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Transform-Invariant Generative Ray Path Sampling for Efficient Radio Propagation Modeling

Jérome Eertmans, Enrico M. Vitucci, Vittorio Degli-Esposti, Nicola Di Cicco, Laurent Jacques, Claude Oestges

arXiv:2603.01655v2cs.LGeess.SP

TL;DR

Ray tracing is accurate but its exhaustive path search grows exponentially, limiting efficient modeling in complex scenes. This paper uses Generative Flow Networks to sample ray paths intelligently, with replay, exploration, and physics-based masking refinements. In urban street-canyon environments, the framework preserves accuracy while achieving linear per-sample inference complexity and substantial speedups, but generalization to different morphologies remains limited.

  • Problem

    Exhaustive point-to-point ray tracing faces exponential growth in candidate paths, while existing learned channel models depend heavily on specific environments, materials, and frequencies.

  • Method

    The framework uses Generative Flow Networks to transform path enumeration into sequential intelligent sampling within a physics-based ray tracing pipeline.

  • Results

    The model maintains physical interpretability and coverage accuracy while achieving linear inference complexity per sample and substantial speedups in complex urban street-canyon scenarios.

  • Takeaways & Limitations

    The approach can reduce memory and computation for high-order interactions by avoiding storage and evaluation of millions of physically invalid path candidates.

  • Takeaways & Limitations

    Generalization to substantially different deployment scenarios may require retraining or exhaustive ray tracing, and the fixed scene embedding capacity can hinder high-order relational modeling.

Abstract

from arXiv · show

Ray tracing has become a standard for accurate radio propagation modeling, but suffers from exponential computational complexity, as the number of candidate paths scales with the number of objects raised to the interaction order. This bottleneck limits its use in large-scale or real-time applications, forcing traditional tools to rely on heuristics that reduce path candidates at the cost of potentially reduced accuracy. To overcome this limitation, we propose a machine-learning-assisted framework that replaces exhaustive path searching with intelligent sampling via Generative Flow Networks. Applying these generative models to this domain presents challenges, particularly sparse rewards due to the rarity of valid paths, which can lead to convergence failures and trivial solutions when evaluating high-order interactions in complex environments. To ensure robust learning and efficient exploration, our framework incorporates three key components. First, an \emph{experience replay buffer} captures and retains rare valid paths. Second, a uniform exploratory policy improves generalization and prevents overfitting to simple geometries. Third, a physics-based action masking strategy filters out physically impossible paths before the model considers them. Validated on idealized street-canyon scenarios, our model achieves substantial speedups over exhaustive search -- up to $10\times$ faster on GPU and $100\times$ faster on CPU -- while maintaining high coverage accuracy and successfully uncovering complex propagation paths. However, out-of-distribution evaluations on real-world Manhattan street geometries reveal that generalizing to substantially different urban morphologies requires further advancement in model capacity or alternative training strategies. Source code, tests, and a tutorial are available at https://github.com/jeertmans/sampling-paths.

1 Introduction

Ray tracing offers physically accurate radio propagation modeling but becomes computationally prohibitive as candidate paths grow exponentially with interaction order. The paper proposes machine-learning-assisted path sampling with invariant modeling and training refinements to reduce search costs while preserving physical accuracy.

  • Motivation: Ray tracing captures complex propagation interactions accurately, but its candidate paths grow exponentially with reflections, diffractions, and scene complexity.Most candidates are invalid or insignificant, creating substantial computational waste.
  • Motivation: Existing machine-learning propagation models often learn final channel characteristics, making them dependent on environmental conditions, materials, frequencies, or specific scenarios.They may also require extensive retraining for new environments or conditions.
  • Approach: The proposed framework samples ray paths instead of predicting final channel characteristics, reducing computational burden while retaining ray tracing’s physical accuracy.It is designed as a machine-learning-assisted enhancement to the ray tracing process itself.
  • Approach: Scene-transformation invariance across translation, rotation, and scaling supports robustness across environments, while the framework targets scenes of arbitrary complexity and size.The model is intended to learn general ray-propagation principles rather than memorize configurations.
  • Approach: The generative model treats path generation as sequential decision-making, prioritizing promising candidates without requiring computationally expensive ground-truth paths.The approach reduces simulation time during training and inference while preserving physical interpretability.
  • Contributions: Training stability and exploration are improved through successful experience replay, uniform exploratory policy, and physics-based action masking.The refinements address sparse rewards, overfitting, generalization, and physically impossible actions; experiments report faster convergence and improved discovery of high-order paths.

2 Related Work

Prior work applies machine learning either to direct channel-characteristic prediction or to components of the propagation pipeline. The paper identifies generalization, physical interpretability, and computational overhead as barriers motivating intelligent sampling.

  • Existing approaches: Machine-learning radio-propagation methods include direct channel-characteristic predictors and surrogates for specific simulation components.The reviewed literature includes MLP, CNN, NeRF, channel impulse response, neural scene representation, and differentiable simulation approaches.
  • Limitations: Direct field-learning models face a chaotic geometry-to-field mapping, requiring large networks and datasets and often risking overfitting.Small geometry or frequency changes can strongly alter fields through constructive and destructive interference.
  • Limitations: Most direct-learning models lack invariance to rotation or scaling and remain tightly coupled to training frequencies and material properties.A model trained for a 2.4 GHz office environment is unlikely to transfer directly to a 28 GHz street canyon.
  • Limitations: End-to-end black-box models discard ray-geometric information needed for sensing, localization, and beam management.The relevant information includes which reflections and diffractions contribute to the received signal.
  • Limitations: Neural inference can be slower than optimized geometric ray tracing in simple scenes, so practical assistance requires negligible inference overhead relative to the physics engine.This limitation is especially relevant for real-time use.
  • Motivation for this work: The proposed direction is to guide a physics engine by learning valid ray paths rather than predicting received-signal impact, decoupling learning from frequency and material properties.The precise field calculations remain with a robust physics-based solver.

