WiNeRT: Towards Neural Ray Tracing for Wireless Channel Modelling and Differentiable Simulations
Authors: Tribhuvanesh Orekondy
Pratik Kumar
Shreya Kadambi
Hao Ye
Joseph Soriaga
Arash Behboodi
Venue: ICLR 2023 poster (2023)
Keywords: neural rendering
wireless
ray tracing
nerf

Abstract: In this paper, we work towards a neural surrogate to model wireless electro-magnetic 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., time-of-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). NeRF-based 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).

Published as a conference paper at ICLR 2023
WINERT: TOWARDS NEURAL RAY TRACING FOR WIRELESS CHANNEL MODELLING AND DIFFEREN- TIABLE SIMULATIONS
Tribhuvanesh Orekondy, Kumar Pratik, Shreya Kadambi, Hao Ye,
Joseph Soriaga, Arash Behboodi Qualcomm Al Research"
ABSTRACT
In this paper, we work towards a neural surrogate to model wireless electro- magnetic propagation effects in indoor environments. Such neural surrogates pro- vide 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., time- of of-flights, power of each path) in an environment spatial configuration (e.g.. placement of transmit and receive antennas). NeRF- based 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 se- quence of 'local' evaluations (e.g., using co-ordinate neural networks). In similar spirit, we model the time-angle channel impulse response (the global wireless sig- nal) as superposition of multiple paths. The wireless characteristics power) of each path is a result of multiple evaluations of neural network that learns implicit ray-surface interaction properties. We evaluate our approach in multi- ple indoor scenarios and demonstrate that our model achieves strong performance (e.g., <0.33ns error in tmeeof-flight predictions). Furthermore, demonstrate that our neural surrogate whitens the 'black-box' wireless simulators, and thus enables inverse rendering applications (e.g., user localization).
function of the sensor's 's
INTRODUCTION
Realistic simulations of physical processes vital many scientific and engineering disciplines. In this paper, we focus on simulation of wireless electromagnetic (EM) signals within propagation environment. The physics of such EM wave propagation between 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-iie offffights 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 statistical sense where 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 detailed CAD representation of the environment along with the material propertics, and numerous wireless configuration parameters (c.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 bottleneck for closed-loop design pipelines, as wireless configura- tions cannot be quickly mapped to propagation characteristics. Second, because they are non-
"Qullcomm AI Research initiative of Qualcomm Technologies, Inc

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differentiable, they are not amenable with inverse physical design formulations, for example op- timizing 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 real- world 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 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 caatinuous-functiin neural networks (Sitzmann et al. 2019 Tancik al., 2020 Mildenhall et al. (2200. 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 out- going ray. By shooting out a number of rays and evaluating the network at relevant spatial regions the environment, we estimate the wireless characteristics as a set of transmit-receive paths, each path encodes attributes such 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 high- dimensional time-angle measurement signals.
We demonstrate that our neural wireless simulator reasonably renders the wireless propagation as- pects by evaluating two datasets which captures 50-100 m² indoor propagation scenes. Interest- ingly, we find that the 3D-struccure-aware implicit formulation is a strong inductive bias and helps generalization to significant inference-time distributions shifts. Finally, we demonstrate the poten- tial 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 closedllop design pipelines of wireless communication systems.
RELATED WORK
Physics-based Neural Simulations. There exists wide body of literature to model physi- cal processes using advances in neural networks (Djeumou et al.) 2022, Karniadakis et al. [02T] Raissi et al. 2017). As simulating physical processes can be expensive and can also present non- differentiable black-boo'' in design pipelines, recent literature addresses how to work towards neural surrogates, such as for particle simulation SSanchez-Gozzale et al.) (020),, mesh simulations (Pfaff ctt al. 2020), design of particle accelerators (Shirobokov et al. (020), and inverse kinematics (Sun al. 2021. In this paper, we are particularly interested in specific physical process wireless EM-wave propagation. Although this has received limited recent attention (Xia et al.] (020)) in a 3DD oblivious setting, is unclear whether these extend 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 central topic in wireless communication (Jakes & Cox 1994; Lee 1982, Rappaport et al., (022), 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 re- lation is modelled as 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 "heahee et al. 2019 Ye et al., 2018 Yang et al.) 2019] Sheea et al.2001) Orekondy 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 trac- ing wireless rays, however, unlike classical ray tracers, 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 encoding the signal in the latent space of generative model (Kingmal & Welling 2013: Goodfellow et al. 2014). A more recent link of work encodes the signal in the pa-
rameters of co-ordinate ML.P (Par al. 2019
Sitzmann et al. 2020 Tancik et al.] 2020 Fathony
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render, ()
Floor laout
perray's
Ray marching
Launch rays
fo
fa
Aggregate
A rays
Spatell 00 co-ordiaates
Nex
Figure 1: Approach Overview. We learn a forward simulator renderø()) that maps an environment config- uration to a wireless channel ft,. Here, f;; is a set of wireless propagation paths between 2nx- (green rays in right image), each path encoding certain channel attributes e.g.. path gain.
et al. 2020), thereby mapping co-ordinates (e.g., spatial, temporal) to the signal intensity values (e.g.. pixel intensity, amplitude). In specific case where the signal is 2D RGB image, recent works (Schwarz et al., 2020); Niemeyer & Geiger 2021 Mildenhall et al. 2020) show promising results by additionally employing image-based differentiable rendering paradigms (Drehin et al., 1988 Liu et al., (219)) to recover 3D properties of the scene. Inspired by this idea, our work neurally represents wireless scene by tackling a set of orthogonal challenges, such as dealing with sparse high-din digh-dimensional signals and particularly modelling reflection and transmission effects. Consequently, we work towards the first 3D-aware neural 'wireless' scene representation model.
APPROACH
In this section, we begin with some preliminaries to the subsequent formulation of the neural wire- less ray tracing problem. We then provide an initial overview of our approach Sec. 3.1 and then dive deeper into specific technical aspects of wireless ray marching in Sec. 3.2
Preliminaries: Wireless Channels Scattering, reflection and diffraction are among the main effects
in electromagnetic propagation. A general mathematical description of wireless channel, seen as linear time varying system, given by its impulse response T'e Viswanath (2005); Rappaport 1996). A general model can be written (Samimi & Rappaport] 2016):
(t, O D
Σ as (t) TM (x())) (t)))
(1)
where (t+ is the complex gain, Tt(t) is the delay (time-of-flighl) of path k, @x(t) is azimuth and elevation angle of departure (AoD), and (t) is azimuth and elevation angle of arrival (AoA). Going forward, we use Φ2 = (@x, ; (x)) ) as a shorthand to collectively represent all angles. Intu- itively equation [ represents each path as dirae function in time-angle space. The task of channel modeling can, therefore, be reduced to predicting channel attributes (ax(t) Th((), (f)) for a given environment map, and transmit and receive location. See Sec. A.I for detailed discussion.
Forward Model: renders. The general goal of our forward model is to run a wireless ray simulation given a certain configuration of the propagation environment. More specifically, as shown in Figure 1. the model takes three configuration parameters as input: a 3D representation of the environment F and the spatial co-ordinates of the transmitter Inx and receiver Tx devices. The model predicts the wireless scene as:
h == { {(ax, Tki Φz)) renderg Txx, F)
where the output is a variably-sized set of K paths. Each path uy encodes three channel attributes: gain ag, time-of-flight тı and angles di.- With these predicted channel attributes, can obtain time-angle impulse response (the "channel') to characterize the wireless propagation effects.
