Full-waveform inversion is a powerful geophysical imaging technique that infers high-resolution subsurface physical parameters by solving a non-convex optimization problem. However, due to limitations in observation, e.g. limited shots or receivers, and random noise, conventional inversion methods are confronted with numerous challenges, such as the local-minimum problem. In recent years, a substantial body of work has demonstrated that the integration of deep neural networks and partial differential equations for solving full-waveform inversion problems has shown promising performance. In this work, drawing inspiration from the expressive capacity of neural networks, we provide a new deep learning approach aimed at accurately reconstructing subsurface physical velocity parameters. This method is founded on a re-parametrization technique for Bayesian inference, achieved through a deep neural network with random weights. Notably, our proposed approach does not hinge upon the requirement of the labeled training dataset, rendering it exceedingly versatile and adaptable to diverse subsurface models. Furthermore, uncertainty analysis is effectively addressed through approximate Bayesian inference. Extensive experiments show that the proposed approach performs noticeably better than existing conventional inversion methods.