Differentiable and Learnable Wireless Simulation with Geometric Transformers
Authors: Thomas Hehn
Markus Peschl
Tribhuvanesh Orekondy
Arash Behboodi
Johann Brehmer
Venue: ICLR 2025 Poster (2025)
Keywords: inverse problems
learning to simulate
wireless channel modeling
geometric deep learning
equivariance
inverse problems
electromagnetic signals
diffusion models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We solve forward and inverse problems on simulated and real-world data using equivariant neural simulation surrogates that exploit geometric symmetries found in wireless signal propagation..

Abstract: Modelling the propagation of electromagnetic wireless signals is critical for designing modern communication systems. Wireless ray tracing simulators model signal propagation based on the 3D geometry and other scene parameters, but their accuracy is fundamentally limited by underlying modelling assumptions and correctness of parameters. In this work, we introduce Wi-GATr, a fully-learnable neural simulation surrogate designed to predict the channel observations based on scene primitives (e. g., surface mesh, antenna position and orientation). Recognizing the inherently geometric nature of these primitives, Wi-GATr leverages an equivariant Geometric Algebra Transformer that operates on a tokenizer specifically tailored for wireless simulation. We evaluate our approach on a range of tasks (i. e., signal strength and delay spread prediction, receiver localization, and geometry reconstruction) and find that Wi-GATr is accurate, fast, sample-efficient, and robust to symmetry-induced transformations. Remarkably, we find our results also translate well to the real world: Wi-GATr demonstrates more than 35% lower error than hybrid techniques, and 70% lower error than a calibrated wireless tracer.

Published as a conference paper at ICLR 2025
DIFFERENTIABLE AND LEARNABLE WIRELESS SIMULATION WITH GEOMETRIC TRANSFORMERS
Thomas Hehn, Markus Peschl, Tribhuvanesh Orekondy, Arash Behboodi, Johann Brehmer" Qualcomm AI Research
ABSTRACT
Modelling the propagation of electromagnetic wireless signals is critical for de- signing modern communication systems. Wireless ray tracing simulators model signal propagation based the 3D geometry and other scene parameters, but their accuracy is fundamentally limited by underlying modelling assumptions and correctness of parameters. In this work, we introduce Wi-GATr, fully-learnable neural simulation surrogate designed to predict the channel observations based on scene primitives (e. g., surface mesh, antenna position and orientation). Recog- nizing the inherently geometric nature of these primitives, Wi-GATr leverages an equivariant Geometric Algebra Transformer that operates on a tokenizer specif- ically tailored for wireless simulation. We evaluate our approach tasks (i. c., signal strength and delay spread prediction, receiver localization, and geometry reconstruction) and find that Wi-GATr is accurate, fast, sample-efficient and robust to yymmery-innuuced transformations. Remarkably, we find our results also translate well to the real world: Wi-GA1 demonstrates more than 35% lower error than hybrid techniques, and 70% lower error than a calibrated wireless tracer.
range of
INTRODUCTION
Modern communication is wireless: more and more, communicate via electromagnetic waves through the antennas of various devices, leading to progress in and adoption mobile phones, au- tomotive, AR/VR, and IoT technologies (Chen et al., 2021; Dahlman al., 2020). All these innova- tions build upon electromagnetic (EM) wave propagation. Therefore, modelling and understanding wave propagation in space is core research area in wireless communication, and remains crucial as we are moving toward new generations of more efficient and spatially-aware wireless technologies.
How one model the influence of spatial environments on EM waves transmitted by one antenna and received by another? Model-based approaches (specifically wireless ray tracing methods) are popularly used, which identify physical propagation paths and their corresponding characteristics (c. g., power, phase) between the two antennas. The propagation paths are naturally influenced by the environment, causing phenomena such as reflection, diffraction, and transmission. Although popular, model-based approaches have shortcomings: Due to their approximate nature, they suffer from a sim-to-real gap non-trivial environments, and in most cases they are neither learnable nor differentiable.
Hybrid and fully-learnt approaches attempt to address some shortcomings of model-hased meth- ods. Hybrid approaches (Orekondy et al., 2022b; Hoydis et al., 2023) retain some aspects of model- based techniques (c. g., evaluating physical propagation paths), and introduce learnable components (e. g.. neural evaluation of materials). The learnable parameters help with incorporating real mea- surement data to reduce the sim-to-real gap. However, similar to model-based approaches, they once again Nottle-necked by availability of accurate and efficient implementations of explicit physical models. In contrast, fully-learni approaches (Hehn et al., 2023; Levie et al., 2021; Lee & Molisch, 2024), propose end-to-end neural surrogate that consumes visual representation of the scene and predicts spatial wireless characteristics. The core idea, which we too leverage in this work, is to enable the neural surrogate implicitly learn the physical models that best explain the
WWrk was completed while an employee at Qualcomm Technologies Netherlands B, QQullcomm Al Research is initiative of Qualcomm Technologies, Inc.

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observations. While there have been some advances in this direction, find the choice of architec- tures ill-suited and inefficient for the task.
Existing fully-learnt surrogates largely rely CNN-based architectures and are limited to lossy 2D representations of the scene (i. c., top-down binarized images). We argue that wireless propagation is inherently a lireedimmesiooal geometric problem: directional signal is transmitted by oriented transmitting antenna, the signal interacts with oriented surfaces in the environment, and the signal eventually impinges an oriented receiving antenna. For this reason, it critical for neural surrogates to model and flexibly represent 3D geometric aspects in the propagation environment.
Learning neural surrogates directly 3D data presents two significant challenges. The first chal- lenge involves representing the diverse 3D geometric input data such that neural network can ef- ficiently learn the signal propagation interactions among scene elements. The second challenge is ensuring that the neural network can generalize to unseen scenes, given that the received wireless signal remains invariant under rotations, translations, and reflections of the entire scene.
In this work, we aim to tackle these challenges with novel fully-learnt approach to model wireless signal propagation in 3D. We propose Wi-GATr, a backbone based on Geometric Algebra Trans- formers (GATr) (Brehmer et al., 2023) for surrogate models utilizing 3D geometric representations and strong geometric inductive biases. A key component is a new tokenizer that defines the input representation for the diverse, geometric 3D data of wireless scenes. Furthermore, this architecture is E33)ccquivariant with respect to the symmetries of wireless propagation, but maintains the scala- bility of transformer architecture.
A geometric, fully-learnt treatment of modelling the influence of simulation parameters (e. g., scene mesh antenna position, orientation) on resulting wireless measurements (c. g., receive signal strength) offers many advantages. First, the underlying inductive biases of Wi-GATr enable sample- efficient learning with the resulting predictions being robust certain variations, such as the choice of coordinate frame. Second, since Wi-GATr is fully differentiable w.r.t. all its simulation parame- ters, we can leverage it to tackle inverse problems (c.g.. localization). Finally, Wi-GATr can be cas- ily adapted into a generative framework and thereby allowing probabilistic inferences.
We present a comprehensive evaluation of Wi-GATr across various tasks and use cases. To facilitate large-scale evaluation and training. we introduce two novel wireless datasets, Wi3R and WiPTR, each comprising thousands of indoor scenes with varying complexity. Our results show that Wi- GATr outperforms competing approaches. For instance, on Wi WiPTR, Wi-GATr achieves an MAE of 0.64 dB using only 10% of the training dataset, surpassing PLVIT (1.28 dB) and Transformer (0.69 dB) trained on the full data. Furthermore, we demonstrate the versatility of our surrogate by leverag- ing the diffcrentiability of Wi-GATr for receiver localization, achieving accuracy up to 60 cm. Ad- ditionally, we show how Wi-GATr can serve a generative diffusion model trained on joint simula- tion parameters, enabling the reconstruction of various scene variables, including the intricate mesh geometries. We also train and evaluate Wi-GATr on the real-world measurement dataset DICHA- SUS (Euchner et al., 2021), where it achieves state-of-the-art results. Specifically, Wi-GATr outper- forms both hybrid approaches (with >35% reduction in error) and a calibrated wireless ray tracer (with 700% reduction in error). Overall, our results indicate that our proposed approach Wi-GATr takes a promising step towards a sample-efficient and robust fully-learnt simulator that can effec- tively leverage the underlying 3D scene parameters to simulate wireless propagation effects.'