Key Idea: Implicit Representation Network fo- Our approach recursively constructs the channel by using a learnt function fe F x u + u (r+1) As shown in Figure given an initial ray u (-=0) model the final state as evaluation of interactions that the ray undergoes with the environment F. Intuitively, fe models the local interaction of any given ray l either in free-space, or in particular when it is incident on an interacting surface. In the latter case of ray-surface interaction, we leverage a co-ordinate MI. to predict the transformation (e.g., attenuation, rotation) to the incident ray.
Representing Environment F. We primarily focus indoor propagation environments in this paper, where the environment is 3D geometric representation. Specifically, we consider the envi- ronment represented 3D mesh composed of F faces and V vertices, where each face corresponds
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Environment Mesh
_
Ap 0
Ray Trangle intersection
(r+1)
Ray-Surface fä
d
Reception/ Tomination?
Free-space Interaction fd
(1-8)
Og
gain
Ray
Figure 2: Renderer: Ray Marching Steps. At each step r of the simulation, we learn the transformation
introduced on a ray " c.g., reflection off a particular surface. The final transformation is a result of Iearnt (green blocks) and non-learnabee (blue blocks) evaluations.
to some surface on a wall. We consider mesh structure with two subtleties: (a) we represent walls as flattened polygon and thereby do not explicitly consider its thickness; and (b) we do encode materials of the corresponding wall faces, but rather learn the properties implicitly from data.
3.1 OVERVIEW: NEURAL RENDERING
In this section, we present an overview the three steps in our approach (as shown in Fig. !
Ray Launching. We begin by shooting out fixed set of K rays from the transmitter location 2 (r=0) = (vk)). We launch the rays ooni-directionally from the transmitter co-ordinate, agnostic to the environment and location of the receiver location. Direction d of each ray is oriented in the direction of a unique vertex of a ico-sphere centered at Tx.. use the number of sub-divisions of the ico-sphere trade-off between computational complexity and accuracy.
Ray Marching. The crux our approach involves 'marching' the ray and accounting for inter- actions (e.g., transmission) with various surfaces of the environment. A key aspect here is using neural network to make local evaluations: mapping an incident ray with some direction and power to updated outgoing attenuated ray. The neural network is hence tasked to learn complex non- linear characterization of the surface materials at spatial co-ordinate. We further elaborate on the ray marching procedure in the next section.
Ray Aggregation and Reception. Of the K rays launched from the ray launching step, we are now interested in the subset of the rays that impinges the receiver. We model the reception sphere with a specific radius, which can be tuned to achieve a desired level of precision. To mitigate double-counting of received rays, filter rays by associating them with unique interaction path.
3.2 RAY MARCHING
We now dive deeper into the ray marching step, which tracks the evaluation of each ray as it propa- gates in the environment and hits various surfaces. We walk through the steps as shown sequentially in Fig. We begin with a set of geometric rays us (r=0) originating at the transmitter co-ordinate Txx- addition the channel attributes of each ray (see Eq. (). we also consider in this section an
additional set of meta-attributs (e.g., origin æg. direction d+) that helps us with the ray marching steps (elaborated in Sec. A.2)
Cay-Environment Intersections. For each ray, we evaluate its first interaction with the environ- ment (e.g., first wall hits). Representing the ray geometry as p(t) x + ttd(") we are primarily interested in a solution >> for which the ray is incident on some surface. This location helps us
(r+1) for the subsequent step. determine the relay (i.e., new origin) æ
Ray-Surface Interaction. While the previous step solves for where the ray is incident in the environment, crucial next step is determining attributes of the outgoing ray result of this interaction. We specifically focus on determining two attributes in this step: the new direction (r+1) and gain ag
(r+1) Popular non-neural simulators, such Remcom (2022). look-up frequency-

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dependent material properties (e.g., conductivity, permittivity) at the incidence point from standard databases (ITU-R P.2040-2) to calculate the attributes of the outgoing ray. However, it is unclear how to calculate the attributes with imprecise knowledge of the surfaces (e.g., unknown thickness and material types of each layer in a wall) or when the material properties of a layer have not been previously empirically analyzed. Our solution is instead predict the attributes using learnt
(r+1) and direction d' (see fg in Fig. 2 The ray-surface interaction network fa used in our experiments is a ReLU MLP with 3 layers (with 64- hidden units). Similar to NeRF (Mildenhall et al. 2020), we split the network into learning incident direction-indedd inteeendenndeeed and dependent features by concatenating direction d(d) with bottlenecked outputs of the penultimate layer in the network (See Sec. A.3 fore more details). The network predicts an attenuation factor s and a rotation matrix A (4-dim Euler-Rodrigues parameterization), which is then used to determine the updated gain (ak (r+1) sa(") and direction (d. = )dd").
network as a function of the incident location zkk
Reception/Terminattii check. For some special cases, we halt ray marching for subset of rays. Namely, when ray k impinges on a reception sphere of a pre-specified radius (30em in our experiments). This prevents future version of the already received ray being potentially incorrectly received at a future iteration. In addition, for computation reasons, we also terminate ray marching the ray exits the region of interest (e.g., ray exiting the environment).
Free-space interaction. While the previous steps modeled the interaction of material properties of the environment on wireless propagation, we now switch focus to free-space. In this case, we model
propagation of a ray using the empirically- which represents the power at the received at the receive antenna P, as a function of the power fed into transmitting antenna Pt, and the distance travelled by the ray d. We learn the remaining scalar parameters G (antenna gain constant), À (attenuation factor), and dg (reference distance).
Friis' Equation: (r(d)) P.G ( (d ) (d 2 dg)
3.3 TRAINING
Over the previous sections, we walked through our approach on predicting channel h render Lxx, F). We train the model in supervised setting, with ground-truth time-angle impulse response measurements. Importantly, we rely only on final measurements (i.e., at r R) for training and do not use any intermediate information (e.g., interaction data through a ray tracer).
Set-based Channel Loss. and ground-truth
We compare two sets of multi-path channels: predictions h { (u)) { to provide a supervisory signal for training. We evaluate the set- based loss as: Lehan (h, h) E, d(u,, (n(()), which has two key ideas: (a) correspondence II: we associate each ground-truie truth path u, with a predicted path ún = I[()). To perform such an associa- information and thereby pair paths launched in approximately the same direction; and (b) inter-path distance d(u,, ig): to compare two paths, we use mean square er- ror for valuur- attributes (e.g., time-of-flights and cosine distances between angular- aetribss (e.g., direction of arrival). For the latter, we represent angles as unit vectors in cartesian coordinates.
tion, we
Training and Implementation Details. We train approach for 100 epochs using Adam opti- mizer with learning rate of 10-3, We found it crucial to not aggregate rays (Sec. 3.1) in the training steps, as it led vanishing gradients due negligible number of rays that contributed towards gra- dient updates. We model the reception sphere as a fixed-sized sphere of radius 30cm. Additional implementation details are provided in Sec. C.4
EXPERIMENTAL RESULTS
In this section, we discuss experimental analysis of our neural simulator approach. We begin by dis- cussing the preliminaries: the choice of datasets and the evaluation metrics to compare simulations. The section concludes by discussing overall performances and highlights certain benefits of neural simulations, such as running controllable simulations outside of training conditions.
