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Optical metasurfaces consist of densely arranged unit cells that manipulate light through various light confinement and scattering processes. Due to its unique advantages, such as high performance, small form factor and easy integration with semiconductor devices, metasurfaces have been gathering increasing attention in fields such as displays, imaging, sensing and optical computation. Despite advances in fabrication and characterization, a viable design prediction for suitable optical response remains challenging for complex optical metamaterial systems. The computation cost required to obtain the optimal design exponentially grows as the design complexity increases. Furthermore, the design prediction is challenging since the inverse problem is often ill-posed. In recent years, deep learning (DL) methods have shown great promise in the area of inverse design. Inspired by this and the capability of DL to produce fast inference, we introduce a physics-informed DL framework to expedite the computation for the inverse design of metasurfaces. Addition of the physics-based constraints improve generalizability of the DL model while reducing data burden. Our approach introduces a tandem DL architecture with physics-based learning to alleviate the nonuniqueness issue by selecting designs that are scientifically consistent, with low error in design prediction and accurate reconstruction of optical responses. To prove the concept, we focus on the inverse design of a representative plasmonic device that consists of metal gratings deposited on a dielectric film on top of a metal substrate. The optical response of the device is determined by the geometrical dimensions as well as the material properties. The training and testing data are obtained through Rigorous Coupled-Wave Analysis (RCWA), while the physics-based constraint is derived from solving the electromagnetic (EM) wave equations for a simplified homogenized model. We consider the prediction of design for the optical response of a single wavelength incident or a spectrum of wavelength in the visible light range. Our model converges with an accuracy up to 97% for inverse design prediction with the optical response for the visible light spectrum as input. The model is also able to predict design with accuracy up to 96% and optical response reconstruction accuracy of 99% for optical response of a single wavelength of light as input.