2 BACKGROUND AND RELATED WORK
Wireless signal propagation. How do wireless signals propagate from a transmitting antenna (Tx) to receiver antenna (Rx) in a (static) 3D environment? While the system is fundamentally described by Maxwell's equations, for many realistic problems the ray approximation of geometric optics suffices (Keller, 1962). It approximates the solution to Maxwell's equations as a sum of planar waves propagating all directions from Tx. Each planar wave is represented by various attributes (e. g., power, phase, delay) since transmission. As a ray reaches an object- that is, it intersects with its mesh the interaction is modelled as reflection, refraction, or diffraction. During such interactions, the power, phase, polarization, and propagation direction of the wave can
ray, characterized
OOr Wi-GATr code is available https: //gi ht///ggtttut thub. com /uaa comm-A A --r-ssrrrhhhheeeeeee /wi -ATTT.
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Wi GATri
Receiber x
Tokanizo
GATH
Transmitter
V
Lth. Boba)
Geometry
Wireless Ray Trecer
Scene paremnetrrs V
Channels ht
(a) Wi-GATr: Learnable and Differentiable
GATr. Diftusion model
68.0 dBm
45 dBmn
floooo celling,
(b) Generative medel formolation
(c) Conditional generation: geometry
(d) Conditional genenaion: Receiver
Figure 1: Wi-GATr geometric neural surrogate for wireless simulation. (a): Akin to ray tracer, Wi- GATr forward-model maps the 3D geometry, transmitter, and receiver properties to desired channel infor- mation, such as the received power. Wi-GATr is differentiable and can provide useful gradients with respect to its inputs. (b): In probabilistic approach based on diffusion, Wi-GATr can reconstruct 3D environments (c) and antenna positions (d) from the wireless signal.
change in complex, material-dependent ways. In addition, new rays can emanate from the point of interaction. After multiple interactions, the rays eventually reach the receiving antenna. The Tx and Rx are then linked by a connected path p of multiple rays. The effects on the received signal are described by the channel impulse response (CIR) h(r) Ep dpō(T Tp), where Up C is the complex gain and Tp the delay of the paths at the receiver (Tse & Viswanath, 2005).
Wireless channel models and simulation. Wireless propagation models play a key role in design and evaluation of communication systems, for instance by characterizing the gain of competitive de- signs in realistic settings or by network performance in base station placement for maximal cov- erage. Statistical approaches (3GPP TR 38.901) represent propagation as a generative model where the parameters of probabilistic model are fitted to measurements. On the other hand, wireless ray- tracing approaches (Remcom; Amiot et al., 2013; Hoydis et al., 2022) approximate wave propaga- tion using geometric optics principles: propagation paths between a transmit and receive antenna estimated, and corresponding per-path characteristics (c. g,, time-of-flight) are calculated. They are sufficiently accurate in many use cases and do not require expensive field measurement collec- tion campaigns.
Neural wireless simulations. Both statistical and ray-tracing simulation techniques are accompa- nied by their own shortcomings, subsequently mitigated by their neural counterparts. Neural sur- rogates for statistical models (Ye et al., 2018; Shea et al., 2019; Dörner et al., 2020; Orekondy al., 2022a) reduce the amount and cost of measurements required. Neural ray tracers (Orekondy et al., 2022b; Hoydis et al., 2023; Zhao et al., 2023) address the non-differentiability of simula- tors using NeRI-like strategy (Mildenhall al., 2020) by parameterizing the scene using spatial MLP and rendering wireless signals using classic ray-tracing or volumetric techniques. While these techniques can be faster than professional ray tracers, they are similarly bottlenecked by expensive bookkeeping and rendering steps (involving thousands of forward passes). In contrast, we propose a framework simulate wireless signals with a single forward pass through a geometric transformer that is both sample-efficient and generalizes to novel scenes.
Geometric Algebra Transformer. The growing field of geometric deep learning (Bronstein al., 2021) aims to incorporate structural properties of problem into neural network architectures and algorithms. central concept is equivariance to symmetry groups (Cohen, 2021): a network f(x) is equivariant with respect to group G if its outputs transform consistently with any symmetry transformation g E G of the inputs, f(g- = g f(x), where denotes the group action. The Geometric Algebra Transformer (GATr) (Brehmer et al., 2023) is an E(3)-equivariant architecture for geometric problems, where E(3) is the three dimensional Euclidean group. We will show that this representation is particularly well-suited for wireless channel modelling. Second, GATr is transformer architecture (Vaswani et al., 2017). It computes the interactions between multiple tokens through scaled dot-product attention. With efficient backends like
(Dao al., 2022),
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the architecture is scalable to large systems, without any restrictions the sparsity of interactions like in paasagg-passing networks.
3 THE WIRELESS GEOMETRIC ALGEBRA TRANSFORMER (WI-GATR)
Our goal is to model the wireless channel, i. e., the interplay between the transmitted wireless sig- nal, the 3D environment, and the transmitting and receiving antennas. We first outline the specific problem setup before we provide the details of how we tokenize the environment in geometric alge- bra primitives that provide the foundation of our Wi-GATr backbone.
3.1 PROBLEM SETUP
fully-learnt forward model. The fundamental problem in simulating wire- less propagation is to model an accurate forward process: mapping simulation parameters v to the 'wireless channel' h which describes the transformation between the transmitted signal and the re- ceived signal. In this work, we propose fully-learnable surrogate ho (1b) for the forward process. The core idea is to implicitly learn physical propagation models that best explain the observations.
Wireless simulation:
Simulation inputs: parameters vb. A simulation scenario is defined by large set of parameters bb. Similar to prior works, we consider three groups of representative parameters: (a) scene geome- try and materials F: parameterized as a 3ID mesh, with each face associated with discrete material class; (b) transmit antenna properties æ!": parameterized by 3D position and orientation; (c) receive antenna properties ,x" also parameterized by 3D position and orientation. Note that in our setting, we assume that antenna radiation patterns are shared between all receivers and transmitters, respec- tively, and as a consequence, they need be learned implicitly. To the best of our knowledge, we present the first fully-learnt surrogate that works on the full 3D geometry (as opposed to 2D/2.5D image representations).
Simulation outputs: wireless channel h. For fixed set of simulation inputs, we want the sim- ulator to predict the wireless channel h. Since we propose a learnable model, we rely on channel measurements for supervised training which can be collected using conventional devices (e. g mo- bile phones). Measuring the exact channel challenging (Euchner et al., 2021), which why we describe the signal transformation by averaged statistics of the channel. Similar to prior works, we learn to predict realevalued, scalar channel characteristics h; non-coherent received power (Section received power, and band-limited delay spread (both Section 5.4).
5.1),
3.2 WI-GATR BACKBONE
We want a learnable forward- for simulations (tb),, that can effectively leverage a wireless scene ab to reason about spatially-varying channels h g., receive power). Our core insight is that the forward-process is inherently geometric: electromagnetic waves are emitted by an oriented trans- mit antenna, interact (e. g., reflect and refract) multiple times with various surfaces in space and fi- nally reach an oriented receive antenna at a different location. To exploit this inherent geometric na- ture of propagation, we propose 'Wi-GATr', Wireless Geometric Algebra Transformer (Wi-GATr) backbone. Wi-GATr consists of two main components: a tokenizer and network architecture. The tokenizer embeds the information of the wireless scene into geometric algebra while the network learns to model the channel. We now elaborate on these individual components.
Wireless GA tokenizer. The tokenizer takes as input information characterizing simulation pa- rameters &b and outputs a sequence of tokens that can be processed by the network. A key challenge of the tokenizer is efficiently representing a variable set of diverse data types. our case, the data types range from 3D environment mesh F, which features three-dimensiona, objects with dielectric material properties such as buildings, to antennas xx and xt characterized through a point-like po- sition, antenna orientation, and additional information about the antenna type, and the character- istics of the channel h.