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4.1 EXPERIMENTAL SETUP: DATASETS, EVALUATION METRICS, AND BASELINES
We train and evaluate our algorithm using ground-truth data from wireless ray tracing packages. We collect two datasets, where each dataset contains channel measurements (i.c., gains, time-of- flights. angles) for different distributions of environments (c.g.. floor layout). We keep the wireless configuration fixed to using omni-directional antennas at both the transmitter and receiver operating 3.5GHz carrier frequency. Now we further elaborate the datasets used in our experiments.
Dataset 1: W13ROOMS, We create a synthetic dataset which gives us greater control over many aspects over the generation process. Using a 10m (0mx5nm 55m3mm hull, randomly synthesize interior brick walls such that the eventual configuration consists of three rooms inter-connected with Im doorways. We import the environment into open-source wireless propagation toolbox (Amiot et al. 2013) and collect 41.6K channels, of which ~37% of measurements are used for training.
Dataset 2: WIINDOOR. We use the indoor floorplans from the RPLAN dataset (Wu et al. 2019) which is popularly used to model indoor scenes (Nauata et al. 2020 layouts represent real-world single floor houses, with 4-8 rooms and 65 120m areas. Each floorplan is further accompanied with room semantics such as whether a certain area is a living room, bed room, bathroom, etc. We these semanties selectively sample transmit/receiver locations (e.g.. locations are not outside the boundary) and to determine wall materials (e.g., external facing walls are bricks, where internal facing walls are dry plaster walls). We use a commercial ray tracer Remcom 'Wireless Insite' (Remcom, 2022) with ray tracer X3D to collect measurements in the RPLAN environment. Similar to the earlier dataset, we collect 42.5K measurements, of which ~36% are used to train the model.
2021 Para et al. 2021) ). These
Train and Test Regimes. For the training dataset, we collect measurements by sampling transmitter ("Tx') from ~0 locations (XY plane at an elevation of 2.8m) and similarly, receiver ("Rx') from 60x30 locations (but with elevation of 2m). We then create three challenging test sets (see Fig. A2 for an illustration) with novel Tx-Rx locations: (a) Checkerboard where train and test Rx locations form a checkerboard pattern on the same XY plane at 2m elevation; (b) Generalization-z: where move the test Rx locations in (a) to novel clevation (==1. 1.0m for ThrceRooms and 2=2.5m for RPLAN); and (c) Generalization-iiee where we sample test Rx locations on a diagonal XYZ plane. Such regimes let us validate the generalization performance under distribution shifts.
Evaluation Metries. We consider three evaluation metrics to evaluate our approach: (i) Overall prediction error ('Overall'): We follow similar formulation our loss (Sec. 3.3 with one key difference - we find correspondences II by solving a linear-sum assignment problem. The eventual error aggregates all attributes relevant for the path (e.g., gain, angles). Intuitively, this measures the distance between two sets (sets of multi-dim paths in our case), using a similar metric common in set prediction tasks (Fan et al. 2017, Zhang et al. 2019). (ii) Geometry prediction error (* ('Geometry'): : We follow formulation similar to (i), but now focus on two specific features that captures the geometrical accuracy of the path time-of-fligh and angles departure and arrival. Intuitively, this metric measures whether the predicted rays take the same GT route between the transmit and receive co-ordinates. (iii) Average Delay Time- MAE ('AvgDelay"): We average the time-of-flige flightttts per path of the channel, weighted by its linear power p(ag). We report the mean absolute error of average delays between the predicted and ground-truth channel attributes.
We propose two reference basclines (i) K-NN (with k=1); which predicts the channel, given the closest match to the input spatial co-ordinates in terms of Euclidean distance (ii) MIP: A seometry-oblivion MLP regressor with 3-hidden layers, each with 128 units. We train the MLP using the same loss as WiNeRT. Additional details of the baselines are provided in Sec. C.4
Baselines.
4.2 OVERALL RESULTS
In this section, we present the overall qualitative and quantitative results of our approach. We com- plement the overall performances with additional analysis in the next section.
Quantitative Results. We report the quantitative results for the two datasets (column groups) and three test sets (row groups) in Table I We observe from the table: (a) by focusing on the overall emors, we find WiNeRT generally outperforms all baselines, with a significant average decrease of

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W13ROOMS
WIINDOOR
Overall
Geometry
AvgDelay
Overall
Geometry
AvgDelay
kNN
0.232
0.212
2.238
0.412
0.396
2.484
checkerboard
MLP
0,287
0.330
2.051
0.373
0,399
1.745
WiNeRT
0.202
0.087
2.029
0.237
0.207
1.546
KNN
0.253
0.226
2.033
0.424
0.428
2.487
gen-z
MLP
0.297
0.350
1.797
0.388
0,421
1.969
WiNeRT
0.217
0.084
1.522
0.285
0.250
1.839
kNN
0.252
0.213
2.118
0.380
0,251
1.377
gen-diag
MLP
0.312
0.322
1.889
0.390
0.315
1.513
WiNeRT
0.229
0.085
1.792
0.369
0.170
0.828
Table 1: Quantitative Results, Comparing errors of our approach (WiNeRT) with baselines, over two datasets
(column groups) and three test regimes (row groups). Lower values are better and the lowest errors are in bold.
GT
kNN
MLP
WiNeRT
Figure 3: Receive Powers. By fixing the transmit location (ax, red cross), we measure the receive power (color at cach point; in dB) predicted at cach location in W13R0OMS dataset. kNN and MLP suffer from memorization and falsely predict highest receive powers around phantom transmit locations (purple star).
-0.071 points compared to kNN and -0.085 with MLP; (b) WiNeRT is especially strong in capturing the geometry (e.g., 59-63% drop in errors w.r. second best W13ROOMS) of the environment, which can be likely attributed to a strong inductive bias enforced by decoupling global rendering from local evaluations; (c) Although WiNeRT has reasonable performance capturing the average delays, the performance gap here (c.g., 1-15% reduction in errors on W13ROOMS) is not especially large compared to other metrics. We attribute this contributions from 'false positive' rays with non-negligible power arising from our dense ray-launghing technique. (d) The contributions of false positives can be mitigated by using a more sophisticated ray launching technique. For instance, by piggybacking ray launch directions from GT channels, we can significantly improve perfor- mances across all metrics e.g., from 1-15% error reduction to 15-20% reduction in average delays on W13ROOMS; (e) Overall, attribute the underperformance of the baselines to poor generaliza- tion performance. For instance, in Figure 3] we illustrate the receive powers (in dB) predicted by all approaches in W13ROOMS, for some placement of the transmitter (red cross in top-right room). We observe in this particular case that the high-power areas in the kNN and MLP baselines are predicted for false phantom location (purple star), which roughly corresponds location in training set. This contrasts predictions by WiNeRT where the high-power areas are cor- rectly concentrated around the transmitter location. As result, we find that simple baselines find challenging to generalize to new unseen spatial co- co-ordinates at inference time.
transmitter
Qualitative Results. We complement the previous quanitative discussions with observations drawn from qualitative analysis. WiNeRT particularly helps for this analysis, as we can recovert interme- diate enyeennironment interaction information. From qualitative examples shown in Fig. a. b), we draw some observations: (a) WiNeRT surprisingly learns ray-surface interactions implicitly. with- out any direct supervision. For instance, we observe multiple reflected paths between Tx and Rx; (b) we also find that our predictions (red rays) are generally consistent with the underlying simula- tion process (green rays) e.g., reflections from adjacent walls, floor and ceiling; and (c) we notice WiNeRT sometimes predicts false positives (e.g., above Ex in Fig. 4b) which we attribute to dense onni-directional ray launching.
