Probabilistic and Differentiable Wireless Simulation with Geometric Transformers

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 17:33:47

Abstract: Modelling the propagation of electromagnetic signals is critical for designing modern communication systems. While there are precise simulators based on ray tracing, they do not lend themselves to solving inverse problems or the integration in an automated design loop. We propose to address these challenges through differentiable neural surrogates that exploit the geometric aspects of the problem. We first introduce the Wireless Geometric Algebra Transformer (Wi-GATr), a generic backbone architecture for simulating wireless propagation in a 3D environment. It uses versatile representations based on geometric algebra and is equivariant with respect to E(3), the symmetry group of the underlying physics. Second, we study two algorithmic approaches to signal prediction and inverse problems based on differentiable predictive modelling and diffusion models. We show how these let us predict received power, localize transmitters, and reconstruct the 3D environment from the received signal. Finally, we introduce two large, geometry-focused datasets of wireless signal propagation in indoor scenes. In experiments, we show that our geometry-forward approach achieves higher-fidelity predictions with less
data than various baselines.

Main paper: 
 introduction: 
 Modern communication is wireless: more and more, we communicate via electromagnetic waves through the antennas of various devices, leading to progress in and adoption of mobile phones, automotive, AR/VR, and IoT technologies [12, 16] . All these innovations build upon electromagnetic wave propagation. Therefore, modelling and understanding wave propagation in space is a core research area in wireless communication, and remains crucial as we are moving toward new generations of more efficient and spatially-aware wireless technologies.Wireless signal propagation follows Maxwell's equations of electromagnetism and is often accurately modelled by state-of-the-art ray-tracing simulation software. However, these simulators take substantial time to evaluate for each scene, cannot be fine-tuned on measurements, and are (usually [29] ) not differentiable. This limits their usefulness for solving inverse problems.In contrast, neural models of signal propagation can be evaluated cheaply, can be trained on real measurements in addition to simulation, and are differentiable and thus well-suited for solving inverse problems. Several such approaches have been proposed recently, often using image-based representations of the inputs and outputs and off-the-shelf vision architectures [6, 23, 34, 35, 44, 46, 51, 52] . However, wireless surrogate modelling faces various challenges. Realistic training data is often scarce, requiring surrogate models to be data efficient. Wireless environments can consist of complex meshes. Finally, input and output data consist of a variety of data types, including the shape of extended 3D objects, point coordinates and spatial orientation of antennas, and information associated with the transmitted signal.Submitted to 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Do not distribute. In this work, we present a new approach to modelling wireless signal propagation. It is grounded in the observation that wireless propagation is inherently a geometric problem: a directional signal is transmitted by an oriented transmitting antenna, the signal interacts with surfaces in the environment, and the signal eventually impinges an oriented receiving antenna. We argue that it is critical for neural surrogates to model and flexibly represent geometric aspects (e. g. orientations, shapes) in the propagation environment. We therefore develop surrogate models based on flexible geometric representation and strong geometric inductive biases.We first propose the Wireless Geometric Algebra Transformer (Wi-GATr), a backbone architecture for wireless signal propagation problems. A key component is a new tokenizer for the diverse, geometric data of wireless scenes. The tokens are processed with a Geometric Algebra Transformer (GATr) network [9] . This architecture is equivariant with respect to the symmetries of wireless channel modelling, but maintains the scalability of a transformer architecture.Second, we study Wi-GATr models as differentiable, predictive surrogates for the simulator (see Fig. 1a ). Here the network predicts observables such as the received power as a function of transmitter position, receiver position, and 3D environment. We show how this enables forward modelling, and in addition, inverse problem solving due to Wi-GATr's differentiability.Next, we propose an alternative, more versatile probabilistic approach to prediction and inference tasks: training Wi-GATr diffusion models (Fig. 1b ) on the joint distribution of transmitter, receiver, channel information, and 3D environment. At test time, the model can be flexibly conditioned on any available information to predict the received power, localize a transmitter or receiver (Fig. 1c ), or even reconstruct the (full or partial) 3D geometry from the wireless signal (Fig. 1d ).To enable machine learning development for wireless problems, we finally introduce two new datasets, Wi3R and WiPTR. Each dataset consists of thousands of indoor scenes of varying complexity and include all the geometric information that characterizes a wireless scene.Finally, we demonstrate the predictive and the probabilistic models on these datasets. Our experiments show that the Wi-GATr approach gives us a higher-fidelity predictions than various baselines, generalizes robustly to unseen settings, and requires up to 20 times less data for the same performance than a transformer baseline. 
background and related work: 
 Wireless signal propagation. How do wireless signals propagate from a transmitting antenna (Tx) to a 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 [31] . It approximates the solution to Maxwell's equations as a sum of planar waves propagating in all directions from Tx. Each planar wave is represented as a ray, characterized 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 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(τ ) = p a p δ(τ -τ p ), where a p ∈ C is the complex gain and τ p the delay of the incoming rays [53] .Maxwell's equations and in extension ray propagation are highly symmetric. The received signal does not change under rotations, translations, and reflections of the whole scene, as well as the exchange of transmitter and receiver. The latter property is known as reciprocity [37] .Wireless simulators. Wireless propagation models play a key role in design and evaluation of communication systems, for instance by characterizing the gain of competitive designs in realistic settings or by optimizing systems performance as in base station placement for maximal coverage.Statistical approaches [2] represent propagation as a generative model where the parameters of a probabilistic model are fitted to measurements. On the other hand, wireless ray-tracing approaches [1, 5, 29] are increasingly popular due to their high accuracy and because they do not require expensive field measurement collection campaigns.Neural wireless simulations. Both statistical and ray-tracing simulation techniques are accompanied by their own shortcomings, subsequently mitigated by their neural counterparts. Neural surrogates for statistical models [19, 40, 42, 56] reduce the amount and cost of measurements required. Neural ray tracers [29, 41, 58] address the non-differentiability of simulators using a NeRF-like strategy [38] by parameterizing the scene using a spatial MLP and rendering wireless signals using classic ray-tracing or volumetric techniques. While these techniques are 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 to simulate wireless signals with a single forward pass through a geometric transformer that is both sample-efficient and generalizes to novel scenes.The growing field of geometric deep learning [11] aims to incorporate structural properties of a problem into neural network architectures and algorithms. A central concept is equivariance to symmetry groups [15] : a network f (x) is equivariant with respect to a group G if its outputs transform consistently with any symmetry transformation g ∈ G of the inputs,f (g • x) = g • f (x), where • denotes the group action. Of particular interest to us is the Euclidean group E(3) of isometries of 3D space, that is, transformations that leave Euclidean distances invariant. This group includes spatial translations, rotations, reflections, and their combinations. As we argued above, the physics of wireless signal propagation are invariant under this group.GATr. The Geometric Algebra Transformer (GATr) [9] is an E(3)-equivariant architecture for geometric problems. Among equivariant architectures, it stands out in two ways. First, it uses geometric (or Clifford) algebras [14, 22] as representations. For a rigorous introduction to these algebras, we refer the reader to Dorst [20] . From a practical machine learning perspective, these algebras define embeddings for various geometric primitices like 3D points, planes, or E(3) transformations. We will show that this representation is particularly well-suited for wireless channel modelling. Second, GATr is a transformer architecture [54] . It computes the interactions between multiple tokens through scaled dot-product attention. With efficient backends like FlashAttention [17] , the architecture is scalable to large systems, without any restrictions on the sparsity of interactions like in message-passing networks.Diffusion models. Diffusion models [25, 48, 50] are a class of generative models that iteratively invert a noising process. They have become the de-facto standard in image and video generation [26, 45] . Recently, they have also shown to yield promising results in the generation of spatial and sequential data, such as in planning [30] and puzzle solving [28] . Aside from their generative modelling capabilities, diffusion models provide a flexible way for solving inverse problems [13, 36] through multiplication with an appropriate likelihood term [48] . Furthermore, by combining an invariant prior distribution with an equivariant denoising network, one obtains equivariant diffusion models [33] . 