To support all of these data types, propose new tokenizer that outputs a sequence of geometric algebra (GA) tokens. Each token consists of a number of elements (channels) of the projective geometric algebra G3,0,1 1 in addition to the usual unstructured scalar channels. We define the GA
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precisely in Appendix A. Its main characteristics are that each element is 16-dimensional vector and can represent various geometric primitives: 3D points including an absolute position, lines, planes, and so The richly structured space of projective geometrie algebra G3,0,1 is thus ideally suited to represent the different elements encountered in a wireless problem. It naturally supports scalars for wireless scalar fields (e. g., power, delay spread), points for positions (e. g-, antennas, mesh vertices), vectors for orientations (c. g., antennas), and planes for surfaces (c. g., mesh faces). Details of our tokenization scheme are specified in Appenndix B.
Network. After tokenizing, we process the input data with a Geometric Algebra Transformer (GATr) (Brehmer et al., 2023). This architecture naturally operates on our C3.0,1 parameterization of the scene. It is equivariant with respect to permutations of the input tokens as well as E(3), the symmetry group of translations, rotations, and reflections. These are exactly the symmetries of wireless signal propagation, with one addition: wireless signals have an additional reciprocity symmetry that specifies that the signal is invariant under an role exchange between transmitter and receiver. We will later show how we can incentivize this additional symmetry property through data augmentation. 2 Finally, because GATr is transformer, it can process sequences of variable lengths and scales well to systems with many tokens. Both properties crucial for complex wireless scenes, which can in particular involve a larger number mesh faces.
3.3 APPLICATIONS BEYOND FORWARD MODELLING
In the previous section, we proposed a Wi-GATr backbone for sewwadd-simulttion he (v)) We now highlight two additional benefits of the proposed backbone. First, exploiting differentiability of he to solve inverse problems. Second, adapting the Wi-GATr within a generative model formulation and thereby allowing probabilistic inferences (either for forward, or inverse problems).
Inverse problems solved by gradient descent. The differentiability of surrogate models make them well-suited to solve inverse problems. More generally, the differentiability enables gradient- based optimization of input data ei. simulation parameters) to minimize a given loss. For instance, the surrogates for receiver localization. Given 3D environment F, transmitters {>." ) and corresponding signals {hr). and a random initialization æti we can compute the gradients of the predicted channels V Vanho [F, 228 xx()) Using gradient descent, we iteratively refine the receiver position to solve 5xth = arg min w E. F, xr xx - -11122
Probabilistic inference with diffusion models. While a predictive model of the signal serve as a versatile neural simulator, it has two shortcomings. First, solving an inverse problem through gradient descent requires sizable computational cost for every problem instance. Second, predic- tive models are deterministic and do not allow us to model stochastic forward processes or express the inherent uncertainty in inverse problems. To overcome this, we draw inspiration from the in- verse problem solving capabilities of diffusion models using guidance (Chung et al., 2022; Gloeck- ler et al., 2024). We hereby follow the DDPM framework and use a Wi-GATr model as score estima- tor (denoising network) to learn a joint distribution Pa (F, x²", x h) between 3D environment mesh F, transmitter "X receiver ax and channel h, for single transmitter-receiver pair (see Fig. 1b). See Appendix F.2 for more details.
NEW DATASETS WITH DIVERSE SCENE GEOMETRY
To show the importance of geometry and symmetries in wireless signal propagation, we require dataset that comprises complex signal interactions with diverse geometries. While several datasets of wireless simulations and measurements exist (Orekondy et al., 2022b; Zhang et al., 2023; Alkhateeb, 2019; Alkhateeb et al., 2023), they either do not include geometric information, lack scene diversity or the signal predictions are not realistic. Therefore, we generate two datasets that feature indoor scenes and channel information at frequency of 3.5 GHz using Wireless InSite, state-of-the-ar ray-tracing simulator (Remcom). We focus on indoor scenes as transmission plays stronger role than outdoors. The datasets provide detailed characteristies for each path between Tx and Rx, such
We also experimented with a reciprocityccquivariant variation of the architecture, but that led to a marginally worse performance without a significant gain in sample efficiency. 'The datasets available https: / / gi thub. com/ /Qua comm- A Al-research/wi InSim.
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Ground truth
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Figure 2: Qualitative signal prediction results. We show a single floor plan from the WiPTR test set. The black lines indicate the walls and doors, the colors show the received power as function of the transmitter location (brighter colours mean stronger signal). The transmitting antenna is shown coordinates transmitter receiver are fixed to same height. compare the ground truth predictions (top left) the predictions from different predictive models, each trained on only 100 wi PTR floor plans. Wi- GATr is able to generalize to this unseen floor plan even with such small training set.
black cross. The
as gain, delay, angle of departure and arrival at Tx/Rx, and the electric field the receiver itself, which allows users to compute various quantities of interest themselves. See Appendix D for more details.
Wi3R dataset. Our first dataset focuses diversity of room layouts and their impact on the signal. We take 5000 layouts from Wi 3Rooms (Orekondy et al., 2022b) and randomly sample 3D positions and Rx positions. In Appendix D we provide more details and define training, validation, and test splits as well as an dutto--distribution set to test the robustness of different models.
WiPTR dataset. To increase complexity, we generate second, more varied, realistic dataset that includes material properties, varying number of faces, and varying layout dimensions. The floor layouts are based the P ProcTHOR-109 Ok dataset for embodied Al research (Deitke et al., 2022). This dataset consists of 12k different floor layouts, split into training, test, validation, and OOD sets described in Appendix D, WiPTR stands among wireless datasets in terms of geometric diversity of scenes, which enables the geometric deep learning community to study symmetries in electromagnetic signal propagation. Additionally, since is based ProcTHOR-10k, it can provide wireless signals as novel modality for embodied AI research.
5 EXPERIMENTS
In our experiments, we provide empirical evidence for the benefits of the inductive bias of our equivariant architecture, as well as the inference speed, differentiabilityy and expressiveness of fully- learnt neural simulation surrogates on simulated and real-world data. In addition, we demonstrate probabilistic inference through a diffusion model.
5.1 SIGNAL STRENGTH PREDICTION ON SIMULATED DATA
On the simulated data, we focus on the prediction of the time-averaged non-coherent received power h = Ep 2a where lapl is the received power of each independent ray tracing path We train surrogates ho(F, xxx _x)) that predict the power function of the Tx position and orientation "x Rx position and orientation xx and 3D environment mesh F, on both the Wi38 and WipTh PTR datasets. All models trained with reciprocity augmentation, i.e., randomly flipping Tx and Rx

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Wi3R dataset
WiPTR datasct
Wi-GATr (ours)
Wi-GATr (ours)
Transf.
SEGNN
PLVIT
Transf.
PLVIT
In distribution
Unseen Rx positions
0.63
1.14
0.92
4.52
0.39
0.62 0.69
1.27
Unseen floor plans
0.74
1.32
1.02
4.81
0.41
1.28
Symmetry transformations
Rotation
0.74
78.68
1.02
4.81
0.41
38.51
1.28
Translation
0.74
64.05
1.02
4.81
0.41
4.96
1.28
Permutation
0.74
1.32
1.02
4.81
0.41
0.69
1.28
Reciprocity
0.80
1.32
1.01
10.15
0.41
0.69
1.28
Out of distribution OOD layout
7.03
14.06
2.34
5.89
0.43
0.86
1.23
Table 1:
Signal prediction results. We show the mean absolute error on the received power in dB (lower is better, best in bold). Top: In-distribution performance. Middle: Generalization under symmetry transforma- Generalization out-of-distribution settings. In almost all settings, Wi-GATr is the highest- fidelity surrogate model.
tions. Bottom:
labels during training. This improves data efficiency, especially for the transformer baseline.
Baselines. addition to our Wi-GATr model, described in Sec. 3, we train several baselines. The first is vanilla transformer (Vaswani al., 2017), based on the same inputs and tokenization of the wireless scene, but without the geometric inductive biases. ablate the effect of our tokenization scheme, also test transformer without it, which represents the scene as simple sequence of 3D positions. Next, we compare to the E(3) eqquivaaaa SEGNN (Brandstetter et al., 2022b) on W13R. On Wi PTR, training this model was impractical it required too much memory (>800 GB). As state-of-the-art method for image-based wireless channel modelling, we train a PLVIT model (Hehn al., 2023). We have also considered WiNeRT (Orekondy al., 2022b) and Sionna RT (Hoydis et al., 2022) as baselines, but found them unsuited for the problem setup as explained in Appendix E. More details on our experiment setup and the baselines are also given in Appendix E.