4.3 ANALYSIS
In the previous section, we evaluated the overall performance WiNeRT and found promising results. Now, we take closer look approach and investigate generalization benefits.

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Precktion
Pradccoo
(a) W13ROOMS
(b) WIINDOOR
(c) W13ROOMS (novel F)
Figure 4: Qualitative results. (a. b) Evaluation on WiNeRT on the environment seen during training. (c) We the previously trained model and re-render re-configured floormap F.
Train set Test sel
Ray-surface interactions
(b) Attenuation: Reflection
Aftennation: Transmission
Figure 5: Evaluating Ruy-se interaction MLP. We display cut-out of the 3R00MS represented as wireframe, with a specific focus on particular wall. (a) find a train-test distribution shift ray-surface incidence points (b, c) Evaluation of the MLP at various incidence points.
What does the ray-surface interaction learn? We begin by investigating the ray-surface network (f) in Fig. 2 in isolation. The network is tasked to map an incident ray (gain ains direction dnn) to an outgoing ray (Avat: d.ut). To accurately make this prediction, the network needs learn direction- and material-dependerr properties at the incident location Zine which poses two challenges. First, the network does not have explicit supervision to learn these properties. Rather, the network needs to implicitly learn these properties by optimizing over number of channel measurements. Second, specific to our case, the measurements collected involve sparse ray-surface interactions i.e., prac- tise we cannot expect for paths in the training measurements to interact densely with all possible surfaces. For instance, consider Fig. 5h, which show the incident points Dine for a particular wall (black edges) that recover from the underlying ray tracing tool. Here, we observe that the implicit training set interactions (red markers; never used during our training) are localized to a ~550mm band (15% area of the wall). However, test-time, the network tasked to generalize to interactions for a different distribution of incidence points (purple markers). In spite of the challenges we find the ray-surface network associates meaningful information surface co-ordinates. For instance, we show the attenuation factor predicted for the reflected (Fig. 5bb) and transmitted co-ordinates (Fig. (5)) for rays arriving from a fixed 2yx co-ordinate (placed 2=8m). We find that the network learns smooth material- and direction-dependen function over the surface. Over the next experiments, we exploit these locally learnt properties and evaluate WiNeRT rendering in novel scenarios.
Controllable synthesis: Predicting in Novel Environment Configurations. The previous exper- iments focused on evaluating approaches for novel locations of transmit and receive co-ordinates at simulation time. Now, we consider novel test-time environments by simulating approaches on re-configured layouts F' of the train-time environment F, such as by randomly editing placement of interior walls. Overall, we find that WiNeRT remarkably extrapolates ronment, with the overall error unchanged with WiNeRT (0.202 on vs. 0.203 on F"; more results Table A2). Furthermore, by observing the results qualitatively in Figure 4c we find the predicted interactions remain consistent with the ground- truth simulated rays in novel environment configura- tions. This is partieularly appealing for simulation use-cases which require modelling dynamic objects (e.g., moving vehicle), as WiNeRT allows re-configuring environment without retraining.
the reconfigured envi-
Controllable synthesis: Simulating Higher-order Interactions. In this experiment, evaluate the ability of approaches to generalize to different numbers interactions (denoted by r in Sec. 3 inference time. With WiNeRT, have the ability control the number of interactions
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test-time (i.e., by unrolling fo for fewer or more steps). We briefly summarize our observations here (see Table A4 for more details). WiNeRT exhibits promising results: while the baselines struggle with a simpler task of lower-order interactions (e.g.. 0.22-0.58 overall errors at r=0), WiNeRT's performance improves (from 0.20 to 0. 02). A better performance is natural in this particular setting, easier task than original (predicting only line-of-sight
since the model is required to perform component), For higher-order interactions, we observe performances of all approaches degrades, but WiNeRT outperforms the baselines. In particular, even r we find the geometric-errors of WiNeRT (0.27) comparable to baselines in their originally trained setting (r=1, 0.21 errors).
How fast are the simulations? We investigate the wall-clock simulation times of WiNeRT and baselines and compare them with wireless ray tracers. In the specific case of WiNeRT, we have some control over the time-accuracy trade-offss at test-time by varying the density of initial rays launched (see Sec. B.I). Overall, we find that WiNeRT demonstrates speed-ups of 11-22x over PyLayers and 6-22x over Wireless Insite. While the baselines are even faster (538-687x with MLP and 79-97x with KNN), it is achieved at the price of higher errors and poor generalization capabilities (Sec. 4.2) Overall, we find WiNeRT presents reasonable time-accuracy trade-offs compared to baselines. See Sec. for additional details.
Exploiting differentiability: User Localization via inverse (differentiable) rendering. Over the previous sections we focused on forward simulations. Now, we study a proof-of-concept for lever- aging our differentiable simulator for inverse problems, such as for user localization: determining user location Txx from observed channel habs We solve for Txx, by performing gradient spa- tial coordinate æxx
ukn that minimizes the channel loss renderg (Zx aitka, F.). This is possible with WiNeRT, since we backpropagate through the neural simulation of the channel. We evaluate over 100 test examples and find encouraging results, with a median error of 0.58m in W13ROOMS (a 150m² volume) and 1.21 in WIINDOOR 300m² volume). See Sec. C.4 for more details.
5 CONCLUSION, LIMITATIONS, AND BROADER IMPACT
In this paper, we proposed the first neural forward model for wireless ray tracing-based simula- tions. Such models are particularly appealing as they help alleviate some drawbacks of classical non-neural simulators (e.g.. better handling model-measuremen mismatches, non-differentiability). Towards this goal, we proposed WiNeRT which tasks an MLP to learn how surfaces in 3D environ- ment influence propagation of wireless rays, such as by predicting attenuation factor of a reflective component. Overall, we find promising results indicating neural simulators closely capture propa- gation effects. As neural simulators are additionally differentiable, we further show that they can be used to optimize inverse problems such as user localization.
Limitations and Future Work. This paper presents the first step towards realizing neural sur- rogate for simulating propagation of wireless rays. While we find promising results - in terms of empirically mimicking the simulator's performance while simultaneously reducing complexity many important steps remain to realize our over-arching goal of differentiable wireless ray tracing. Our approach is designed to capture linear effects of the channel in line with standards (3GPP TR 38.901 ITU-R P.2040-2) and extending to non-linear effects (e.g., amplifier saturations) remains an open-problem. Additionally, while our focus is primarily reflection and transmission properties of ray-surface interactions (capturing majority of receive power) which are increasingly relevant for high-frequency transmissions, other properties (e.g., scattering, diffraction) require investigation to model simulations across wider radio- andi-frreuuncyy spectrum. Finally, our surrogate's 's performance is currently upper-bounded by the underlying simulator's 's performance, motivating studies into learnt calibration of the surrogate model with real-world measurement data to bypass simulation accuracy.
Broader Technical Impact. Although our paper focuses on neural simulation of EM waves in the radio-frequency spectrum 100 GHz), we believe working towards this goal complements research in non-radio modalities as well. For instance, to model propagation of acoustic signals in spatial environments, estimating material-dependenn ray-surface interactive properties remains a challenging problem and the proposed research direction potentially complements existing tech- niques. More generally, we believe that radio signals require modelling both ray (e.g., reflection) and physical optic (c.g., interference, diffraction) properties, advances here are intertwined with many modalities across the EM spectrum (e.g., audio, visual).