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 [27] and planning [10] . We will demonstrate similar benefits in modelling wireless signal propagation.3 The Wireless Geometric Algebra Transformer (Wi-GATr)Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: We discuss the main limitations of our work in Sec. 6 and throughout the paper.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.• The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). 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problem formulation: 
 Our goal is to model the interplay between 3D environments, transmitting and receiving antennas, and the resulting transmitted wireless signals. More precisely, we consider wireless scenes consisting of:• The 3D geometry F of the environment. We specify it through a triangular mesh with a discrete material class associated with each mesh face.• A set of transmitting antennas t i for i = 1, . . . , n t . Each t i is characterized by a 3D position, an orientation, and any antenna characteristics. We will often focus on the case of a single Tx and then omit the index i.• Analogously, a set of receiving antennas r i for i = 1, . . . , n r .• The channel or signal h ij between each transmitter i and each receiver j, which can be any observable function of the CIR.In this setting, we consider various downstream tasks:• Signal prediction is about predicting the signal received at a single antenna from a single receiver, p(h|F, t, r) with n t = n r = 1. This is exactly the task that ray-tracing simulators solve. Often, the signal is modelled deterministically as a function h(F, t, r).• Receiver localization: inferring the position and properties of a receiving antenna from one or multiple transmitters, r ∼ p(r|F, {t i }, {h i }), with n r = 1.• Geometry reconstruction or sensing: reconstructing a 3D environment partially, inferring p(F u |F k , t, r, h), where F u and F k are the unknown and known subsets of F , respectively.The latter two problems are examples of inverse problems, as they invert the graphical model that simulators are designed for. They are not straightforward to solve with the simulators directly, but we will show how neural surrogates trained on simulator data can solve them.Core to our approach to this family of inference problems is the Wireless Geometric Algebra Transformer (Wi-GATr) backbone. It consists of a novel tokenizer and a network architecture.Wireless GA tokenizer. The tokenizer takes as input some subset of the information characterizing a wireless scene and outputs a sequence of tokens that can be processed by the network. A key challenge in the neural modelling of wireless problems is the diversity of types of data involved. As we argued above, a wireless scene consists of the 3D environment mesh F , which features threedimensional objects such as buildings and trees, antennas t and r characterized through a pointlike position, an antenna orientation, and additional information about the antenna type, and the characteristics of the channel h. 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 or orientated planes is described in Appendix A.To support all of these data types, we propose a 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 G 3,0,1 in addition to the usual unstructured scalar channels. We define the GA precisely in Appendix A. Its main characteristics are that each element is a 16-dimensional vector and can represent various geometric primitives: 3D points including an absolute position, lines, planes, and so on. This richly structured space is ideally suited to represent the different elements encountered in a wireless problem. Our tokenization scheme is specified in Tbl. 1.Network. After tokenizing, we process the input data with a Geometric Algebra Transformer (GATr) [9] . This architecture naturally operates on our G 3,0,1 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 exception: 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. 1Finally, because GATr is a transformer, it can process sequences of variable lengths and scales well to systems with many tokens. Both properties are crucial for complex wireless scenes, which can in particular involve a larger number of mesh faces.The Wi-GATr backbone can be used either in a predictive or probabilistic ansatz. We begin with the predictive modelling of the measured channel information as a function of the complete 3D environment and the information characterizing the transmitter and receiver, h θ (F, t, r). This regression model is trained in a supervised way on simulated or measured wireless scenes.Forward prediction. The network thus learns a differentiable, deterministic surrogate for the simulator model h sim (F, t, r). At test time, we can use the network instead of a simulator to predict the signals in unseen, novel scenes. Compared to a simulator based on ray tracing, it has three advantages: it can be evaluated in microseconds rather than seconds or minutes, it can be finetuned on real measurements, and it is differentiable.Inverse problems. This differentiability makes such a surrogate model well-suited to solve inverse problems. For instance, we can use it for receiver localization. Given a 3D environment F , transmitters {t i }, and corresponding signals {h i }, we can find the most likely receiver position and orientation as r = arg min r i ∥h θ (F, t i , r) -h∥ 2 . The minimization can be performed numerically through gradient descent, thanks to the differentiability of the Wi-GATr surrogate.While a predictive model of the signal can serve as a powerful neural simulator, it has two shortcomings. Solving an inverse problem through gradient descent requires a sizable computational cost for every problem instance. Moreover, predictive models are deterministic and do not allow us to model stochastic forward processes or express the inherent uncertainty in inverse problems.Equivariant diffusion model. To overcome this, we draw inspiration from the inverse problem solving capabilities of diffusion models using guidance [13] . In this case, we formulate the learning problem as a generative modelling task of the joint distribution p θ (F, t, r, h) between 3D environment mesh F , transmitter t, receiver r, and channel h, for a single transmitter-receiver pair. Concretely, we follow the DDPM framework and use a Wi-GATr model as score estimator (denoising network).By using an invariant base density and an equivariant denoising network, we define an invariant generative model. See Appendix B for a detailed description of our diffusion model and the discussion of some subtleties in equivariant generative modelling.Unifying forward prediction and inverse problems as conditional sampling. A diffusion model trained to learn the joint density p θ (F, t, r, h) does not only allow us to generate unconditional samples of wireless scenes, but also lets us sample from various conditionals: given a partial wireless scene, we can fill in the remaining details, in analogy to how diffusion models for images allow for inpainting. To achieve this, we use the conditional sampling algorithm proposed by Sohl-Dickstein et al. [48] : at 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 p θ (h|F, t, r)), receiver localization (from p θ (r|F, t, h)), geometry reconstruction (from p θ (F u |F k , t, r, h)), or any other inference task in wireless scenes. We thus unify "forward" and "inverse" modelling in a single algorithm. Each approach is probabilistic, enabling us to model uncertainties. This is important for inverse problems, where measurements often underspecify the solutions.In principle, the unconditional diffusion objective should suffice to enable test-time conditional sampling. In practice, we find that we can improve the conditional sampling performance with two modifications. First, we combine training on the unconditional diffusion objective with conditional diffusion objectives. For the latter, we randomly select tokens to condition on and evaluate the diffusion loss only on the remaining tokens. Second, we provide the conditioning mask as an additional input to the denoising model. See Appendix B for details.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [NA] Justification: Our paper does not include theoretical results.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.• All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.• Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.• Theorems and Lemmas that the proof relies upon should be properly referenced. 
