In many inverse problems (IPs) in science and engineering, optimization of design parameters (e.g., sensor placement) with experimental design (ED) methods is performed due to high data acquisition costs when conducting physical experiments, and often has to be done up front due to practical constraints on sensor deployments. However, existing ED methods are often challenging to use in practical PDE-based inverse problems due to significant computational bottlenecks during forward simulation and inverse parameter estimation. This paper presents Physics-Informed Experimental Design (PIED), the first ED framework that makes use of PINNs in a fully differentiable architecture to perform continuous optimization of design parameters for IPs. PIED utilizes techniques such as learning of a shared NN parameter initialization, and approximation of PINN training dynamics during the ED process, for better estimation of the inverse parameters. PIED selects the optimal design parameters for one-shot deployment, allows exploitation of parallel computation unlike existing methods, and is empirically shown to significantly outperform existing ED benchmarks in IPs for both finite-dimensional and function-valued inverse parameters given limited budget for observations.