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REPRODUCIBILITY STATEMENT
To ensure reprodueibility, we take a number of steps. On the dataset side, we use either publicly available indoor layouts (e.g.. RPLAN) or synthetically generate layouts with known random seeds (0 and 10 in our case). We further elaborate on the simulation settings to recreate our dataset in Section 4.1 and Section B We plan release the simulated data measurements. On the implemen- tation side, we provide specific training details in Section C.4 and further elaborate on the detailed architecture Section
ETHICS STATEMENT
The data used in our paper corresponds simulated data of physical processes (EM wave propa- gation). Since this does not involve any human subjects personally identifiable information, we believe there no conflict in this regard.
ACKNOWLEDGEMENT
We thank Hanno Ackermann for discussions and feedback on the paper. We additionally thank nu- merous colleagues for insightful discussions: Thomas Hehn, Fabio Valerio Massoli, Maziar Raissi, Afshin Abdi, June Namgoong, Taesang Yoo, and Akash Doshi.
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Appendix
A APPROACH
A.1 BUILDING CHANNEL MODELS
This section accompanies the text in Section 3
Channel models are defined either in a statistical way by defining distribution over channel at- tributes or in deterministic way using ray tracing. Statistical channel models are inadequate for applications involving positioning, sensing and challenges of communication higher frequencies (e.g., mm Wave at 30-300 GHz (Rappaport et al.) 2022)). Inspired by similar techniques in com- puter graphics (Glassner 1989). traditional ray tracing approaches (see for example (McKown & Hamilton 1991 Ikegami et al. 1991; Walfisch & Bertoni 1988)) approximate propagation of clec- tromagnetic waves by modeling interactions of each ray with objects its paths. These interactions include for example reflection, diffraction and penetration. Although this is more efficient than solving Maxwell equations, ray tracing methods need a detailed knowledge of the environment and generally slow for prototyping. They generally utilize hard coded and mathematically tractable models for example knife-edge model for diffraction (Lee) 1982 Rappaport] 1996). These abstrac- tions suffer from mismatches and require occasional tedious fine-tuning and calibration with real data. Improving these models while remaining tractable for rapid simulation rounds is not straight- forward. Finally, they are non-differentiable and cannot be integrated into closed loop design pipeline. We plan to tackle these issues by building neural surrogate of physics-based wireless ray tracer in this paper.
A.2 REPRESENTING RAY ATTRIBUTES
We represent the k-th ray (among K rays) the r-th iteration of rendering u (r). For notation convenience, we drop the sub- and super-script for the rest of the section. We characterize the wireless ray analogous to the concept of an optical ray (such as with geometric direction, intensity). In additional to the wireless attributes (see Equation 2). we further include meta-level attributes that helps us propagate and render the eventual ray received the receiver co-ordinate Iyx- We briefly describe these attributes here and elaborate on how they are obtained or updated over the next sections. The ray contains the attributes:
u (
d
Prx
Cupd
(a) Channel Attributes
(b) Ray Geometry
(c) State
which as shown can be grouped into three categories: (a) Wireless Channel Attributes. Exactly as discussed earlier in the section (see Equation 2). it contains the attributes to construct the wireless channel time-angle impulse response (Equation T} (b) Ray Geometry. We additionally include geometrical representation of the ray, which helps determine how propagate the ray through the environment. Specifically, we represent the geometry of the ray using the line equation: p(t) + td, where the origin and d is unit-vector encoding the ray direction. We are interested in two particular solutions of t in this equation: t, for which the ray intersects with a surface (mesh face in our case) and lx for which the ray is tangential to a sphere around some receiver of radius Prx- (c) Ray state. To help with subsequent updates to the ray at future iterations, we track two binary variables. aapa denotes whether the ray has be updated in the next iteration. oxx denotes whether the ray has impinged on a reception sphere of predefined radius.
A.3 RAY MARCHING: DETAILS
Bay-Ennironment Intersections. For each ray, we are interested in their first interaction with the environment (e.g., first wall it hits, impinging the receiver). For this,
interested in the solutions to the line equation representing the geometry of the ray: p(t) = =! + ta.. In particular, we interested in two solutions of t: (a) Ray-Face intersection. The smallest value of t > 0 for which p(t) lies on a surface (a triangular mesh face in our case). For this, we perform ray- triangle intersections with each face in the environment and find the corresponding solution t L.-
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= Face xx (11
New complex gain
Spatial MLP
Surface normal nj b, sdf(x; (1+3))
(r+1)
(r)
Dirc.
Sj
0k
S 5
Xex)
MLP
Pi
A = r(p:)
k
(++1) Ad
E3)(3)
New direction
Figure A1: Ray-surface interaction network fo
(r+11) = +1,d (a) Ray-Rx intersection. In This helps us estimate the new relay location: Tg parallel, we are also interested in positive solutions t for which the ray hits the receiver if were modeled as sphere of radius Prx. In this case, we obtain the value of t as the projection of æyw on p(t):
Lrz max((, (Ørz - 2 (r))
(3)
Prz [1 (1)) 4,
(4)
Consequently, at the end of ray-environment, we analytically estimate the first intersections of the ray with both the environment and (potentially) the receiver.
Ray-Surface Interaction.
the ray u (originating at 2r() and travelling in direction d hits wall at #k (r+1) (as estimated in the previous step), we are now interested in characterizing the outgoing ray with origin at Zi (++1) Specifically, we are interested in estimating the new direction d(r+2) (does the ray penetrate the wall? or reflect?) and the corresponding change in gain that arises (i.e., loss of power, change of phase). This is a complex problem and typically requires in- depth knowledge of the surface (e.g., which material) as well as it's specific EM properties (e.g., frequency-den deeeedeet effects). Our solution is to instead learn these properties by associating spatial regions in the environment with EM-specific properties. Towards this, we delegate the association neural network show in Figure Al The key idea is to associate spatial co-ordinates (or sets of co-ordinates, given by face on which they lie) with EM properties. We achieve this by mapping spatial properties (e.g., face corresponding x (r+1)) EM properties (e.g., gain factor).
Specifically, our neural network is:
v; = spatial.net n,, b.)
(5)
$ P: directionalnet((w)
(6)
which consists of a spat i _net to encode EM properties specific to a spatial region, but inde- pendent of the incidence direction. This network takes as inputs the one-hot encoding of the face f. on which the relay point x(++1) (r+1) lies and the surface normal vector at that point n.. In addition, we also provide the network a 3-dim conditioning vector of signed distances
b. (sdf((E..r fi), sdf(arrz fi), sdf(x((+1) f()
where sdf(æ; is the signed distance function between co-ordinate and face f. We find it cru- cial to condition the network on these values to help predict E.M-roperties for relevant outgoing components (e.g., reflective, transmission).
The output of the network is a gain factor 8;, such that the new gain of the ray u (r+1) is a (r+1) = s.a?") Since the gain magnitudes can be represented in either linear logarithmic scale, we predict both additive and multiplicative factors of the gain in practise (a) (r+1) 51,10 (r) + 91,2). In parallel, the network also predicts the rotation ray incident with direction d( on f. undergoes. We characterize rotations using a 4-dim rotation p: using Euler-Rodrigues parameterization. This parameterization encodes the axis of rotation and about which d(") rotates by angle &. We repre- sent the rotation by a 3x3 SO(3) matrix A and the new outgoing direction of ray l is given by d(++1) Ad(r)
Reception/Termination check. For some special cases, we halt ray marching for subset of rays. Namely, when ray l impinges reception sphere of radius under meters. This prevents
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(Taint Tx locations
[lest] Tx locations (lest Rx locations
(tram Rx locasions
(a) Testset 1: "Checkerboard"
(b) Testset 2: "Generalization-"
(c) Testset 3: "Generalizatoon-dia"
Figure A2: Train and test regimes: We consider disjoint subsets of train (blue markers; identical in all figures) and (orange markers) co-ordinates of transmit and receive locations.
future version of the ray being potentially being incorrectly received once again. In addition, for computation reasons, we also terminate ray marching if the ray exits the region interest (e.g., ray exiting the environment).
