Physics-informed neural networks (PINNs) has been shown to be an effective tool for solving partial differential equations (PDEs). PINNs incorporate the PDEs residual into the loss function, seamlessly integrating it as part of the neural network architecture. This novel methodology has exhibited success in tackling a wide range of both forward and inverse PDE problems. However, a limitation of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose two advanced methodologies: PINNs utilizing an H1 loss function and parallel physics-informed neural networks (P-PINNs). The former involves modifying the loss function with the H1 norm, while the latter entails solving the coupled Schrödinger-Korteweg–de Vries (Sch-KdV) equation separately with two parallel networks. These advancements are designed to enhance both accuracy and training efficiency. We have extensively tested these two advanced methods through a series of experiments and demonstrated the accuracy and effectiveness in approximation of the Sch-KdV system.