new datasets: 
 While several datasets of wireless simulations and measurements exist [3, 4, 41, 57] , they either do not include geometric information, are not diverse, are at a small scale, or the signal predictions are not realistic. To facilitate the development of machine learning methods with a focus on geometry, we generate two new datasets of simulated wireless scenes. 2 Both feature indoor scenes and channel information generated with a state-of-the-art ray-tracing simulator [1] at a frequency of 3.5 GHz.They provide detailed characteristics for each path between Tx and Rx, such as gain, delay, angle of departure and arrival at Tx/Rx, and the electric field at the receiver itself, which allows users to compute various quantities of interest themselves. See Appendix C for more details.Wi3R dataset. Our first dataset focuses on simplicity: each of 5000 floor plans has the same size and number of rooms, and all walls have the same material across layouts. They differ only in their layouts, which we take from Wi3Rooms [41] , Tx positions, and Rx positions. In Appendix C we define training, validation, and test splits as well as an out-of-distribution set to test the robustness of different models. is also suited for the integration with embodied AI research.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes]Justification: See Appendix D.Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.• If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.• Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.• While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 
predictive modelling: 
 We focus on the prediction of the time-averaged non-coherent received power h = p |a p | 2 , disregarding delay or directional information that may be available in real measurements. We train predictive surrogates h θ (F, t, r) that predict the power as a function of the Tx position and orientation t, Rx position and orientation r, and 3D environment mesh F , on both the Wi3R and WiPTR datasets.All models are trained with reciprocity augmentation, i. e., randomly flipping Tx and Rx labels during training. This improves data efficiency slightly, especially for the transformer baseline.In addition to our Wi-GATr model, described in Sec. 3, we train several baselines. The first is a vanilla transformer [54] , based on the same inputs and tokenization of the wireless scene, but without the geometric inductive biases. Next, we compare to the E(3)-equivariant SEGNN [8] , though we were only able to fit this model into memory for the Wi3R dataset. In addition, we train a PLViT model, a state-of-the-art neural surrogate for wireless scenes [24] that represent wireless scenes as an image centered around the Tx position. Finally, we attempt to compare Wi-GATr also toWiNeRT [41] , a neural ray tracer. However, this architecture, which was developed to be trained on several measurements on the same floor plan, was not able to achieve useful predictions on our diverse datasets with their focus on generalization across floor plans. Our experiment setup and the baselines are described in detail in Appendix D.Signal prediction. 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. Despite the performance of a vanilla transformer degrades substantially. All methods but SEGNN struggle to generalize to an OOD setting on the Wi3R dataset. This is not surprising given that the training samples are so similar to each other. On the more diverse WiPTR dataset, Wi-GATr is almost perfectly robust under domain shift.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 of the E(3)-equivariant SEGNN, which performs similarly well for a small number of training samples. This confirms that equivariance is a useful inductive bias when data is scarce. But Wi-GATr scales better than SEGNN to larger number of samples, showing that our architecture combines the small-data advantages of strong inductive biases with the large-data advantages of a transformer architecture.Inference speed. One of the advantages of neural surrogates is their test-time speed. Both Wi-GATr and a transformer are over a factor of 20 faster than the ground-truth ray tracer (see Appendix D).Receiver localization. Next, we show how differentiable surrogates let us solve inverse problems, focusing on the problem of receiver localization. We infer the Rx position with the predictive surrogate models by optimizing through the neural surrogate of the simulator as discussed in Sec. 3.3.The performance of our surrogate models is shown in Fig. 4 and Appendix D. 3 The two neural surrogates achieve a similar performance when only one or two transmitters are available, a setting in which the receiver position is highly ambiguous. With more measurements, Wi-GATr lets us localize the transmitter more precisely.Next, we experiment with our probabilistic approach. We train diffusion models on the Wi3R dataset.In addition to a Wi-GATr model, we study a transformer baseline, as well as a transformer trained on the same data augmented with random rotations. Both models are trained with the DDPM pipeline with 1000 denoising steps and samples from with the DDIM solver [49] . Our setup is described in detail in Appendix D.Signal prediction, receiver localization, and geometry reconstruction as conditional sampling.In our probabilistic approach, signal prediction, receiver localization, and geometry reconstruction are all instances of sampling from conditional densities: h ∼ p θ (h|F, t, r), r ∼ p θ (r|F, t, h), andF u ∼ p θ (F u |F k , t,r, h), respectively. We qualitatively show results for this approach in Figs. 1 and 5 . All of these predictions are probabilistic, which allows our model to express uncertainty in ambiguous inference tasks. When inferring Rx positions from a single measurement, the model learns multimodal densities, as shown in the middle of Fig. 5 . 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 D.2. Canonicalized scenes Signal pred. We quantitatively evaluate these mod- Our analysis is in many ways a first step. The range of materials in our datasets is limited and we only experimented with measurements of the non-coherent total received power, which is a stable signal, but offers less spatial information than measurements of the time delay or angular information. More importantly, we only considered idealized inference tasks. For instance, our receiver localization problem assumed perfect knowledge of the room geometry and materials.Nevertheless, we hope that we were able to highlight the benefits of a geometric treatment of wave propagation modelling. Augmenting or replacing the image-based or general-purpose representations and architectures prevalent in wireless modelling with geometric approaches has the potential of improving data efficiency, performance, and robustness. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper.• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [No] Justification: We provide an estimate of epistemic uncertainties for some of our experiments.Due to compute restrictions, we were not able to provide meaningful estimates of epistemic uncertainties for the other experiments yet.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).• The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)• The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean.• It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.• For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).• If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
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crowdsourcing and research with human subjects: 
 Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?Answer: [NA]Justification: The paper does not involve crowdsourcing nor research with human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.• According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?Answer: [NA]Justification: The paper does not involve crowdsourcing nor research with human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.• We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.• For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.As representation, Wi-GATr uses the projective geometric algebra G 3,0,1 . Here we summarize key aspects of this algebra and define the canonical embedding of geometric primitives in it. For a precise definition and pedagogical introduction, we refer the reader to Dorst [20] .Geometric algebra. A geometric algebra G p,q,r consists of a 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 a base vector space R d to lower orders (scalars) and higher-orders (bi-vectors, tri-vectors, . . . ). The algebra combines all of these orders (or grades) in one 2 d -dimensional vector space. From a basis for the base space, for instance (e 1 , e 2 , e 3 ), one can construct a basis for the multivector space. A multivector expressed in that basis then reads, for instance for d = 3, x = x ∅ + x 1 e 1 + x 2 e 2 + x 3 e 3 +x 12 e 1 e 2 + x 13 e 1 e 3 + x 23 e 2 e 3 + x 123 e 1 e 2 e 3 .The geometric product is fully defined by bilinearity, associativity, and the condition that the geometric 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 a scalar, the inner product between the vectors, and a bivector, which is related to the cross-product of R 3 . In particular, the conventional basis elements of grade k > 1 are constructed as the geometric product of the vector basis elements e i . For instance, e 12 = e 1 e 2 is a basis bivector. From the defining properties of the geometric products it follows that the geometric product between orthogonal basis elements is antisymmetric, e i e j = -e j e i . Thus, for a d-dimensional basis space, there are d k independent basis elements at grade k.Projective geometric algebra. To represent three-dimensional objects including absolute positions, we use a geometric algebra based on a base space with d = 4, adding a homogeneous coordinate to the 3D space. 