Free-space interaction. While the previous steps modeled the interaction material properties of the environment wireless propagation, we now switch focus to free-space. In this case, we model propagation of a ray using the empirically-adjusee Friis' Equation:
dg A
(r(d) P.K
do
which represents the power at the received the receive antenna P. as a function of the power fed into transmitting antenna Pt and the distance travelled by the ray d. We learn the remaining scalar parameters K (constant representing of antenna gains), (wavelength of signal), and do (reference distance).
B DATASET: ADDITIONAL DETAILS
B.1 TRAIN AND TEST REGIMES
Figure A2 accompanies the text in Section 4.1
B.2 SIMULATION FOR WILNDOOR DATASET: DETAILS
We created 3 different floor-plans in Wireless Insite where 2D floorpplan layout and semantic labels of each room are picked from House GAN++ dataset and mapped into 3D layout where the scale and dimensions are determined based on practical floor-plan scenarios. All layouts are scaled to 10mx 10m with ceiling height 3m. All the inner walls and floor materials are layered dielectrics with specific permittivity, conductivity & roughness. These have finite reflection and transmission coefficients. The reflection coefficient corrected if the surface is not smooth while the transmission coefficients are unaffected by surface roughness.
Materials. Propagation characteristies are naturally affected by the medium and we create a dataset with fairly diverse set of materials. Layered dielectric with two layers separated by free-space of 89cm chosen for all inner walls and the outer-walls were made of thicker materials of concrete. Doors were created using free space except the balcony door which was created using glass with a small thickness. The balcony walls were laid out using brick walls. The propagation factor and index of reflection functions of the permittivity (c) and conductivity (a) of medium. In Table Table AI, we present the relative permittivity and conductivity.
Antenna and Transceiver configuration. Omnidirectional beam patterned antenna with polariza- tion perpendicular to the z axis is setup for all receive and transmit antennas. Location, Orientation of the antenna set relative to global reference such that they rotated about the axis by 90deg and placed at height 2.8m. All antennas employ the same configuration with no transmission loss.
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thickness(cm)
permittivity
conductivity a (S/m)
Layered d1y,
1.3
2.8
0.013
Brick
12.5
4.44
0.0001
Concrete
30
5.31
0.015
Glass
2.4
Table A1: Material properties
Overall
Geometry
Avg. Delay
KNN
0.264
0.288
1.479
MLP
0.280
0.378
1.191
WiNeRT
0.203
0.114
1.297
Table A2:
Quantitative results. For trained ap-
proach evaluated on reconfigured floormap F'
Prediction
GT
Table A3: Qualitative results
Simulation. We currently run the simulation using the shoot and bounce model where a geometric path is drawn from every point on the transmitter field pattern the receive point. This also includes transmission through surfaces allowing it to model transmittance and reflection. Rays are first traced from the source points with the rays reflecting specularly from the building walls. The rays that hit building walls are reflected specularly and continue to be traced up to the maximum number of reflections and transmissions.
The spatial separation of rays is set to 0.75%. The geometric path traced by the ray undergoes upto 6 specular reflection and 3 transmittance with path loss threshold set to -70dBm.
Total received power of all paths is determined as the of time averaged power of group of correlated paths. A set of ray paths that interact with similar set faces and follow nearly same path defined as group.
C EVALUATION: ADDITIONAL DETAILS
C. CONTROLLABLE SYNTHESIS: GENERALIZATION TO RECONHIGRRED FLOORMAPS
Table A2 accompanies the discussions in Section 4.3 where we evaluate WiNeRT model trained
in one environment F and evaluated in a reconfigured environment F*
C.2
CONTROLLABLEE SYNTHESIS: LOWER- AND HIGHER- INTERACTIONS
See Table A4
which accompanies the discussions in Section
Overall (DoD)
Geometry
Avg. Delay
#interactions r
1"
2
0
1*
2
3
KNN
0.22
0.33
0.50
0.55
0.31
0.21
0.29
0.33
1.30
2.24
2.96
3.40
MLP
0.58
0.46
0.61
0.67
0.34
0.33
0.37
0.41
0.98
2.05
2.93
3.48
WiNeRT
0.12
0.25
0.44
0.51
0.00
0.09
0.21
0.27
0.03
2.03
2.43
2.8
Table A4: Low- and Higher-Order Interactions. We vary the number of ray-surface interactions (denoted by 7) for model trained using singlc-order interactions (r=1, denoted by the table).
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Roo oms
wilndoor
10a
Simulator (PyLayers)
Simulater (Insite)
WINeRT
WiNeRT
WiNeRT (Oracle ray launch)
WiNeRT (Oracle ray launch)
Simulation time (secards)
Simulation time (secords)
10-1
KNN
10-1
KNN
10-1
10
10-1
0.16
0.22
0.20
0.30 Error (Overall)
Error (Overall)
Figure A3: Simulation Time. Comparing wall-clock time vs, accuracy performances of our approach (WiN- cRT) against baselines (MLP, kNN) and wireless ray tracing softwares (PyLayers and Insite). The 'Oracle ray launch' variant, which utilizes known ray launch directions at lest-time, indicates approximate performance upper-bound of approach.
C.3 SIMULATION TIME
In Sections 4.2 and 4.3 we found our proposed approach WiNeRT achieves reasonable performance compared with non-differentiable and non-neural simulator packages. Additionally, we demon- strated that WiNeRT capable of generalization (c.g., to novel elevations, re-configured floor- plans) and can be used for inverse problems. In this section, we additionally discuss run oun-time per- formance of WiNeRT and compare against baseline approaches as well as the simulator package.
Experimental Setup. The end-goal of the experiment is to analyze the simulation time (specifi- cally wall-clock times) of the proposed WiNeRT approach and contrast against both the simulator softwares (PyLayers, Wireless Insite) and proposed baselines (MLP, kNN). We first remark that the implementations fundamentally vary between the approaches and hence an ideal wall-clock timing comparison is not possible. For instance, some approaches (WiNeRT, MLP, kNN) use a PyTorch implementation which can be run on GPU whereas the wireless ray tracing simulation packages are either proprietary (e.g., Wireless Insite) or developed exclusively for CPU (e.g.. PyLayers) and thereby limiting the choice of hardware on which they can be run. Nonetheless, we keep sim- ulation settings consistent when possible: by running the exact simulations used for the overall results (setting 'checkerboard'; see Scction 44I) and furthermore estimating wall-clock times per simulation (batch size of 1) over N individual simulations with maximum of reflection and transmission (i.e., r=1). For all approaches, we report only the mean simulation time over the mul- tiple simulations, we found the variances low (o² < 3.5 x 10-3). When possible, we also report corresponding accuracy ('overall prediction error'; see Sec. 4.1.. We evaluate Py PyTorch-based imple- mentations (WiNeRT, MLP, kNN) over N = 8K simulations using pretrained models (specifically the ones for reporting IT on a Nvidia A100 GPU. In the case of WiNeRT, we are able to control time-accuracy trade-off to some degree at test-time by varying the number of launched rays K (see 'Ray Launching" in Sec. 3.T) as function of the number of subdivisions of the ico-sphere. We choose 1-5 sub-divisions and additionally an 'oracle ray" launch strategy to depict lower-bound on the time-accuracy values.
