale dynamics, in many cases, underlying low-rank structures exist, which describe the bulk of the motion. These structures tend to be spatially large and temporally slow and may contain most of the energy in a given flow. The extraction and parsimonious representation of these low-rank dynamics from high-dimensional data is a key challenge. Inspired by the success of physics-informed machine learning methods, we propose a spectrally informed approach to extract low-rank models of fluid flows by leveraging known spectral properties in the learning process. We incorporate this knowledge by imposing regularizations on the learned dynamics, which bias the training process toward learning low-frequency structures with corresponding higher power. We demonstrate the effectiveness of this method to improve prediction and produce learned models, which better match the underlying spectral properties of prototypical fluid flows.