4 We use a basis (e 0 , e 1 , e 2 , e 3 ) with a metric such that e 2 0 = 0 and e 2 i = 1 for i = 1, 2, 3. The multivector space is thus 2 4 = 16-dimensional. This algebra is known as the projective geometric algebra G 3,0,1 .Canonical embedding of geometric primitives. In G 3,0,1 , 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 k = 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 to Tbl. 1 in Brehmer et al. [9] , or to Dorst [20] .Formally, we employ the standard DDPM framework [50] to train a latent variable model p θ (x 0 ) = p θ (x 0:T )d x 1:T , where x 0 = [rsrp, tx, rx, mesh] denotes the joint vector of variables following the dataset distribution p data (x 0 ). In DDPM, the latent variables x 1:T are noisy versions of the original data, defined by a discrete forward noise process q(x t |x t-1 ) = N x t ; √ 1 -β t x t-1 , β t I and β i > 0. We approximate the reverse distribution q(x t-1 x t ) with p θ (x t-1 |x t ) = x0 q(x t-1 |x t , x0 )p θ (x 0 |x t , t), where q(x t-1 |x t , x 0 ) is a normal distribution with closed-form parameters [25] . The forward and backward distributions q and p form a variational autoencoder [32] which can be trained with a variational lower bound loss. Using the above parametrization of p θ (x t-1 |x t ), however, allows for a simple approximation of this lower bound by training on an MSE objective L = E xt,x0 ||f θ (x t , t) -x 0 || 2 which resembles denoising score matching [55] .To parametrize p θ (x 0 |x t , t), we pass the raw representation of x t through the wireless GA tokenizer of Wi-GATr and, additionally, we embed the scalar t through a learned timestep embedding [43] . The embedded timesteps can then be concatenated along the scalar channels in the GA representation in a straightforward manner. Similar to GATr [9] , the neural network outputs a 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 dimensionality as x 0 .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 [33] . Because the group of translations is not compact, we cannot define a translationinvariant base density. Previous works have circumvented this issue by performing diffusion in the zero center of gravity subspace of euclidean space [27] . 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 in the convention where the zaxis represents the direction of gravity and positive z is "up"; we therefore provide this direction of gravity as an additional input to our network.Masking strategies. To improve the performance of conditional sampling, we randomly sample conditioning masks during training which act as an input to the model, as well as a mask on the loss terms. Namely, we sample masks from a discrete distribution with probabilities p = (0.2, 0.3, 0.2, 0.3) corresponding to masks for unconditional, signal, receiver and mesh prediction respectively. If we denote this distribution over masks as p(m), the modified loss function thenreads as L = E m∼p(m),xt,x0 ||m ⊙ f θ (x m t , t, m) -m ⊙ x 0 || 2 ,where x m t is equal to x 0 along the masked tokens according to m. In all layouts, the interior walls are made of brick while exterior walls are made of concrete. The The Tx and Rx locations are sampled uniformly within the bounds of the floor layouts (10m × 5m × 3m).WiPTR dataset. Based on the floor layouts in the ProcTHOR-10k dataset for embodied AI research [18] , we extract the 3D mesh information including walls, windows, doors, and door frames.The layouts comprise between 1 to 10 rooms and can cover up to 600 m 2 . We assign 6 different dielectric materials for different groups of objects (see Tbl. For SEGNN, we use representations of up to ℓ max = 3, 8 layers, and 128 hidden features. The model has 2.6 • 10 5 parameters. We selected these parameters in a scan over all three parameters, within the ranges used in Brandstetter et al. [8] .The PLViT model is based on the approach introduced by Hehn et al. [24] . We employ the same centering and rotation strategy as 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 Inference speed. To quantify the trade-off between inference speed and accuracy of signal prediction, we compare the ray tracing simulation with our machine learning approaches. For this purpose, we evaluate the methods on a single room of the validation set with 2 different Tx locations and two equidistant grids at z ∈ {2.3, 0.3} with each 1637 Rx locations. Figure 7 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 our method that was used to 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 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 can be reduced significantly. For example, the configuration 3/2/1 shows a significant increase in inference speed, but at the same time we can already see that the simulation results do not match the ground truth anymore. This effect is even more pronounced for the case of Wireless InSite 3/1/1. Our machine learning solutions outperform all tested configurations of Wireless InSite in terms of inference speed, while at the same time keeping competitive performance in terms of prediction accuracy (MAE) compared to the data generation simulation itself in a simpler configuration setting.In addition, the differentiability of ML approches enables them to solve inverse problems and such as finetuning to real-world measurement data. Finetuning, often referred to as calibration, remains challenging for simulation software and will likely lead to increased MAE as the ground truth is not given by Wireless InSite itself anymore.Experiment setup. For all conditional samples involving p(F u |F k , t, r, h), we always choose to set F k to be the floor and ceiling mesh faces only and F u 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 F k acts only as a mean to break equivariance. Since F is always canonicalized in the non-augmented training dataset, this allows for direct comparison of variational lower bounds in Tbl. 3 with the non-equivariant transformer baseline.Models. For both Wi-GATr and the transformer baseline, we follow similar architecture choices as for the predictive models, using an equal amount of attention layers. To make the models timestepdependent, we additionally employ a standard learnable timestep embedding commonly used in diffusion transformers [43] and concatenate it to the scalar channel dimension.Optimization. We use the Adam optimizer with a learning rate of 10 -3 for the Wi-GATr models.The transformer models required a smaller learning rate for training stability, and thus we chose 3 • 10 -4 . In both cases, we linearly anneal the learning rate and train for 7 • 10 5 steps with a batchsize of 64 and gradient norm clipping set to 100.Evaluation. We use the DDIM sampler using 100 timesteps for visualizations in Fig. 5 and for the error analysis in Fig. 8 . To evaluate the variational lower bound in Tbl. 3, we follow [39] and evaluate L vlb := L 0 + L 1 + . . . L T , where L 0 := -log p θ (x 0 |x 1 ), L t-1 := D KL (q(x t-1 |x t , x 0 )||p θ (x t-1 |x t )) and L T := D KL (q(x T |x 0 ), p(x t )). To be precise, for each sample x 0 on the test set, we get a single sample x t from q and evaluate L vlb accordingly. Table 3 reports the mean of all L vlb evaluations over the test set.Additional results. Fig. 8 , shows the quality of samples from p θ (h|F, t, r) as a function of the amount of available training data, where we average over 3 samples for each conditioning input. It is worth noting that diffusion samples have a 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 a 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 distribution of received power versus the distance of transmitter and receiver in Fig. 9 and color each point according to 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/transmitter 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 [21] could overcome this limitation by allowing to condition on multiple receiver and received power measurements.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 contribute to better and more accessible means of communication.We also experimented with a reciprocity-equivariant variation of the architecture, but that led to a marginally worse performance without a significant gain in sample efficiency.We are preparing the publication of the datasets.Neither the SEGNN nor PLViT baselines are fully differentiable with respect to object positions when using the official implementations from Refs.[7,24]. We were therefore not able to accurately infer the transmitter positions with these architectures.A three-dimensional base space is not sufficient to represent absolute positions and translations acting on them in a convenient form. See Brehmer et al.[9], Dorst[20], Ruhe et al.[47] for an in-depth discussion. 
 
