Gravity waves (GWs) and their associated multi-scale dynamics are known to play fundamental roles in energy and momentum transport and deposition processes throughout the atmosphere. We describe an initial, two-dimensional (2-D), machine learning model – the Compressible Atmosphere Model Network (CAMNet) - intended as a first step toward a more general, three-dimensional, highly-efficient, model for applications to nonlinear GW dynamics description. CAMNet employs a physics-informed neural operator to dramatically accelerate GW and secondary GW (SGW) simulations applied to two GW sources to date. CAMNet is trained on high-resolution simulations by the state-of-the-art model Complex Geometry Compressible Atmosphere Model (CGCAM). Two initial applications to a Kelvin-Helmholtz instability source and mountain wave generation, propagation, breaking, and SGW generation in two wind environments are described here. Results show that CAMNet can capture the key 2-D dynamics modeled by CGCAM with high precision. Spectral characteristics of primary and SGWs estimated by CAMNet agree well with those from CGCAM. Our results show that CAMNet can achieve a several order-of-magnitude acceleration relative to CGCAM without sacrificing accuracy and suggests a potential for machine learning to enable efficient and accurate descriptions of primary and secondary GWs in global atmospheric models.