The capabilities of modern precision nanofabrication and the wide choice of materials [plasmonic metals, high-index dielectrics, phase change materials (PCM), and 2D materials] make the inverse design of nanophotonic structures such as metasurfaces increasingly difficult. Deep learning is becoming increasingly relevant for nanophotonics inverse design. Although deep learning design methodologies are becoming increasingly sophisticated, the problem of the simultaneous inverse design of structure and material has not received much attention. In this contribution, we propose a deep learning-based inverse design methodology for simultaneous material choice and device geometry optimization. To demonstrate the utility of the proposed method, we consider the topical problem of active metasurface design using PCMs. We consider a set of four commonly used PCMs in both fully amorphous and crystalline material phases for the material choice and an arbitrarily specifiable polygonal meta-atom shape for the geometry part, which leads to a vast structure/material design space. We find that a suitably designed deep neural network can achieve good optical spectrum prediction capability in an ample design space. Furthermore, we show that this forward model has a sufficiently high predictive ability to be used in a surrogate-optimization setup resulting in the inverse design of active metasurfaces of switchable functionality.