 Appendix:  


Decision: The reviewers have indicated that the diffusion generator approach to modeling multi-path channels based on geometry and tokens is an interesting and potentially useful research direction.  Applying a group equivariant convolutional NN for this classical inherently linear reciprocal channel ray modeling application is logical.  The paper overall seems preliminary, and some key issues and questions have been raised by the reviewers. 

There are a number of important issues raised such as scalability, use in a realistic scenario, claims of much lower complexity compared to ray tracing, only exploring power prediction, and limitations of experiments and comparisons.

Simulations are carried out for 3D indoor models at 3.5 GHz (Cellular 5G uses this frequency), where the wavelength is 86mm. Consequently, even very small unmodeled objects and perturbations in the environment or antenna position can cause significant channel state fluctuation. Note also that ray tracing models are only appropriate for relatively higher frequencies, and not for all wireless communications bands.  It may be useful to indicate the frequency bands of interest and discuss applicability, scalability, 3D modeling, and other issues in this context.

The results do not consider changes in the environment, e.g., doors, people. This impacts the forward and inverse modeling accuracy and statistical variation observed.  Similarly, small calibration errors in antenna response (and across multiple antennas), placement, and timing can also lead to significant channel changes.

Mesh-based methods have well known tradeoffs in mesh size versus modeling accuracy, that are not fully considered.  This impacts scalability of mesh methods and is relative to the wavelength of interest and the environmental complexity and extent.

The pending release of the indoor floor scenes dataset could be a useful contribution to the community.

Reviews: review: paper_summary: paper_summary: The paper presents the Wireless Geometric Algebra Transformer (Wi-GATr), a new architecture for simulating wireless signal propagation in 3D environments. This model utilizes geometric algebra to handle the geometric complexities of wireless scenes and ensures E(3) equivariance to respect the symmetries of the physical problem. The authors introduce two datasets, Wi3R and WiPTR, to benchmark their model. Wi-GATr outperforms existing baselines in terms of prediction fidelity and data efficiency, and it can solve both forward (signal prediction) and inverse (receiver localization and geometry reconstruction) problems in wireless communication.
strength_weakness: strength_weakness: 1. Limited Real-World Testing: While the model performs well on the introduced datasets, its application in real-world, dynamic environments remains underexplored. Additional experiments in more varied and complex real-world scenarios, such as urban or industrial settings, would strengthen the paper.

2. Scalability and Computational Load: The paper could provide more detailed insights into the computational requirements and scalability of Wi-GATr. Understanding the model's performance with larger datasets and more complex environments would be valuable for practical deployment.

3. Generalizability Across Frequencies: The model is tested at a specific frequency (3.5 GHz). Evaluating its performance across different frequencies and under various signal conditions would provide a more comprehensive understanding of its robustness and generalizability.


4. Detailed Case Studies: While the paper presents strong experimental results, including more detailed case studies or examples of practical applications, such as network design or optimization in real-world environments, would illustrate the model's impact and practical benefits.
questions: questions: 1. The paper focuses on indoor scenes for evaluation. Have you considered testing Wi-GATr on datasets that include outdoor or mixed environments to assess its robustness and generalizability?

