Neural networks became prominent in engineering problem-solving because of their efficiency and flexibility. Their use is restricted by their need for humongous training datasets, however, which restricts applicability in low-data environments. Physics-Informed Neural Networks (PINNs) overcome this limitation by including governing differential equations and boundary conditions without large datasets. This paper suggests a hybrid PINNWZ methodology in solving nonlinear Ordinary (ODEs) and Partial Differential Equations (PDEs). The method uses a discontinuity detector to label local solution topologies, dividing them into smooth and non-smooth scales. Automatic differentiation addresses the smooth regions, while a Weighted Essentially Non-Oscillatory (WENO-Z) scheme is modified to effectively identify discontinuities. Numerical simulations demonstrate improved approximations for the viscous and inviscid Burgers equations relative to standard discrete-time PINNs, especially for large time steps. The outcomes demonstrate the hybrid PINNWZ’s capability to enhance the accuracy of the solution in discontinuous problems and thereby make it a valuable contribution to the solution of nonlinear differential equations.