We introduce a new type of neural network--the dynamic wave expansion neural network (DWENN)--for path generation in a dynamic environment for both mobile robots and robotic manipulators. Our model is parameter-free, computationally efficient, and its complexity does not explicitly depend on the dimensionality of the configuration space. We give a review of existing neural networks for trajectory generation in a time-varying domain, which are compared to the presented model. We demonstrate several representative simulative comparisons as well as the results of long-run comparisons in a number of randomly-generated scenes, which reveal that the proposed model yields dominantly shorter paths, especially in highly-dynamic environments.