2. The paper mentions solving inverse problems like receiver localization and geometry reconstruction. Can you provide more detailed examples or case studies where these capabilities significantly outperform traditional methods?

3. Given the computational complexity, how does Wi-GATr perform in real-time applications? Can it be integrated into real-time wireless network management systems, and if so, what are the challenges?

4. The datasets introduced are focused on indoor environments. Have you considered incorporating additional data sources, such as satellite imagery or LIDAR data, to enhance the model's accuracy and applicability in different environments?

5. How does the model adapt to changes in the environment, such as new constructions or changes in material properties? Is there a mechanism for updating the model dynamically to reflect these changes?
limitations: limitations: The authors briefly discussed the limitation in Sec. 6.
paper_summary: paper_summary: The motivation for this work is that modeling the propagation of electromagnetic signals is critical for designing modern communication systems. Ray tracing simulators are not suitable for inverse problems or integration as channel models in designing communication systems.

In this context, the goal of this paper is to model the interplay between the 3D environment F, transmitting and receiving antennas (each characterized by a 3D position, orientation, and specific antenna characteristics) represented by t and r, respectively, and the signal h between each transmitter and receiver. The 3D geometry F is represented by a triangular mesh, where each triangle is assigned a material type from predefined classes, modeling both the shape and materials of the environment. Once the model is learned, three tasks can be performed:
1. Prediction of the received signal p(h∣F,t,r): The model is trained for this task. At test time, the network can predict signals in unseen, novel scenes. This approach is faster, fine-tunable on real measurements. The model obtained is also differentiable. This is referred to as the forward problem.
1. Localization of the receiver p(r∣F,t,h).
1. Sensing the environment p(F∣t,r,h).

The last two tasks are referred to as inverse problems. The model introduced is an adaptation of the Geometric Algebra Transformer called Wireless (Wi-GATr), used for simulating wireless propagation in a 3D environment. The authors also cast this problem as a generative modeling task of the joint distribution p(F,t,r,h) (from which the above three tasks can be accomplished) using Denoising Diffusion Probabilistic Models and Wi-GATr.
strength_weakness: strength_weakness: As per my understanding, the main weaknesses of the work are: 
1. The novelty of the work lies in tokenizing various geometric objects encountered in the wireless communication scene. However, the same tokenization is used in the vanilla transformer, making it unclear if the new tokenization provides any benefit.
1. As the authors point out, the channel is modeled only in terms of time-averaged non-coherent received power, missing crucial information such as time and direction of arrival, which are essential for modeling wireless environments.
1. While the proposed solutions seem general, most results are presented for the single antenna case. Additionally, the dataset includes only transmitting sinusoidal waveforms, which is limiting as it does not cover larger bandwidths. The wave propagation depends on frequency, and non-linearities can occur with wider bandwidths.
questions: questions: In addition to the weaknesses listed above, I have the following questions:

1. The authors should clarify how many transmit antennas were used in receiver localization and sensing, and why they believe the results are promising despite the limitations of using a larger number of antennas and ignoring time and angle of arrival information.
1. The authors should also clarify the challenges involved in their novel tokenization and in developing diffusion-based models with the Wi-GATr model.
limitations: limitations: The authors mention limitations in the Discussion section (Section 6).
paper_summary: paper_summary: Authors proposed transformer based ideas on very well studied area: Wireless environment simulation.
The key idea here is to capitalize Geometric Transformers to simulate radio environments. It true that wireless (directional) signal propagation is a ray tracing approach, meaning a highly directional wireless signal (Ray) may bounce off ambient surfaces or directly reach a receiver. The proposed method inserts geometric shapes in the environment as tokens in a transformer networks. The trained transformer predicts the received power at a given point in 3D space. Transformer is trained and evaluated using two datasets: Wi3R and WiPTR that simulate indoor signal propagation environments. In comparison to the baselines, transformer architecture requires 20 times less data.
strength_weakness: strength_weakness: 1. It is not clear if datasets are of any relevance to real world environment, since the primary challenge of any modelling problem is simulation to reality gap. Since wireless channel modelling is very well studied area, a novel contribution must take in to account such differences rather than results that show the ability of transformer architecture in modelling a wireless environment
2. There are several high fidelity radio propagation modelling software, perhaps it is important to consider datasets generated from such model, current evaluation is very limited and primitive. Fig 2 and 5 are no comparison to robust channel models that are available to wireless researcher and practitioners.
3. The modelling of radio environment is no clear, reviewer is of impression that several affects like diffraction, refraction are not considered in the datasets
4. The paper also has weakness: WiNeRT: Towards Neural Ray Tracing for Wireless Channel Modelling and Differentiable Simulations which is both papers have not considered user mobility: coherence time, coherence bandwidth
5. Current evaluation is only limited to indoor environments
6. Since the prior work has already established neural network architecture are useful modelling, to push the state of the art, it is important to show accurate modelling than yet another architecture to model wireless channel.
7. Upon inspecting table 2, the reviewer is afraid that there might be issue with results here. There is 80dBm difference is accuracy with transformers based modelling, usually such a difference is unacceptable, can author please explain the training of transformer model and why it produced such a large error. The reviewer is concerned that whether such sample point is a fair to benchmark againt
questions: questions: 1. The challenging problem of wireless environment simulation is how the simulation and real world measurement hold together.
2. Since, this is very old problem, the novel solution must show the ability to bridge gap between simulation and real channel traces
3. It is quite important show how well the proposed method simulates various wireless environment, currently such evaluation is missing.
4. Could you please provide any benchmarks with real-world channel traces datasets?
limitations: limitations: In reviewer's opinion authors have not sufficiently addressed all the limitations of the current work. I encourage authors to look at weakness section and update the limitations of the work in the current draft
paper_summary: paper_summary: This paper proposes the use of a transformer architecture to model electromagnetic propagation of physical systems. The approach is claimed to outperform existing methods by (i) computational efficiency (compared to raytracers) and (ii) enabling solving inverse problems. The method is evaluated on a number of benchmark tasks and the paper is accompanied by two new datasets.
strength_weakness: strength_weakness: The main body of the paper lacks the details of the proposed pipeline and transformer architecture. In fact, all of the interesting and technical details are relegated to the appendices. 

One of the main motivations of the paper seems to be that most raytracers are too slow. However, the authors seem to ignore recent projects, such as Instant RM (https://github.com/NVlabs/instant-rm) which can compute coverage maps in a few milliseconds, depending on the desired accuracy.

