This article introduces physics-informed neural networks to simulate THz time-domain spectroscopy (THz-PINNs) with a comprehensive analysis of forward problems in the time domain. The analysis demonstrated the feasibility to accurately capture dispersive behavior of THz waves in the medium over the broad frequency range from about 0.1 THz to 10 THz with THz-PINNs methods with significantly less computation costs compared to finite-difference time-domain (FDTD) methods. Using the auxiliary differential equation (ADE) method to construct new loss functions and incorporating Drude-Lorentz model parameters as training variables improved the simulation accuracy particularly near the sample boundary.