In-distribution and out-of-distributio performance. In Fig. 2 we illustrate the prediction task on a WiPTR floor plan. We show signal predictions for the simulator as well as for surrogate models trained on only 100 floor plans. While the transformer and PLVIT show memorization artifacts, Wi- GATr able capture the propagation pattern, although this floor plan was not seen during training.
In Tbl. 1 we compare surrogate models trained on the full Wi38 and WiPTR datasets. Both when evaluating unseen Rx positions on seen training floor plans as well when evaluating on new scenes unseen during training, Wi-GATr offers the highest-fidelity approximation of the simulator. Wi- GATr as well as the equivariant baselines are by construction robust to symmetry transformations, while the performance of vanilla transformer degrades substantially. All methods but SEGNN struggle to generalize to an OOD setting on the Wi 3R dataset. This is not surprising given that the training samples almost perfectly robust under domain shift.
similar to each other. On the more diverse WiPTR dataset, Wi-GATr is
Data efficiency. Next, we study the data efficiency of the different surrogates in Fig. 3. Wi-GATr is more data-efficient than any other method with the exception the E33)-equivariant SEGNN, which performs similarly well for small number of training samples. This confirms that equivariance is useful inductive bias when data is scarce. But Wi-GATr scales better than SEGNN to larger number of samples, showing that architecture combines the small-data advantages of strong inductive biases with the large-data advantages of a transformer architecture. We find clear benefits of our wireless tokenizer when comparing the transformer baselines with and without the tokenizer.
Inference speed. One of the advantages of neural surrogates is their test- time speed. Both Wi-GATr factor of 20 faster than the ground-truth ray tracer (see Appendix E).
and transformer

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WI3R
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Figure 3: Signal prediction. We show the mean absolute error the received power as a function of the training data on Wi3R (left) and WiPTR (right). Wi-GATr outperforms the transformer and PLVIT baselines at any amount of training data, and scales better to large data or many tokens than SEGNN.
Figure 4: Rx localization error, function of the number of Tx. Lines and error band show mean and its standard error over 240 mea- surements.
(4) Uncandttioaal generation
(b) Receiver localization
(e) Geometry reconstraction
Signal atrer gih 46.8 ifem
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recetver
pcatioos.
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28.7
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62.7 dBm
Figure Probabilistic modelling. We formulate various tasks as sampling from unconditional or condi- tional densities of a single diffusion model. (a): Unconditional sampling wireless scenes p( a x h). (b): Receiver localization as conditional sampling from p(æ" F, x. h) for two different values of and xx". (c): Geometry reconstruction as conditional sampling from p( Fu] Fk, xt" xx",, h)) for two different values of h, keeping w". x"", F2 fixed.
5.2 INVERSE RECEIVER LOCALIZATION
We show how differentiable surrogates let us solve inverse problems, focusing on the problem of re- ceiver (Rx) localization. We infer the Rx position with the predictive surrogate models by optimiz- ing through the neural surrogate the simulator as discussed in Sec. 3.3. The performance of our surrogate models is shown in Fig. 4 and Appendix E.4 The two neural surrogates achieve a similar performance when only one or two transmitters are available, a setting in which the receiver position is ambiguous. With more measurements, Wi-GATr lets localize the transmitter more preciscly.
5.3
INVARIANT DIFFUSION FOR INVERSE PROBLEMS
In the following, consider the problem of signal prediction well as the inverse problems of receiver localization and geometry reconstruction. All three are instances of sampling from condi- tional densities: h po (h F, xx _r*),, xxxx ~ po (a FF ,xx h) and F ~ po (PF(F... x²* xxx,,,) h), respec- tively. We obtain the conditional densities by inpainting during the sampling from the joint densities (Sec. -icch---ickttenn et al. (2015)). To do so, we train a diffusion model using a Wi-GATr back- bone with the DDPM pipeline and 1000 denoising steps. We additionally apply conditional masking strategies to improve inpainting performance. See Appendix C for detailed description of our dif- fusion model, the masking protocol, as well discussion equivariant generative modelling. For
(h
'Niither the SEGNN nor PLVIT baselines are fully differentiable with respect to object positions when using the official implementations from Refs. (Brandstetter et al., 2022a; Hehn et al., 2023). We were therefore able accurately infer the transmitter positions with these architectures.
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comparison, we study a transformer baseline and a transformer trained the same data augmented with random rotations. More details on the model architectures can be found in Appendix E.
probabilistic, We qualitatively show results for this approach in Figs. 1 and 5. All predictions which allows our model to express uncertainty in ambiguous inference tasks. In Fig. 5 we visualize different conditional samples. For each example, sample from the same trained Wi-GATr diffu- sion model, while only varying the conditioning mask. In the case of receiver localization, we sam- ple single batch of points (F, æ²*, .... h))= where F, xxx and h are fixed and B denotes the batch size. In this case, we observe that the model learns multimodal densities corresponding to plausi- ble receiver locations. Furthermore, we see that the model is sensitive to change in signal strength h, with large values of h resulting in receiver locations close to the given transmitter x"x, whereas small values of h result in sample locations that avoid direct line of sight with the transmitter. Sim- ilarly, when reconstructing geometry, the model will sample diverse floor plans as long as they are consistent with the transmitted signal, see the right panel of Fig. 5. Additional results on signal and geometry prediction are given in Appendix E.2.
We quantitatively evaluate these mod- els through the variational lower bound on the log likelihood of test data under the model. To further analyze the ef- fects of equivariance, we test the model both on canonicalized scenes, in which all walls are aligned with the x and y axis, and scenes that are arbitrarily ro- tated. The results Tbl. show that Wi-GATr outperforms the transformer baseline across all three tasks, even in the canonicalized setting or when the transformer is trained with data aug- mentation. The gains of Wi-GATr are particularly clear on the signal predic- tion and receiver localization problems.
Transformer
Wi-GATr (ours)
default
data augm.
Canonicalized scenes
Signal pred. Receiver loc. Geometry reco.
1.62
3,00
15.66
3.64
8,28
14.42
-3.95
-3.61
-2.10
Scenes in arbitrary rotations
Signal pred.
1.62
9.57
17.65
Receiver loc. Geometry reco.
3.64
105.68
14.45
-3.95
389.34
-2.34
Table 2: Probabilistic modelling results. We show variational upper bounds on the negative log likelihood for different condi- tional inference tasks (lower is better, best in bold).
5.4 COMPARISON TO DIFFERENTIABLC RAY TRACING ON WOALDW DATA
The utility of wireless channel simulation is idcally evaluated on real data. Unfortunately, the scarcity of available measurement data limits evaluation to simple scenarios and prevents thorough testing of the geometric generalization capabilities. Yet, we aim to highlight limitations of hybrid ray tracing methods compared to our fully-learnt approach. To this end, we replicate the evaluation setup of Hoydis et al. (2023) on the DICHASUS dataset (Euchner et al., 2021).
The dataset consists of single hallway with receiver arrays at each end and an occluded corri- dor in between. During collec- tion, a robot-mounted transmit- ter was moving through the hall- way, covering most of the area, with accurate location data com-
3.0
Mean Reletive Absolute Error [96]
1 8 &
Wi-GATI tours,
Tran afommer
Mean Aesaltee Emor (aa)
Wi-ATT (ours)
2.5
Neural Materials
Leamndd Materials
28
L
ing from a tachymeter. We train
-B
our model to predict the average received power and the average delay spread for each receiver ar- ray, using the same data splits and evaluation scheme as in Hoy- dis et al. (2023). We report the mean absolute (logarithmic) error on the average received power and the relative absolute error the average delay spread (see equa- tions 44& & 45 in Hoydis et al. (2023)). For this purpase, we can consider each array as a single receiver without modifications our Wi-GATr backbone. As baselines, their hybrid ap-
1000
Tx positions in training set
Tx positions in training set
Figure 6: Signal prediction on real measurements. As a function of the training data size, we show the mean absolute (logarithmic) error on the received power (left) and the relative absolute error the delay spread (right).