It is unclear why [29] is cited as a non-differentiable raytracer although it is, to my knowledge, the only raytracer that actually is. Instant RM is also differentiable and calibration results for both tools were already demonstrated. To be honest, I have the impression that the authors tried to cover up the fact that [29] is a powerful *differentiable* raytracer that enables solving inverse problems. 

Although the authors claim that channel impulse responses can be generated, this is not demonstrated in the paper. I think that this claim should be removed unless the authors demonstrate that it is actually feasible.

The description of scene geometry recovery is incomprehensible to me.

In Figure 7, the Wi-GATr is around 20ms for inference for a tiny indoor scene. The authors should compare this against Instant RM which can probably run even faster and is differentiable.

It would be good to get confidence information (e.g., standard deviation) in Fig. 3 and Fig. 4.
questions: questions: The authors should comment on the scalability of the proposed method to scenes with millions of triangles.

How well would a neural-network-based positioning pipeline do on the task used to generate Figure 4?
limitations: limitations: The paper lacks a detailed comparison to the capabilities of the differential raytracer from [29]. In fact, I feel that for the individual tasks, more baselines should be included. 

Scalability to very large datasets and extremely complex scenes is unclear. 

It is unclear whether the method generalizes to electromagnetic environments that are nonreciprocal (e.g., containing certain nonreciprocal metasurfaces). 

It is unclear whether the method generalizes to scenarios in which ray-tracing is inaccurate (e.g., scenarios at low carrier frequencies).
paper_summary: paper_summary: The paper proposes the use of the Wireless Geometric Algebra Transformer (Wi-GATr) to model signal propagation. Based on the Wi-GATr network, it introduces a differentiable prediction model and a diffusion model. Compared to traditional statistical and ray-tracing methods, the proposed approach not only addresses conventional signal prediction problems but also tackles inverse problems such as receiver localization and 3D environment reconstruction. Experimental results demonstrate the effectiveness of this method. Furthermore, the authors present two large-scale wireless signal propagation datasets.
strength_weakness: strength_weakness: 1) There is a gap between the challenges and the solutions. The author asserts that wireless surrogate modelling faces challenges like data scarcity and diverse data types. However, the lack of analysis on these issues makes the proposed solutions appear abrupt. It is recommended to provide insights that lead to the proposed solutions of this paper.
2) The innovation is somewhat limited. Wi-GATr primarily extends the GATr method into a wireless setting, with equivariance being a pre-existing property of the original framework. Apart from tokenizing input data, did the paper introduce any additional advancements? It would be beneficial for the authors to highlight these aspects.
3) The experimental evaluation is not sufficiently convincing. For more details, please refer to the "Questions*" part.
4) There are some typographical errors in the paper. For example, lines 56 and 57 do not correspond to Figure 1. Additionally, the abstract mentions transmitter localization, but the main text describes receiver localization, among other discrepancies.
questions: questions: 1) The contributions of Wi-GATr appear to be limited. What are the specific contributions of Wi-GATr, and how do they compare in detail with those of GATr?
2) In the background section, the authors mention a series of neural statistical models [19, 40, 42, 56] and neural ray tracers [29, 41, 58], claiming the proposed method is "sample-efficient and generalizes to novel scenes". However, the evaluation section only compares two methods, which seems relatively weak. What considerations are made when choosing the comparison methods?
3) In qualitative experiments, the proposed method is compared with ViT. Why is it not compared with the benchmarks used in quantitative experiments—SEGNN and PLViT?
4) The experimental results in Figure 5 lack the necessary explanations. For instance, does the "Geometry reconstruction" on the right side lack comparison with ground truth?
5) This paper introduces both differentiable prediction models and diffusion models, claiming that diffusion models overcome the limitations of differentiable prediction models. Could the author conduct experiments to compare these two models and validate this claim?
limitations: limitations: Sufficiently discussed.

Comments: title: Thanks for your response.
comment: I would like to thank the authors for their response, which did address some of my concerns. I did appreciate the additional results which demonstrated the benefits of the tokenization scheme. Based on the response and update, I am going to keep my initial score. Thank you and all the best.
**[Real-world datasets] Authors may find the following resources valuable for further evaluation**

Thanks for pointing out the resources. However, they do not apply since they are either simulated, small-scale, or designed for a different task. This further strengthens our argument that real-world data is lacking.

* https://nvlabs.github.io/sionna/ Sionna is a simulator not a real-world dataset. It has been sufficiently discussed in our rebuttal.
* https://www.deepsig.ai/datasets/ As you mention, these are also simulated datasets. On this website, the authors advise against using this data unless for historical or educational purposes.
* https://www.deepsense6g.net/: While being real-data, DeepSense6G tackles a different problem, i.e., beam and blockage prediction at mmWave. For this purpose, much simpler models are sufficient (see https://arxiv.org/abs/2401.17781, published at ICC 2024). Further, only a subset of the scenarios contain 3D information from LiDAR, and none of them contain meshes and material information. It is a different problem and not suitable to show the benefits of equivariant architectures.

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**Could you please present a comparative analysis Sionna simulator vs Remcom.**

We have provided a detailed comparison to Sionna throughout this rebuttal. The summary is that Sionna currently provides simplistic models for physical effects, e.g., lack transmission, single difraction only, no thickness of materials modeled. See Figs. R2 and R3 in the rebuttal pdf for quantitative and qualitative comparisons.

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**In the interest of community as a whole, it might be a good idea to use accessible and open-simulators, since we may not access to them**

With our plans to publish our dataset, we want to make our high fidelity simulations freely available to the community.

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**[Mobility is not considered] Could you please elaborate, I couldn't understand the above**

Mobility is an important problem, but it is not the only problem in wireless channel modeling. We tackle a different problem.

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**[Indoor vs outdoor scenarios] Since, reviewer thinks other way, could you please provide any references in the draft supporting the above**

This is a quote from a very recently accepted challenge at ICASSP 2025 (https://indoorradiomapchallenge.github.io/index.html)
> it's necessary to develop models tailored for indoor environments. In such cases, the refracted electromagnetic field components through obstacles play a more significant role in radio signal propagation, as opposed to the outdoor scenarios that is dominated by reflected field components. Therefore, accurate indoor radio map estimation requires accounting for the larger variety of construction materials and their electromagnetic properties.

