  In recent years, scientific machine learning, particularly physic-informed
neural networks (PINNs), has introduced new innovative methods to understanding
the differential equations that describe power system dynamics, providing a
more efficient alternative to traditional methods. However, using a single
neural network to capture patterns of all variables requires a large enough
size of networks, leading to a long time of training and still high
computational costs. In this paper, we utilize the interfacing of PINNs with
symbolic techniques to construct multiple single-output neural networks by
taking the loss function apart and integrating it over the relevant domain.
Also, we reweigh the factors of the components in the loss function to improve
the performance of the network for instability systems. Our results show that
the symbolic PINNs provide higher accuracy with significantly fewer parameters
and faster training time. By using the adaptive weight method, the symbolic
PINNs can avoid the vanishing gradient problem and numerical instability.