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Published as a conference paper at ICLR 2025
proaches named Learned Materials and Neural Materials.
In Figure 6, we show the performance in both prediction tasks with respect to the number of trans- mitter locations used during training. Each tramsmitter location corresponds to two data samples, one per receiver. The Learned Materials approach maintains consistent performance due to model bias, making it robust with limited data but less flexible with larger datasets. Neural Materials also benefits from model bias with little data but, thanks to its flexible material model, can leverage data to overcome other model and geometry limitations. Our Our fully-learn! approaches Wi-GATr and the Transformer outperform both hybrid models as the dataset size grows, while their drawback of higher variance in the small data regime can be stabilized through pretraining on WiPTR (indicated by "pt"). We find that using Transformer with the Wi-GATr tokenization scheme slightly outper- forms GATr in the small data regime. However, we attribute this to simplicity of the dataset, where symmetry transformations of 3D space are neither observed nor required.
6 CONCLUSION
We developed a class of neural surrogates grounded in geometric representations and strong induc- tive biases. They are based on the Wi-GATr backbone, consisting of a new tokenization scheme for wireless scenes together with -(3)-qquivariant transformer architecture. In our experiments, we demonstrated the benefits of the inductive bias, inference speed, differentiability, and expressiveness of our fully-learnt equivariant neural simulation surrogate simulated and real- world data. In ad- dition, we showed how our backbone can be used in a diffusion model for probabilistic inference.
Limitations. Our paper presents step towards learnt wireless simulations, which has natural trade-offs when compared to model-based and hybrid ray tracers. Unlike model-based approaches (which require training data), Iearnable approaches depend on reasonably amounts of train- ing data and hence involve data collection overhead. Furthermore, model-based approaches are highly configurable (e. g-, scene parameterization, antenna patterns, controlling precision), whereas our approach is less configurable with many configuration choices learnt implicitly using training data. Finally, emphasize that we are not proposing drop-in replacement of model-based ray tracer, but rather take a step towards the overarching goal of learning wireless simulations from data.
Acknowledgements.
We thank Suresh Sharma, Pim de Haan, Maximilian Arnold, and Jens Petersen for fruitful discus- sions.
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A GEOMETRIC ALGEBRA
As representation, Wi-GATr uses the projective geometric algebra G3,0.1. Here we summarize key aspects of this algebra and define the canonical embedding of geometric primitives in it. For precise definition and pedagogical introduction, we refer the reader to Dorst (2020).
Geometric algebra. A geometric algebra Gpa.r consists of vector space together with a bilinear operation, the geometric product, that maps two elements of the vector space to another element of the vector space.
The elements of the vector space are known as multivectors. Their space is constructed by extending base vector space IR to lower orders (scalars) and higher-orders (bi-vectors, tri-vectors, ...). The algebra combines all of these orders (or grades) in one 2 pi-dimensional vector space. From basis for the base space, for instance (e1, 62; C3), one can construct basis for the multivector space. A multivector expressed in that basis then reads, for instance for d = 3, Zg + I161 + D262 + I363 + 12221e2 + 13 + #235263 + T 1123616263.
The geometric product is fully defined by bilinearity, associativity, and the condition that the geo- metric product of a vector with itself is equal to its norm. The geometric product generally maps between different grades. For instance, the geometric product of two vectors will consist of scalar, the inner product between the vectors, and a bivector, which related to the cross-prodde pross-product In particular, the conventional basis elements of grade k: are constructed as the geometric prod- uct of the vector basis elements e;. For instance, 61 = 6162 is a basis bivector. From the defining properties of the geometrie products it follows that the geometric product between orthogonal basis elements is antisymmetric, eze; -eees Thus, for d-dimensional basis space, there are (c) inde- pendent basis elements at grade k.
IR3
Projective geometric algebra. To represent stree-dimensional objects including absolute posi- tions, we use a geometric algebra based on a base space with 4, adding a homogeneous coordi- nafe to the 3D space.² We use a basis (eo, e,, e2 eg) with metric such that eg and ef 1 for i 1, 2, 3. The multivector space is thus 2 = 16-dimensional. This algebra is known as the pro- jective geometric algebra G3,0,1.
Canonical embedding of geometric primitives. In G3.,, we can represent geometric primitives as follows:
Scalars (data that do not transform under translation, rotations, and reflections) are represented as the scalars of the multivectors (grade 0).
Oriented planes are represented as vectors (k 1), encoding the plane normal as well as the
distance from the origin.
Lines or directions are represented as bivectors (k 2), encoding the direction as well as the shift from the origin.
Points or positions are represented as trivectors (k= 3).
For more details, we refer the reader Tbl. in Brehmer al. (2023), or to Dorst (2020).
B WIRELESS GEOMETRIC ALGEBRA TOKENIZER
Table 3 outlines how the individual scene parameters are embedded into Geometric Algebra.
C DIFFUSION MODEL
Formally, we employ the standard DDPM framework Song et al. (2021) to train a latent vari- able model pe (xx) variables following the dataset distribution Paata (xp). ). In DDPM, the latent variables X1.7 are noisy versions of the original data, defined by discrete forward noise process ((x{(x(11)
) where Xy = [F, ax xxx h) denotes the joint vector of
A diceedimensional base space is not sufficient to represent absolute positions and translations acting them in a convenient form. See Dorst (2020); Ruhe al. (2023); Brehmer et al. (2023) for in-depth discussion.
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N/ (xx: v1 -, 5:1) and 3i > O. We approximate the reverse distribution q(x;-1|xx (xx) with po(x (xx) = (N(x;11; po oT). The variance o² is typically chosen to match the vari- ance of (xx+-1|xx, Xo), which has been shown to follow a flosed-form normal distribution Ho et al. (2020). Furthermore, instead of parametrizing 14 to predict the mean ju of ((;; 1 Xo), opt to predict the denoised sample 8n- = fe (xx, t). For a discussion of these different parametrizations we refer (Ho al., 2020, Section 3.2). The forward and backward distributions q and p form varia- tional auto-encoder Kingma& & Welling (2014) which can be trained with a variational lower bound loss. Using the above parametrization of ( {x), however, allows for simple approximation of this lower bound by training
MSE objective & = Ex,,x No [Il (x,, t) xoll3] which resem- bles denoising score matching Vincent (2011). We provide an outline of the training algorithm in Algorithm I, which closely follows the proposed scheme of Ho et al. (2020).
To predict denoised samples using fa(x,, 1), we pass the raw representation of x through the wire- less GA tokenizer of Wi-GATr and, additionally, we embed the scalar through learned timestep embedding Peebles & Xie (2022). The embedded timesteps can then be concatenated along the sealar channels in the GA representation et al. (2023), the neural network outputs prediction in the GA representation, which is subsequently converted to the original latent space. Note that this possibly simplifies the learning problem, as the GA representation is inherently higher dimensional than our diffusion space with the same dimen- sionality as Xg.
straightforward manner. Similar to GATr Brehmer
Related Work. Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021) are a class of generative models that iteratively invert noising process. They have become the de-facto standard in image and video generation (Ramesh et al., 2022; Ho et al., 2022). Recently, they have also shown to yield promising results in the generation of spatial and sequential data, such as in planning (Janner et al., 2022) and puzzle solving (Hossieni et al., 2024). Aside from their generative modelling capabilities, diffusion models provide a flexible way for solving inverse problems (Chung et al., 2022; Lugmayr et al., 2022; Gloeckler et al., 2024) through multiplication with an appropriate likelihood term (Sohl-Dickstein et al., 2015). Furthermore, by combining an invariant prior distribution with an equivariant denoising network, one obtains equivariant diffusion models (Köhler et al., 2020). These yield a sampling distribution that assigns equal probability to all symmetry transformations of an object, which can improve performance and data efficiency in symmetry problems like molecule generation (Hoogeboom et al., 2022) and planning (Brehmer et al., 2024). We demonstrate similar benefits in modelling wireless signal propagation.