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**Could you please provide a little more information, on how many experiments are performed and what is OOD data looks like.**

We have provided details about our OOD data in our paper (Sec. 4) and its appendix (Appendix C).
**... The computation of coverage maps is typically done using shoot-and-bounce, whose complexity is almost independent of the number of primitives in a scene.**

**This is incorrect.** The complexity of shoot-and-bounce (SBR) algorithms is *not* independent of the number of primitives in a scene if you include refraction and diffraction. In fact, it highly depends on the geometry of the scene.
* Refraction: At every refraction event, the incoming ray is split into two outcoming rays, thus increasing the complexity. One that is reflected and one that is refracted.
* Diffraction: Diffraction first requires finding edges in the scene. The number of edges increases with the number faces in the scene unless neighboring faces are entirely parallel (which they are not in practice).

As a result, it is common practice to limit the number of diffractions and refractions during ray tracing to find a trade-off between accuracy and scalability. For example, Sionna resorts to only modeling a single diffraction event at the end of an interaction chain. InstantRM (a fast SBR ray tracer) does not support diffraction. As mentioned before, none of them model refractions.

In contrast, the run time and scalability of our method is independent of the exact geometry of the mesh.

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**... In fact, adding refraction *could be added* [to ray tracers] without any problem.**

**Our method should be compared to the actual methods available today**, even if ray tracers can in theory be extended to include refraction.
As mentioned above, even if they only predict coverage maps, the extension of ray tracers is not trivial. From personal interaction with Sionna RT authors, we have learned that implementing differentiable transmission efficiently is challenging.


Even more so: To the best of our knowledge, there are no works that ever showed the feasibility of optimization of the transmitter or receiver position using a ray tracer, neither on real or on simulated data. Sionna RT have claimed that it is possible, but we have not found an experiment to show it (only transmitter orientation). We show we can solve receiver localization.
We thank the reviewer for taking the time to read our rebuttal. We would like to clarify one thing: we did not mean to imply that outdoor evaluations are unnecessary or that indoor measurements are more critical – on the contrary, we would find an analysis on outdoor data just as valuable.

However, we want to stress that the symmetries of the underlying physics are the same in indoor and outdoor problems. We therefore believe that our high-level findings about the benefits of geometric, symmetry-aware machine learning models should also apply to outdoor scenes.
Thank you very much for your response.

**Benchmarks with real-world data ... is missing (common concern - reviewers LEVe, 2AZ5)**
Authors may find the following resources valuable for further evaluation

https://nvlabs.github.io/sionna/ : Very large scale simulated dataset generation feasible
https://www.deepsig.ai/datasets/ :  Simulated/Emulated dataset for use
https://www.deepsense6g.net/: Large scale real world datasets available


**We use the commercial ray-tracer "Remcom Wireless InSite" (see Sec. 4 in the paper) that is popular in the literature ([41, 3]). Please let us know your concrete concerns or missing features of this software.**

In the interest of community as a whole, it might be a good idea to use accessible and open-simulators, since we may not access to them. Could you please present a comparative analysis Sionna simulator vs Remcom.  Also, how closely, a particular scenario has been simulated, scatters, reflections, material properties, mobility scenarios(I believe they are currently missing) 


Modelling of radio environment is not clear ... diffration/refraction not considered
This is incorrect. All simulated paths include diffraction, transmission, and reflection effects; see Table 4 in Appendix C for details. We will clarify in the main paper.

**Mobility is not considered
Mobility is an interesting and relevant wireless channel modeling research problem. However, this complements our work that studies on generalization and symmetries of novel 3D scenes.**
Could you please elaborate, I couldn't understand the above


**Evaluation limited to indoor environments (common concern - reviewers LEve, 2AZ5)**
Evaluation on outdoor scenarios would make a great addition. However, the physical phenomenons (e.g., diffraction, reflections) of wireless signals would largely remain the same as a function of the 3D geometry in both indoor or outdoor settings. Nonetheless, in this paper, we choose indoor scenarios since we believe it is a more challenging scenario to evaluate influence of 3D scenes towards wireless characteristics. For instance, heights of surfaces play a bigger role (since signals can reflect off floors and ceilings) and transmissions through surfaces have significant contributions towards receive powers in non-LOS regions.

*Since, reviewer thinks other way, could you please provide any references in the draft supporting the above*


**Table 2 ... 80 dBm error ... explain large error of baseline**
Thank you for highlighting this result. It is one of the main arguments of our paper. The baseline is strong on in-domain data (e.g., MAE of 1.32 dB on the unseen floor plans, see Table 2). However, the generalization to out-of-distribution data is a significant failure mode (e.g., MAE of 78.68 dB on rotated data). In contrast, our proposed Wi-GATr architecture improves the robustness under domain shifts by incorporating the symmetries of the problem. By construction, it is robust to translations and rotations of the scene; we also find an improved robustness to other out-of-distribution test sets.
Could you please provide a little more information, on how many experiments are performed and what is OOD data looks like.
Thanks a lot for the clarifications; I have two remaining comments:

- Scalability remains to be a real issue. The statement that the proposed scheme is faster than ray tracers for large scenes seems incorrect. The computation of coverage maps is typically done using shoot-and-bounce, whose complexity is almost independent of the number of primitives in a scene. 

- The fact that Sionna RT, Instant RM, or any other ray tracer does not implement refraction does not justify the proposed approach. In fact, adding refraction could be added without any problem. Thus, I do not see a strong motivation apart from speed (which is a bit thin since fast raytracers do exist).
The unclear effects in real-world scenarios and outdoor environments reduce the contribution of this paper. The authors claim that indoor rooms are more critical and challenging. However, this does not mean that evaluating their work in outdoor scenarios is unnecessary. I acknowledge this work's novelty, and it's ok for me that the work is not evaluated on real-world data, but I think the experiments on the simulation data are still insufficient.
We thank the reviewer for the response and are happy to hear they appreciated our new comparisons.


We would like to correct one detail: we also ran quantitative experiments with Sionna-RT. These results are included in the rebuttal text and in Fig. R2 of the rebuttal pdf. Based on those results, we conclude that Sionna-RT (in its current state) is not capable of solving the tasks discussed in the paper due to lack of transmission models.
This response clearly distinguishes this work from GATr and highlights its innovative aspects. Additionally, the authors have refined the previously unclear expression and added Sionna-RT as a new competitor. However, this comparison is included only in the qualitative and not the quantitative experiments. It is also suggested that comparisons with the naive transformer be included in all experiments. Based on the above, I am inclined to raise my score to Borderline Accept.