Results. We present the time-aceuracy in Figure A3 and observe: (i) WiNeRT (orange mark- ers) is significantly faster than the simulators (blue line), demonstrating speed-ups of 1 11-22x over PyLayers (Amiot et al., (013) and 6-22x over Wireless Insite (Remcom) 2022). Although the simu- lators are approximately
the accuracy, we find that WiNeRT can make reasonable trade-offs on accuracy to boost simulation times in certain scenarios; (ii) The baselines we propose in this paper (MLP and kNN) are faster. MLP (green marker) is the fastest with speed-ups of 538-687x. which can be largely attributed to simple architecture (3-layer ReLU MLP with 128 hidden units). kNN (red marker) is the second fastest with 79-97 speed-ups the simulators.
upper-bound
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1.0
Density
0.2
wi3Rooms (Median 0.58 m)
wilndoor (Median 1.21 m)
0.0
8
Localization error (in m)
Figure A4: User Localization. We backpropagate through our trained forward model to solve for the position of the receiver.
While these baselines offer much faster simulation times, their generalization capabilities remain unclear as they suffer from memorization (see discussion Fig.
C.4 USER LOCALIZATION VIA INVERSE RENDERING
In this section, provide additional details to complement the discussion the user localization experiment in Section 4.3 For the user localization task, the problem to determine user location Txx from observed channel hobs- We solve for Trx. by performing gradient on spatial coordinate = that minimizes the channel loss rende I (xxx, 2xx ukn F.). This is possible with WiNeRT, since can backpropagate through the neural simulation the channel. We optimize for Tyx using SGD with momentum (Ir=0.01, momentum=0.9, 2000 iterations) with two additional considerations: (a) we constrain T₁x lie in valid ranges (positive, upper-bounded by Fmmax by clamping the values at each iteration; and (b) to prevent solutions in local minimas, we take the result which yields the minimum loss over five random initializations of Tyx. We present the CDF of localization errors over 100 test examples in A4
D IMPLEMENTATION: ADDITIONAL DETAILS
In this section, we provide additional implementation details and hyperparameter choices of ap- proaches discussed in the paper.
D.1 WINERT
Ray-surface Interaction f/. We follow an MI.P architecture (see (see Figure A simi- lar to NeRF approaches (Mildenhall al. 2020 Verbin al., 2022). We decompose the parameters into view-independent ('spatial MLP') and view-dependent ('directional MLP') sets. Given ray incident at spatial co-ordinate 2; in direction du, the spatial MLP (2 hidden layers, units) takes three inputs: (a) the face ft (1-hot index) which 24 lies; (b) the surface normal n. of face fü; and (c) 3d vector of signed-distane
Architecture:
values between the face and Tu. Txx, and ok- We find (c) provides information (e.g., 3ix and 2 the same side of wall) to condition the network predict attributes related to either reflection or transmission components. The directional MLP (1 hidden layer, 64 units) takes two inputs: a 32-dim bottleneck vector produced by the spatial MLP: and (ii) a 3-dim unit vector representing the incidence direction dę. The final output are scaling and (r)
additive co-efficients for the gain magnitude (i.e., dg P; for rotation (based on Fuler-Rodrigues formulation). The rotation parameters p; are mapped to a l(pp.) transform the incident to outgoing ray dy := Adg.
(r+1) 814 82) and 4-dim parameters
3x3 rotation matrix
Renderer: Ray Launching. In the first step of the renderer, we launch K rays from co-ordinate Ex uniformly in all directions. To achieve this, center ico-sphere with sub-divisions and
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Published as a conference paper at ICLR 2023
choose as directions the vectors from Tu towards the ico-sphere vertices (10.2K vertices with 5 sub-divisions). Since we know the exact co-ordinates between Tu and Exx, we manually include the line-of-sight direction resulting in total of K rays.
Renderer: Ray Marching. The core step of the renderer ray marching (detailed in Figure 2)
We elaborate technical implementation details step-by-steny using as reference Figure 2 We
drop sub- and super-seripts for rest of the paragraph for notational convenience. (a) Ray-Triangle intersection: For given ray p for which the ray intersects with each face the mesh. For some face with coordinates (a. b, c). this entails solving for & such that p = o + td aa + 36 + 7c (under constraints and 0 < a, 3, 7 < 1). We calculate valid solutions using Cramer's 's rule for all faces in the mesh and only consider (if one exists) the minimum positive solution corresponding to the first ray-triangle intersecting point. (b) Ray-Surface interaction: Given the solution from the previous step (i.e., on which spatial co-ordinate the ray incident on the surface), we are now interested in estimating the outgoing ray from that co-ordinate. For this, we leverage an MLP that maps incident gain, direction, and certain face properties to outgoing gain and direction. More details of this MI.P are discussed above under the 'Architecture: Ray-surface Interaction'. (c) Reception Termination: Per ray, stop ray marching steps if is either received (hits a reception sphere of fixed size of 30cm) or leaves the region of interest (e.g.. penetrates exterior wall is shot into infinity). In other cases, continue with ray marching steps.
íd, are interested in the minimum finite solution of /
B +
Ray Aggregation. At the end of ray marching steps (over R iterations), we determine the final state of the K rays. We are now interested in small subset of these K rays that is received at receiver at fixed co-ordinate Tyx- Note that perform these steps only test-time. The ray aggregation as a result involves two steps: (a) Ray Filtering: where we determine the subset of rays that arrives Fix by modelling the receiver
Renderer:
sphere of fixed radius of 30em; and (b) Preventing double counting: we find duplicate rays arrive at Uix due to a combination a non-infinitesimally sized reception sphere and a high density of launched rays. We cull such duplicates by grouping rays based on a unique interaction sequence (i.c., IDs of faces it intersects with) and choosing the ray of the shortest length in each group.
Optimization. We perform gradient-descent steps on learnable parameters using Adam with a learning rate of 0.001 with batch size of 1. We observed large gradients (possibly due to single- batch) and hence clip gradient values to 100 during training. The model is trained for 100 epochs and we pick the checkpoint with lowest validation error during training.
D.2 BASELINES
MLP. The MLP baseline extends ideas presented in Tancik et al. (2020); Sitzmann et al. 2020)) where simple MIP used to map co-ordinates to the signal (e.g.. pixel co-ordinate to RGB values). In our paper, the MLP directly maps the spatial co-ordinates Ttx and Txx to channel h4. The MLP contains 3 hidden layers, each with 128 hidden units and ReLU activation. The core idea here is to implicitly learn the geometry of the environment (floormap F), which is common to all train and test examples. Note that in contrast to previous works, this model does not use positional embeddings nor sinusoidal activations, as our initial experiments indicated they learn sighffrey artifacts that is not typically present in our datasets (the wireless channels).
kNN. The kNN baseline (with k=1) works as for given test-example (Ixx, Fin)) find the main
spatially closest training example arg min; Ftx - Tix [a + |xxx + T train la and predict channel hi.