Equivariant generative modelling. A diffusion model with an invariant base density and an equivariant denoising network defines an invariant density, but equivariant generative modelling has some subtleties Köhler et al. (2020). Because the group of translations is not compact, we cannot define translation-invariint base density. Previous works have circumvented this issue by performing diffusion in the zero center of gravity subspace of euclidean space Hoogehoom et al.
Data type
Input parameterization
Tokenization
Channels (G3,0,1 embedding)
3D environment F
Triangular mesh
I token per mesh face
Mesh face center (point)
Vertices (points)
Mesh face plane (oriented plane)
Material classes
One-hot material emb. (scalars)
Antenna x² / /
Position
I token per antenna
Position (point)
Orientation
Orientation (direction)
Receiving/ /trassmitinnng
One-hot type embedding (scalars)
Additional characteristics
Characteristics (scalars)
Channel h
Antennas
1 token per link
Tx position (point)
Rx position (point)
Tx-Rx vector (direction)
Received power
Normalized power (scalar)
Phase, delay.... ...
Additional data (scalars)
Table 3: Wireless GA tokenizer. We describe how the mesh parameterizing the 3D environment and the information about antennas and their links are represented as a sequence of geometric algebra tokens. The mathematical representation of G3,0.1 primitives like points orientated planes is described in Appendix A.
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Algorithm Diffusion Wi-GATr Training
1: Input Ø, Initialize 0
Specify mask probability distribution
2: repeat
3: Xg Pdota (xo)
4:
~~ Uniform (I,,..,;T)
5:
X; g(xx| Xn)
Create noisy sample > Sample binary conditioning mask Apply conditioning per token
6:
m ~ Pmask m; +)
7:
x m X (1 m) x
8:
Take gradient descent step on
V |||m (fo"x)" I m) xo) 1|
9:
until converged
Algorithm Diffusion Wi-GATr Sampling
1: Input m, xi
Specify conditioning mask and conditions
2: XT ~ (O
3:
for prr== = or.t=T...od
4:
= X2 + (1 m) Ox
Apply conditioning per token >. Predict denoised sample Compute predicted sample t 1 from Xo
5: Xy fo m)
6:
X;-1 (xx
7: end for
8:
return 250
(2022). However, we found that directly providing the origin as an additional input to the denoising network also resulted in good performance, at the cost of full E(3) equivariance. We also choose to generate samples the convention where the z-axis represents the direction of gravity and positive is "up"; ; therefore provide this direction of gravity an additional input to our network.
Unifying forward prediction and inverse problems as conditional sampling. A diffusion model trained to learn the joint density po F, ax xx*, hh) does not only allow to generate unconditional samples of wireless scenes, but also lets us sample from various conditionals: given a partial wire- less scene, we can fill in the remaining details, in analogy to how diffuusion models for images al- low for inpainting. To achieve this, we use the conditional sampling algorithm proposed by Sohl- Dickstein et al. (2015): each step of the sampling loop, we fix the conditioning variables to their known values before feeding them into the denoising network. This algorithm lets us solve signal prediction (sampling from pa (h) F, xx x²*)), )), receiver localization (from pe (a** xxx h)), geom- etry reconstruction (from po Ful FR, t0 x²x (e)),))) or any other inference task in wireless scenes. We thus unify "forward" and "inverse" modelling in single algorithm. Each approach probabilistic, enabling us to model uncertainties. This is important for inverse problems, where measurements of- ten underspecify the solutions.
Masking strategies. In principle, the unconditional diffusion objective should suffice to enable test-time conditional sampling. practice, we find that we can improve the conditional sampling performance with two modifications. First, we combine training on the unconditional diffusson ob- jective with conditional diffusion objectives. For the latter, we randomly select tokens to condi- tion on and evaluate the diffusion loss only on the remaining tokens. Second, we provide the bi- nary conditioning mask as an additional input to the denoising model. Non-zero entries in the mask indicate what data we wish to generate, while zero entries are the ones we condition on. We sample masks from a discrete categorical distribution with probabilities & = (0.2, 0.3, 0.2, 0.3) corresponding masks for unconditional, signal. receiver and mesh prediction. respectively. If we denote this distribution over masks as Pmass (m: Φ), the modified loss function then reads as = Emnrppaas (m..)....x.. xo [[]m O fe(x;", m) - m @ xol where x," is equal to xo along the masked tokens according to m. To see how the masks are used during training and inference, we fer to Algorithm and 2. We note that while any arbitrary mask m of adequate dimensionality can be passed into the model, we only test masks m supp (Pmask (m: Φ)). That is, we only sample from the unconditional distribution or a conditional distribution corresponding to signal prediction, receiver localization and geometry reconstruction, while respectively keeping all other tokens fixed.
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D DATASETS
Table 4 summarizes major characteristics of the two datasets. In the following we explain more details on data splits and generation.
Wi3R dataset. Based on the layouts of the Wi3Rooms dataset by Orekondy et al. (2022b), we run simulations for 5000 floor layouts that are split into training (4500), validation (250), and test (250), These validation and test splits thus represent generalization across unseen layouts, transmitter, and receiver locations. From the training set, we keep 10 Rx locations additional test set evaluate generalization only across unseen Rx locations. To evaluate the generalization performance, we also introduce an out-of-distribution (OOD) set that features four rooms in each of the 250 floor layouts. In all layouts, the interior walls are made of brick while exterior walls are made of concrete. The Tx and Rx locations are sampled uniformly within the bounds of the floor layouts (10m x 5m X 3m).
WiPTR dataset. Based on the floor layouts in the ProcTHOR-10k 10k dataset for embodied Al re- search Deitke al. (2022), we extract the 3D mesh information including walls, windows, doors, and door frames. The layouts comprise between to 10 rooms and can cover up to 600 m². We sign different dielectric materials for different groups of objects (see Tbl. 5). The 3D Tx and Rx locations randomly sampled within the bounds of the layout. The training data comprises 10k floor layouts, while test and validation sets each contain 1k unseen layouts, Tx, and Rx locations. Again, we introduce OOD validation set with 5 layouts where manually remove parts of the walls such that two rooms become connected. While the multi-modality in combination with the ProcTHOR dataset enables further research for joint sensing and communication in wireless, our dataset set is also, to the best of our knowledge, the first large-scale 3D wireless indoor datasets suit- able for embodied AI research.
E EXPERIMENTS
E. PREDICTIVE MODELLING
Models. We use an Wi-GATr model that 32 blocks deep and 16 multivector channels in addition to 32 additional scalar channels wide. We use 8 attention heads and multi-query attention. Overall, the model has 1.6 10" parameters. These settings were selected by comparing five differently sized networks on an earlier version of the Wi3R dataset, though somewhat smaller and bigger networks achieved a similar performance.
Our Transformer model has the same width (translating to 288 channels) and depth as the Wi- GATr model, totalling 16.7 - 10 parameters. These hyperparameters were independently selected earlier version of the Wi3R dataset.
by comparing five differently sized networks
For SEGNN, we use representations of up to fmx = 3, layers, and 128 hidden features. The model has 2.6 - 10t parameters. We selected these parameters in scan over all three parameters, within
Wi 3R
WiPTR
Total Channels
5M
>5.5M
Materials
6
Transmitters per layout
1-15
Receivers per layout
200
Up 200
Floor layouts
5k
12k
Simulated frequency
3.5 GHz
3.5 GHz 6
Reflections
Transmissions
3
Diffractions
Strongest paths retained
25
25
Antennas
Isotropic Sinusoid
Isotropic Sinusoid
Waveform
Table 4: Dataset details and simulation settings for dataset generation.
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Published as a conference paper at ICLR 2025
WI3R
Mean Rox localization error [m)
Tranffrmer Wi-GATI (ours)
0.5
Number of transmitters
Figure Rx localization error. function of the number of Tx. Lines and error band show and its standard emror over 240 measurements.
the ranges used in Brandstetter et al. (2022b).
The PLVIT model is based on the approach introduced by Hehn et al. (2023). We employ the same centering and rotation strategy in the original approach around the Tx. Further, we extend the original approach to 3 dimensions by providing the difference in z-direction concatenated with the x-y-distance as one token. Since training from scratch resulted in poor performance, we finetuned a ViT-B- 1-16- model pretrained on ImageNet and keeping only the red channel. This resulted model with 85.4 107 parameters and also required us to use a fixed image size for each dataset that ensures the entire floor layout is visible in the image data.
