  Physics-Informed Neural Networks (PINNs) can be regarded as general-purpose
PDE solvers, but it might be slow to train PINNs on particular problems, and
there is no theoretical guarantee of corresponding error bounds. In this
manuscript, we propose a variant called Prior Dictionary based Physics-Informed
Neural Networks (PD-PINNs). Equipped with task-dependent dictionaries, PD-PINNs
enjoy enhanced representation power on the tasks, which helps to capture
features provided by dictionaries so that the proposed neural networks can
achieve faster convergence in the process of training. In various numerical
simulations, compared with existing PINN methods, combining prior dictionaries
can significantly enhance convergence speed. In terms of theory, we obtain the
error bounds applicable to PINNs and PD-PINNs for solving elliptic partial
differential equations of second order. It is proved that under certain mild
conditions, the prediction error made by neural networks can be bounded by
expected loss of PDEs and boundary conditions.
