WINERT: TOWARDS NEURAL RAY TRACING FOR WIRELESS CHANNEL MODELLING AND DIFFEREN-TIABLE SIMULATIONS

Abstract

In this paper, we work towards a neural surrogate to model wireless electromagnetic propagation effects in indoor environments. Such neural surrogates provide a fast, differentiable, and continuous representation of the environment and enables end-to-end optimization for downstream tasks (e.g., network planning). Specifically, the goal of the paper is to render the wireless signal (e.g., timeof-flights, power of each path) in an environment as a function of the sensor's spatial configuration (e.g., placement of transmit and receive antennas). NeRFbased approaches have shown promising results in the visual setting (RGB image signal, with a camera sensor), where the key idea is to algorithmically evaluate the 'global' signal (e.g., using volumetric rendering) by breaking it down in a sequence of 'local' evaluations (e.g., using co-ordinate neural networks). In a similar spirit, we model the time-angle channel impulse response (the global wireless signal) as a superposition of multiple paths. The wireless characteristics (e.g., power) of each path is a result of multiple evaluations of a neural network that learns implicit ray-surface interaction properties. We evaluate our approach in multiple indoor scenarios and demonstrate that our model achieves strong performance (e.g., <0.33ns error in time-of-flight predictions). Furthermore, we demonstrate that our neural surrogate whitens the 'black-box' wireless simulators, and thus enables inverse rendering applications (e.g., user localization).

1. INTRODUCTION

Realistic simulations of physical processes are vital to many scientific and engineering disciplines. In this paper, we focus on simulation of wireless electromagnetic (EM) signals within a propagation environment. The physics of such EM wave propagation between a transmit and receive point are analytically given by Maxwell equations: the transmitted wave undergoes different interactions with the environment (e.g., reflection), and the receiver gets the wave through multiple paths with different time-of-flights and powers, and from different directions. However, solving the Maxwell equations with boundary conditions requires in-depth knowledge of the propagation environment, hence classically modelling EM propagation is intractable for most engineering applications. Existing techniques make such simulations tractable by trading-off accuracy for speed. At one end of the spectrum, such simulations are represented in a statistical sense where a probabilistic model roughly captures the marginalized distribution over time-of-flights, gains and direction of transmit-receive paths. However, this level of accuracy is insufficient for designing systems that efficiently operate in high frequency bands. This motivates solutions at the other end of the spectrum: wireless ray tracing simulators. Given a detailed CAD representation of the environment along with the material properties, and numerous wireless configuration parameters (e.g., placement of a base station), the simulators generate resulting propagation characteristics. Although wireless ray tracing simulators are appealing, there are a few drawbacks. First, they are generally slow, which poses a bottleneck for closed-loop design pipelines, as wireless configurations cannot be quickly mapped to propagation characteristics. Second, because they are non-differentiable, they are not amenable with inverse physical design formulations, for example optimizing base station placement with the simulator in the optimization loop. Third, they usually require additional fine-tuning with real data as they are not data-driven. Calibrating them with realworld measurements is non-trivial and tedious. Fourth, they cannot generally inter-operate with probabilistic frameworks which have the advantage of better dealing with epistemic uncertainties. We believe neural surrogates provide a natural solution to circumvent many of these drawbacks of classical ray tracing simulators. In this work, we propose a neural wireless simulator ('WiNeRT') by building on recent advances in scenes representation as continuous-function neural networks (Sitzmann et al., 2019; Tancik et al., 2020; Mildenhall et al., 2020) . In particular, central to our approach is learning a network to model ray-surface interactions, i.e., the network transforms an incident wireless ray to an attenuated outgoing ray. By shooting out a number of rays and evaluating the network at relevant spatial regions in the environment, we estimate the wireless characteristics as a set of transmit-receive paths, each path encodes attributes such as time-of-flight and gain. Our approach also addresses some unique technical challenges posed by the non-visual wireless modality, such as dealing with sparse highdimensional time-angle measurement signals. We demonstrate that our neural wireless simulator reasonably renders the wireless propagation aspects by evaluating on two datasets which captures 50-100 m 2 indoor propagation scenes. Interestingly, we find that the 3D-structure-aware implicit formulation is a strong inductive bias and helps generalization to significant inference-time distributions shifts. Finally, we demonstrate the potential of our differentiable forward model in solving inverse problem by tackling the user localization problem after posing it as an inverse rendering problem. Our results indicate that simulator physics for specified environments can be 'distilled' into neural surrogates and thereby presenting first steps towards closed-loop design pipelines of wireless communication systems.

Physics-based Neural Simulations.

There exists a wide body of literature to model physical processes using advances in neural networks (Djeumou et al., 2022; Karniadakis et al., 2021; Raissi et al., 2017) . As simulating physical processes can be expensive and can also present nondifferentiable 'black-box' in design pipelines, recent literature addresses how to work towards neural surrogates, such as for particle simulation (Sanchez-Gonzalez et al., 2020) , mesh simulations (Pfaff et al., 2020) , design of particle accelerators (Shirobokov et al., 2020) , and inverse kinematics (Sun et al., 2021) . In this paper, we are particularly interested in a specific physical process -wireless EM-wave propagation. Although this has received limited recent attention (Xia et al., 2020) in a 3Doblivious setting, it is unclear whether these extend to complex configurations. Consequently, in this work, we work towards the first 3d-structure-aware surrogates for wireless ray tracing simulation. Neural Channel Modelling. Although propagation channel modeling has been a central topic in wireless communication (Jakes & Cox, 1994; Lee, 1982; Rappaport et al., 2022) , there has been a recent trend for fully data-driven models. The main paradigm of these activities is to use machine learning to learn complex distributions, model non-linearities and have differentiable simulators. These works can be categorized as statistical channel models where the channel input-output relation is modelled as a conditional probability distribution. Many works leverage recent advances in generative modelling and use models like generative adversarial networks (GANs) (Goodfellow et al., 2014) or variational autoencoders (VAEs) (Kingma & Welling, 2013) to learn the channel model (O'Shea et al., 2019; Ye et al., 2018; Yang et al., 2019; O'Shea et al., 2019; Orekondy et al., 2022; Ye et al., 2020; Dörner et al., 2020) . In contrast to these works, our approach inscribes within ray tracing channel modeling paradigm, where wireless propagation is precisely modelled by tracing wireless rays, however, unlike classical ray tracers, our model is able to blend in the elements of statistical modeling and is trainable directly on field data. To the best of our knowledge, this work is the first differentiable neural ray tracer for wireless channel modelling. Neural Scene Representations. Representing scenes (or more generally signals) has been widely studied in literature, such as encoding the signal in the latent space of a generative model (Kingma & Welling, 2013; Goodfellow et al., 2014) . A more recent link of work encodes the signal in the parameters of a co-ordinate MLP (Park et al., 2019; Sitzmann et al., 2020; Tancik et al., 2020; Fathony 