Finally, we attempt to compare Wi-GATr also to WiNeRT (Orekondy et al., 2022b), a neural ray tracer. However, this architecture, which was developed to be trained several measurements on the same floor plan, was not able to achieve useful predictions on our diverse datasets with their focus generalization across floor plans.
Optimization. All models are trained on mean squared error between model output and the total received power in dBm. We batch size of 64 (unless for SEGNN, where smaller batch size due to memory limitations), the Adam optimizer, an initial learning rate of 3 and cosine annealing scheduler. Models trained for 5 10s steps the Wi3R dataset and for 2 10 on the WiPTR dataset.
Inference speed. To quantify the tradc-off between inference speed and accuracy of signal pre- diction, we compare the ray tracing simulation with our machine learning approaches. For this pur- pose, evaluate the methods and two equidistant grids at 2 E {2.3, 0.3) with each 1637 Rx locations.
single room of the validation set with different Tx locations
The left panel of Fig. 8 summarizes the average inference times per link with the corresponding standard deviation. While Wireless InSite (6/3/1, i.e., 6 reflections/3 transmissions/1 diffraction) represents method that used generate the ground truth data, it is also by far the slowest approach. Note that we only measure the inference speed of Wireless InSite for each Tx individually
Object
Material name
Rel. Permittivity [1] Conductivity [S/m]
Thickness [cm]
Cciling
ITU Ceiling Board
1.5
0.002148
0.95
Floor
ITU Floor Board
3.66
0,02392
3,0
Exterior walls
Concrete
7.00
0.0150
30.0
2.94
0.028148
1.30
ITU Layered Drywall (3 layers)
Interior walls
8.90
2.94
0.0281 148
1.30
Doors and door frames
ITU Wood
1.99
0,017998
3,0
Windows
ITU Glass 6.27
0.019154
0.3
Table 5: Dielectric material properties of objects in WiPTR.
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12
+
Wireless InSite
+
Sionna (GT mat.)
10a
Sionna (learned mat.)
Transformer
10a
MAE [dB)
WI-GATI (ours)
Tx-Rx
4-
10
2
10-1
10
10-4
10
10
10a
10a
10a
10
Time per link (s)
Number of tokens
Figure 8: Left: Inference wall time signal prediction enror per Tx/Rx prediction the first of the WiPTR validation set. Right: Scaling of the inference wall time as function of number of tokens (mesh faces, antennas, and links) in the scene for Wi-GATr.
without the preprocessing of the geometry. By reducing the complexity, e.g., reducing the number of allowed reflections or transmissions, of the ray tracing simulation the inference time be reduced significantly. For example, the configuration 3/2/1 shows a significant increase in inference speed, already see that the simulation results do not match the ground truth
but at the same time anymore. This effect is even more pronounced for the case Wireless InSite 3/1/1. As of version 0.19, Sionna RT does not support any transmission and only supports first order diffraction, i. e.. the path cannot include other reflection events. Therefore, in indoor scenarios such as WiPTR, where signals propagate through walls, Sionna is inaccurate but very fast due the simplified model and optimized implementation. In addition, it can improve its accuracy by learning more suitable material parameters to compensate for the model simplicity, yet fundamental gap remains. This shows that if the model is incomplete, the ground truth parameters (see Tab. 5) are not necessarily the optimal parameters. Our neural surrogates provide a favorable trade-off in terms of inference speed and prediction accuracy.
For larger scenes, the inference time of Wi-GATr grows quadratically in the number of mesh faces, just like for standard transformer. We show this behaviour in the right panel of Fig. 8. Recently, Suk et al. (2024) have successfully demonstrated possibilities to scale GATr to large mesh sizes.
In addition, the differentiability of ML approches enables them to solve inverse problems and such as finctuning to real-world measurement data. Finetuning, often referred to as calibration, remains challenging for simulation software and will likely lead increased MAE as the ground truth is not given by Wireless InSite itself anymore.
E.2 PROBABILISTIC MODELLING USING DIFFUSION
Experiment setup. For all conditional samples involving p(F,, F,, Fk, ax" x=" h), we always choose to set F& to be the floor and ceiling mesh faces only and F, to be the remaining geometry. This amounts to completely predicting the exterior walls, as well as the separating walls/doors of the three rooms, whereas the conditioning on F3 acts only always canonicalized in the non-auga -uggeeeddt training dataset, this allows for direct comparison of variational lower bounds in Tbl. 2 with the non-equivariant transformer baseline.
mean break equivariance. Since F
Models. For both Wi-GATr and the transformer baseline, we follow similar architecture choices as for the predictive models, using equal amount of attention layers. To make the models timestep- dependent, we additionally employ a standard learnable timestep embedding commonly used in diffusion transformers Peebles & Xie (2022) and concatenate it to the scalar channel dimension.
We use the Adam optimizer with a learning rate of 10-3 for the Wi-GATr models. The transformer models required smaller learning rate for training stability, and thus we chose 10 4. both cases, we linearly anneal the learning rate and train for 7 70 steps with a batchsize of 64 and gradient norm clipping 100.
Optimization.
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Published as a conference paper at ICLR 2025
10
WI3R
MAA [dBm)
Transformer
1
Wi-GAT (ours)
10
100
1000
4500
Training rooms
Figure 9: Mean absolute of received power function of number of training for conditional diffusion model samples.
Evaluation. We use the DDIM sampler using 100 timesteps for visualizations in Fig. 5 and for the error analysis in Fig. To evaluate the variational lower bound Tbl. 2, we follow Nichol & Dhariwal (2021) and evaluate Letb == Lo + L1 + ... LT, where Lo == log pe (xo|x1), Lt-1 D ( (xt-11 Xt, Xo) xo)]|po(x: (x; [x+)) and LT = DK ((xx((X))))))))))))))))))))))( p(x)). p(x)) To be precise, for each sample Xy the test set, get single sample x₃ from and evaluate Lui accordingly. Table
reports the mean of all Luib evaluations over the test set.
Additional results. Fig. 9, shows the quality of samples from po(h ax x²*) function of the amount of available training data, where we average over 3 samples for each conditioning input. worth noting that diffusion samples have slightly higher error than the predictive models. This shows that the joint probabilistic modelling of the whole scene is a more challenging learning task than deterministic forward model.
To further evaluate the quality of generated rooms, we analyze how often the model generates walls between the receiver and transmitter, compared to the ground truth. Precisely, we plot the distribu- tion of received power versus the distance of transmitter and receiver in Fig. 10 and color each point according a line of sight test. We can see that, overall, Wi-GATr has an intersection error of 0.26, meaning that in 26% of the generated geometries, line of sight was occluded, while the true geometry did not block line of sight between receiver and transmitter. This confirms that the diffusion model correctly correlates the received power and receiver/rransmitter positions with physically plausible geometries. While an error of 26% is non-negligible, we note that this task involves generating the whole geometry given only a single measurement of received power, making the problem heavily underspecified. Techniques such as compositional sampling (Du et al., 2023) could overcome this limitation by allowing to condition multiple receiver and received power measurements.
F DISCUSSION
Progress in wireless channel modelling is likely to lead to societal impact. Not all of it is positive. The ability to reconstruct details about the propagation environment may have privacy implications. Wireless networks are ubiquitous and could quite literally allow to see through walls. At the same time, we believe that progress in the development of wireless channel models may help to reduce radiation exposure and power consumption of wireless communication systems, and generally tribute to better and more accessible means of communication.
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Published as a conference paper at ICLR 2025
WI-GAT: intersection Erar.: 0.264
Transfomeer Intersection Erar: 0.261
Tue distribution
1,75
Line of Sight
1.50
1.50
Link distance
1.25
Link disaance
1.25
Link distance
1.25
1.00
1.00
1.00
0.75
0.75
0.75
0.50
0.50
0.25
Ocduded.
Occluded
0.25
Line of Sight
Line of Sight
0.00
Figure 10: A scatter plot of normalized received power versus normalized distance between receiver and transmitter. Each point is colored depending on having line of sight between the receiver and transmitter given the room geometry. Left: The geometry used for calculating line of sight is given by conditional diffusion samples using Wi-GATr. Middle: The geometry used for calculating line sight given by transformer samples. Right: The geometry used for calculating line of sight taken from the test data distribution.
21


Reviews:
[{"review_id": "rGC4bBEBFU", "reviewer": "Reviewer_ToCc", "summary": "The paper presents a learnable approach to tackle the problem of indoor wireless simulation. The proposed architecture is based on a Geometric Algebra Transformer, and a new tokenizer is introduced, allowing to leverage a 3D representation of the scene by taking 3D primitives as input. The model can also be integrated into an inverse problem framework based on diffusion, allowing to retrieve the position of the transmitter, the receiver or the geometry of the scene. Two new datasets for wireless simulation are also presented. Experiments are conducted in synthetic and real settings.", "questions": "- Why does data augmentation lead to poorer results in some cases in table 2 for the transformer baseline ?\n- Are the input coordinates of the transmitter/receiver 2D or 3D for the proposed model ?\n- For Rx interpolation in in-distributions experiments (l. ~329) in table 1, have the floor layouts been seen by the model during training ? If so, this should be explained more clearly, and why this setting is relevant.\n\nMinor remarks:\n- l. 297: |ap|^2 -> ap^2 ?\n- l. 315: while -> While\n- l. 789: The The -> The", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The presentation is clear and the paper is well-written. The problem at hand and coresponding challenges are well introduced to the reader.\n- The quantitative and qualitative results show the superiority of the method in multiple settings, and with regards to multiple variables (number of training samples, number of training rooms / transmitters, etc..\n- The versatility of the model is underlined by its adaptation to the inverse problem setting.", "weaknesses": "- Although the results on synthetic data are convincing regarding the contribution of the proposed architecture, the impact of the proposed architecture w.r.t. the transformer is not so clear on real data, although the authors explain this by the simplicity of the scene. \n- The most competitive baseline (SEGNN) is not evaluated on the WiPTR dataset.", "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": "", "extra_scores": {}}, {"review_id": "u1oc1vSgZ6", "reviewer": "Reviewer_z8wX", "summary": "In this paper, the authors introduce Wi-GATr, a fully learnable neural simulation surrogate designed to predict wireless channels based on indoor scene elements, including surface mesh, antenna position, and orientation. They employ an equivariant Geometric Algebra Transformer with a tokenizer for wireless simulation. The proposed method is validated using two distinct simulated datasets.", "questions": "(1) The novelty in the machine learning aspect of this paper is unclear. It appears that the work mainly leverages the equivariant Geometric Algebra Transformer for channel prediction. The authors should clarify which components in the machine learning section present new contributions.\n\n(2) The definitions of inverse problems lack clarity. For instance, the authors should provide a more detailed discussion and formulation for receiver localization. Additionally, the explanations of probabilistic inference with diffusion models on Page 5 are inconsistent with the discussion of diffusion models on Page 13, as the diffusion models used do not seem to follow the standard DDPM framework. The authors should include the training and sampling algorithms for the diffusion model utilized, as well as discuss the model's input.\n\n(3) It would be beneficial to compare the proposed method on public datasets and with other models related to channel prediction, such as NeRF and diffusion models.\n\n(4) Regarding generalization, the authors primarily validate their approach on two different datasets. It would be helpful to consider cross-dataset scenarios to assess performance in unseen conditions. Additionally, the authors should discuss the impact of parameters, such as varying simulated frequencies and the number of paths, on prediction performance.\n\n(5) The authors should proofread the paper to correct typographical errors, such as “The The Tx and Rx locations are sampled uniformly within the bounds of the floor layouts.”", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1) Wireless channel prediction is essential in wireless systems, and developing a fully learnable neural simulation surrogate for predicting wireless channels is an emerging topic.\n\n(2) The techniques and experimental results in this paper are solid. The authors apply their proposed model not only to channel prediction but also to two inverse problems: receiver localization and scenario generation. Various simulation results further validate the effectiveness of the proposed method.\n\n(3) The authors have developed two new 3D wireless datasets to validate their model, which would be valuable resources for the wireless research community if published.", "weaknesses": "(1) The novelty of this paper in the machine learning component is unclear.\n\n(2) The definitions of inverse problems lack clarity.\n\n(3) This paper lacks of comparisons with public datasets for channel prediction and other state-of-the-art channel prediction models with NeRF and diffusion models.", "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": "", "extra_scores": {}}, {"review_id": "Vp2PCQE5b1", "reviewer": "Reviewer_gkSN", "summary": "The paper introduces Geometric Algebra Transformer (GATr) into wireless channel observation problem and builds a learnable neural simulation surrogate Wi-GATr to predict channel states based on scene primitives. The authors design a Wi-GATr Backbone to exploit the inherent geometric nature of the propagation of wireless signals. Further, they apply this model to probabilistic inference and receiver localization problems. Experimental results show that Wi-GATr outperforms other methods on the two datasets they constructed.", "questions": "1. How were the two datasets generated? Were they extracted from other datasets or were they simulated themselves using other tools. Is this sufficient as one of the contributions of the paper?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. They design a new Wireless Geometric Algebra Transformer (Wi-GATr) backbone, which embeds the information of the wireless scene into geometric algebra while the network learns to model the channel.\n2. They develop a learnable forward-model for channel simulation and an inverse-model for receiver localization based on the differentiable properties.\n3. They build two new datasets with diverse scene geometry.", "weaknesses": "1. The problem of this work is not well identified. The authors only give the formulation of geometric algebra, but do not give any introduction of the wireless channel model. Wireless channels are complex and consist of many parameters. Authors need to specify what information about the channel they want to simulate and predict.\n2. The challenges that need to be addressed are not clearly stated. The authors introduce GATr into this work and build a backbone to make it fit for the wireless channel prediction problem. However, the difficulties and challenges of model transfer are not fully introduced. \n3. The innovation is somewhat limited. In addition to the designed backbone, the rest of the work consists only of two application experiments using the properties of the existing model.  \n4. The explanation for some of the pictures is inadequate. For example, Figure 1 shows the geometric surrogates for modeling wireless signal propagation. However, there is not enough explanation of this figure in the paper. It's hard to get the main point of it.", "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": "", "extra_scores": {}}, {"review_id": "xx42U6IxuH", "reviewer": "Reviewer_HmDB", "summary": "This paper introduces Wi-GATr, a novel learnable neural surrogate for wireless channel simulation that leverages geometric primitives such as 3D surfaces, antenna positions, and orientations. The primary focus is on addressing limitations in wireless signal propagation modeling by integrating geometric algebra transformers (GATr), which enhance efficiency and accuracy. Wi-GATr is shown to outperform existing models in tasks like signal strength prediction and receiver localization, achieving significant error reductions compared to existing methods.", "questions": "Listed above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper presents a new approach, Wi-GATr, which is a neural surrogate for wireless channel modeling using Geometric Algebra Transformers, a technique not widely applied in this field. This originality sets it apart from traditional methods by addressing key limitations in differentiability and scalability. The research is supported by thorough empirical evaluations across both simulated and real-world datasets, showing substantial improvements. \nThe introduction of two new datasets, Wi3R and WiPTR, further enhances the credibility and reproducibility of the results. The methodology and results are clearly presented. In terms of significance, this work makes contributions to both wireless communication and machine learning.", "weaknesses": "1. Adam is commonly used in deep learning applications, particularly image-processing tasks. However, wireless signal modeling involves different characteristics and challenges than image data. The authors would benefit from a more detailed discussion on why Adam was chosen, especially considering the fundamental differences between wireless signal modeling and typical image tasks.\n\n2. Wi-GATr’s generalization capabilities come from the E(3)-equivariant design of the Geometric Algebra Transformer (GATr). It would be valuable for the authors to provide more justification or discussion regarding their contribution and novelty in implementing or improving such a design.\n\n3. While the authors introduced their own datasets (Wi3R and WiPTR), the authors should provide a clearer justification for their choice of benchmarks, and using more widely recognized simulators such as WinProp or Wireless InSite could strengthen their work.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)