Neural networks are highly prized today in solving engineering problems due to the flexibility and efficiency they give. However, the applications of such networks often require large data for training, which is not feasible when data availability is challenging. The physics-informed neural networks (PINNs) bypass this challenge by employing governing differential equations and boundary conditions without requiring large datasets. A study focuses on using PINNs to solve nonlinear equations. Specifically, it considers ordinary (ODE) and partial differential equations (PDE) with general nonlinearities. A hybrid approach of PINN for solving these nonlinear equations will be developed within the discrete-time PINN framework based on a discontinuity indicator classifying the local solution structures as distinguishing between smooth and non-smooth scales. It also resolves the smooth scales automatically with an automatic differentiation scheme and captures the discontinuities by an enhanced WENO-Z (weighted essentially non-oscillatory) scheme. The hybrid PINNWZ (WENO-Z) presents results for the viscous and inviscid Burgers’ equation with better approximations for discontinuous solutions than the discrete-time PINN traditional setting, even for larger time steps.