19


Reviews:
[{"review_id": "nNBAkdjzL5", "reviewer": "Reviewer_emnY", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes to employ a differentiable neural ray tracer for wireless channel modeling. It models the time-angle channel impulse response as a superposition of multiple paths, and the wireless characteristics of each path are a result of multiple evaluations of an implicit neural network. The proposed framework achieves strong performance in multiple tasks.", "summary_of_the_review": "I am not an expert in wireless channel modelling, therefore it is not straightforward for me to evaluate the significance of this work in this field. Given the encouraging results and novelty, I think this paper is above the borderline.", "strength_and_weaknesses": "Strength:   \n1. This paper is well-written and the ideas are clearly illustrated.   \n2. It is interesting to find the similarity between wireless channel modeling and neural ray tracing.   \n3. It is good to see that WiNeRT achieves encouraging results on time-of-flight prediction and user location tasks.    \n\nWeakness:   \n1. For the baseline method, the paper only compares with KNN and MLP. What are the comparative results with other channel modeling methods?    \n2. What is the efficiency of the proposed method?    \n3. Are WI3ROOMS and WIRPLAN two standard benchmarks?   \n4. In figure 4, the legends and the rays have different colors, it is difficult to know which are GT and which are predictions.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "f8AJjezER5", "reviewer": "Reviewer_YoPQ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposed a method for wireless channel modelling inspired from NeRF-based neural scene representation. Different from previous works which explicitly model the ray surface interaction, it tries to learn and map the environment configuration to a wireless channel directly through a neural network. The proposed method is evaluated using multiple indoor scenes and demonstrate significant improvements over the baselines.", "summary_of_the_review": "Please check my comments above. I think the studied problem and the proposed method is somewhat novel, but the experimental evaluations are lacking in some aspects such as lack of ablation studies on network architectures, lack of robustness of the proposed method towards scene noise, lack of real world scenario examples. ", "strength_and_weaknesses": "Strength: The application of using NeRF-based method for wireless channel modelling is somewhat novel and the proposed implicit method for ray surface interaction seems to be effective.\n\nWeakness: The proposed method seems to assume the reconstructed scene is perfect without any noise. However, in practice, such ideal reconstruction is rarely happens especially for sparse input setting. The proposed method is evaluated using only synthetic data which also raise a concern about the robustness of the proposed method when it is applied to real world scenarios. How does the proposed method avoid overfitting when the training and testing are both under ideal configurations? The proposed method is only compared with kNN and MLP which are very simple baseline methods. How about the comparisons with other SOTA that using explicit methods for ray surface interaction modelling?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "Sf5c7x6Q8Mz", "reviewer": "Reviewer_oi9G", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Update; I have read the rebuttal and decided to increase the score.\n--------\n\nThis paper proposes a neural surrogate (WiNeRT) for a ray tracer that models the propagation of wireless signals. WiNeRT is a hybrid heuristic/learned ray tracer that replaces some heuristic ray tracing functionality, especially the Ray-Surface Interaction with a neural network. The authors claim that this is the first wireless neural ray tracer and in addition to simulating ray-tracing results, it will allow solving inverse problems via differentiating through the model. The paper experimentally validates the accuracy of the model against simple baselines and shows initial results on inverse problem of localization.", "summary_of_the_review": "Well-written practical paper that has good empirical evaluation, but would need some more work. Once improved and with detailed end-to-end description of the method it might serve also as a baseline for further works.", "strength_and_weaknesses": "Strengths:\n\n—-------------------\n\nThe paper has many strengths and is tackling an important problem. For the sake of time  I am only writing about weaknesses in this review, since those are the ones that should be actioned upon.\n\nWeaknesses (and questions) (W)\n\n—--------------------------------------------\n\nW: The method is a hybrid method that utilizes both typical ray tracing and neural ray marching features. It would be good if the authors could write more precisely which parts of the algorithm are same/similar as normal full ray-tracing and which parts contain learned functions. As far as I understand the main learned part is the Ray-Surface Interaction, and it would be important to describe how this is normally done to understand what the neural surrogate is approximating.\n\nW: Give more details on the end-to-end model, e.g., in an appendix for reproducibility. As it stands it might be difficult to re-implement or improve the current method.\n\nW: The authors should describe more clearly what are the target use cases for the system. It seems to be that the system requires training data from the same/similar environment from an existing ray tracer, so what would be the biggest reasons for using the neural surrogate since an existing ray tracing simulation is anyway needed?\n\nW: The main reason (in addition to inverting) of using a neural surrogate is to improve the inference time simulation speed and computational complexity. As far as I can see this discussion is missing from the paper and it would be very important to discuss it.\n\nW: (See above). If the computational complexity is not reduced much, then the main benefit of the model is the invertibility, which is briefly discussed in the localization experiment. How good is the localization result? It should be compared to other localization approaches in the same environment (without using the trained model). In addition, it would be important to discuss more use cases of the model inversion and more inversion experiments of the neural surrogate should be performed.\n\nW: The paper might find a relatively narrow audience in machine learning and might be even more suitable for publishing in the wireless domain. For ICLR, the authors should discuss which parts could be generalizable to other domains than wireless ray tracing.\n\nW: While there are extensive tests of generalization in the same space as in training (different Rx and Tx locations, etc.), the generalization to new spaces would be interesting to study further in addition to F′ in Figure A3. It will be important to see also the failure cases where there is a big enough change in the environment in the test set, ie., how general ray tracing algorithm it has learned.\n\nW: The authors do not discuss non-linear effects of the channel, such as doppler shift or spread and how this could be modeled for highly mobile UEs.\n\nW: The baselines 1-NN and MLP should be described more carefully. Are they replacing the ray-marching part or the general processing. What parts of a normal ray-tracer are the baselines utilizing (different from the main model)\n\nW: How is diffraction handled in the model?\n\nW: How is penetration handled in the model?\n\nW: Regarding inverse problems, could the authors state which kinds of inverse problems would be solvable using the model and which would lie outside of the current model. For example, could one design changes in the environment geometry using the current model or is it currently restricted only to the trained geometry.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "94UJMJccxH", "reviewer": "Reviewer_sfBb", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a neural network based solution to heuristically solve a wireless signal (physics) rendering problem. Given the environment set-up and configurations of the transmitter and receivers, the pre-trained network is able to simulate the wireless signal propagation in the confined environment instead of physically computing the ray-tracing function, which costs the computational power extremely. In addition, this approach enables the reverse rendering applications. \n\n", "summary_of_the_review": "In summary, this paper proposes a new topic of simulating wireless signal propagation in a confined configurable environment. The paper systematically defines and solves the problem with comprehensive experiments. ", "strength_and_weaknesses": "Strengths\n1. This paper creates a new area that is simulating / predicting wireless signal propagation problem (or more generally: a ray-tracing simulation). This method explores a new topic and expands the boundary of computer vision applications. \n2. The proposed paper has thorough study with signal propagation problem. It provides detailed mathematical definition of the problem, casts the physics ray-tracing computation to a network simulation problem. It also come with solidate experiments to demonstrate the effectiveness of the proposed method. \n3. The paper also proposes two new databases / datasets that allow the community to benchmark further new methods. This could bring more impact to the community of computer vision. \n\nWeaknesses\n1. This paper does not discuss about the non-linear surface or interactions of wireless signals. It assumes the operations are all linear. The authors are suggested to provide a limitation of the method\n2. In the introduction part, the goal of the proposed method is to reduced the inference time. The authors are suggested to make a comparison between the current method and ray-tracing computation. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster