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.

翻译：尽管麦克斯韦在160年前揭示了电磁波的物理定律，但如何在大型复杂电磁环境中精确建模射频信号的传播仍是一个长期难题。其难点在于射频信号与障碍物之间复杂的相互作用（如反射、衍射等）。受计算机视觉领域利用神经网络描述光场取得巨大成功的启发，我们提出了一种神经射频辐射场——NeRF$^\textbf{2}$，它通过表征连续三维空间场景函数来理解射频信号的传播机制。特别地，在基于少量信号测量值完成训练后，NeRF$^\textbf{2}$能够根据发射机位置，推断任意位置接收到的信号类型与强度。作为物理层神经网络，NeRF$^\textbf{2}$可结合学习获得的统计模型与射线追踪物理模型，生成满足应用层人工神经网络(ANNs)训练需求的合成数据集。由此，我们提出一种融合真实与合成数据集的涡轮学习(Turbo-learning)方法，通过增强训练过程将ANN性能提升约50%。实验结果表明，涡轮学习可使性能提升近50%。此外，我们还展示了NeRF$^\textbf{2}$在室内定位与5G MIMO领域中的强大应用潜力。