Although Maxwell discovered the physical laws of electromagnetic waves 160 years ago, how to precisely model the propagation of an RF signal in an electrically large and complex environment remains a long-standing problem. The difficulty is in the complex interactions between the RF signal and the obstacles (e.g., reflection, diffraction, etc.). Inspired by the great success of using a neural network to describe the optical field in computer vision, we propose a neural radio-frequency radiance field, NeRF$^\textbf{2}$, which represents a continuous volumetric scene function that makes sense of an RF signal's propagation. Particularly, after training with a few signal measurements, NeRF$^\textbf{2}$ can tell how/what signal is received at any position when it knows the position of a transmitter. As a physical-layer neural network, NeRF$^\textbf{2}$ can take advantage of the learned statistic model plus the physical model of ray tracing to generate a synthetic dataset that meets the training demands of application-layer artificial neural networks (ANNs). Thus, we can boost the performance of ANNs by the proposed turbo-learning, which mixes the true and synthetic datasets to intensify the training. Our experiment results show that turbo-learning can enhance performance with an approximate 50% increase. We also demonstrate the power of NeRF$^\textbf{2}$ in the field of indoor localization and 5G MIMO.

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