Neural   network (NN) solvers for partial differential equations (PDE) have been widely used in simulating complex systems in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDEs, without explicitly considering intrinsic physical properties such as mass conservation or energy dissipation. This limitation   can result in unstable or nonphysical solutions, particularly in long-term   simulations. To address this issue, we propose Sidecar, a novel framework   that enhances the accuracy and physical consistency of existing NN solvers by incorporating structure-preserving knowledge. This framework builds upon our  previously proposed TDSR-ETD method for solving gradient flow problems, which satisfies discrete analogues of the energy-dissipation laws by introducing a time-dependent spectral renormalization (TDSR) factor. Inspired by this approach, our Sidecar framework parameterizes the TDSR factor using a small   copilot network, which is trained to guide the existing NN solver in preserving physical structure. This design allows flexible integration of the structure-preserving knowledge into various NN solvers and can be easily extended to different types of PDEs. Our experimental results on a set of benchmark PDEs demonstrate that it improves the existing neural network   solvers in terms of accuracy and consistency with structure-preserving   properties.