3 Principle of Machine-Learning-Assisted Ray Tracing

The framework reframes exhaustive ray-path enumeration as intelligent sequential sampling, using learned scene geometry to prioritize promising branches while retaining conventional physics-based validation. Its modular design targets lower computational cost without replacing the standard ray-tracing engine.

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4 Methodology

The framework models ray-path generation as a sequential GFlowNet decision process, combining canonical geometric representations with replay, exploration, and physics-based action masking. It learns to sample complete paths according to their validity rewards while reducing invalid candidates and exploiting TX–RX reciprocity.

  • Generative Flow Networks: A GFlowNet generates ray paths sequentially by selecting scene objects from partial candidates until a complete path reaches the receiver after K interactions.The root represents TX, intermediate states are partial paths, and terminal states connect to RX.
  • Geometric Representation: The geometric preprocessing maps each scene into a canonical frame, making neural-network inputs invariant to translation, azimuthal rotation, and scaling.The transformation places TX at (0, 0, 0) and RX at (0, 0, 1).
  • Reward: The terminal reward is binary: valid geometrically constrained ray paths receive 1, whereas invalid paths receive 0.Validity is determined by path-tracing and geometry-validation modules, including visibility and reflection constraints.
  • Training Strategies: Sparse rewards worsen exponentially with interaction order and scene size, so successful-path replay and uniform exploration preserve rare valid trajectories and maintain exploration.The replay buffer reinforces discovered valid paths, while the exploratory policy prevents collapse toward narrow sampling behavior.
  • Action Masking: Action masking prunes physically impossible next-object choices, increasing the proportion of valid paths in training batches and strengthening the learning signal.The mask restricts choices to objects capable of leading to valid reflections toward the receiver.
  • TX–RX Symmetry: TX–RX reciprocity is incorporated by augmenting training with reversed path candidates, although this augmentation did not significantly improve performance in the experiments.The method uses data augmentation rather than imposing reciprocity directly on the architecture.

5 Application to Radio Coverage Map Prediction in an Ideal Urban Street Canyon

The framework is evaluated for radio coverage prediction in dynamically varied urban street-canyon scenes with interaction orders K = 1 to K = 3. Training uses differentiable ray tracing, randomized geometries and transmitter–receiver placements, and replayed successful trajectories.

  • Application Setting: Urban street canyons provide multipath propagation with multiple building-facade and ground-plane reflections, making them a canonical evaluation setting.Their geometric regularities, higher-order valid paths, and relevance to urban network planning motivate the choice.
  • Implementation: The implementation uses DiffeRT for differentiable geometric computations and validation, with JAX execution across CPUs, GPUs, and TPUs.Equinox defines the model and Optax handles optimization.
  • Scene Generation: Training scenes are generated by modifying a Sionna RT street-canyon environment with randomized building subsets, ground-plane inclusion, TX/RX regions, and heights.These variations include non-line-of-sight and degenerate configurations such as terminals placed inside buildings.
  • Training Configuration: K = 1 to K = 3 interaction orders are evaluated, using batches of B = 64 path candidates and an additional B = 64 replay-buffer trajectories when replay is active.The replay batch reinforces previously discovered successful samples.

5.3 Results

The evaluation studies component effects, computational efficiency, and coverage-map accuracy using accuracy and hit rate to characterize path-sampling quality. Ablations focus on replay, exploration, masking, flow weighting, symmetry, and training-scene diversity.

  • Evaluation Design: The experiments evaluate architectural and algorithmic components through controlled ablations, alongside efficiency comparisons with exhaustive ray tracing and coverage-map accuracy.Only the component under investigation is varied while other hyperparameters remain fixed.
  • Evaluation Metrics: Accuracy measures the fraction of sampled candidates that are valid, while hit rate measures the fraction of unique valid paths recovered by the sampler.Hit rate requires exhaustive ground truth and depends on the number of samples M; accuracy does not.
  • Replay and Exploration: The replay buffer preserves successfully discovered paths, while the ϵ-greedy policy maintains action-space exploration and helps discover additional valid paths.The replay mechanism enforces a fixed proportion of valid generated paths to reduce collapse.
  • Physics-Based Sampling: Action masking blocks physically invalid actions, including repeated consecutive interactions and objects behind the current reflecting interface.Distance-based flow weighting additionally down-weights objects far from the current ray direction.
  • TX–RX Symmetry: Reciprocity augmentation doubles paths by adding forward and reverse directions, but its effect on convergence and performance is explicitly evaluated rather than assumed.The framework tests whether symmetry reduces the need for unique geometric configurations.

Importance of the Training Scenario.

Training-scenario design affects generalization and hit rates, while runtime comparisons evaluate whether learned path sampling provides practical efficiency over exhaustive enumeration.

  • Scenario design: Training variations compare transmitter-receiver sampling regions and randomized building removal to assess generalization across scene configurations.The scenarios include canyon-only versus whole-scene sampling and per-building geometry randomization.
  • Scenario design: Per-building removal replaces facet removal to avoid unrealistic scenes with large missing portions of building facades.The revised strategy randomly excludes entire buildings rather than individual triangular facets.
  • Efficiency evaluation: Runtime experiments compare the learned sampler with an optimized exhaustive JAX baseline across interaction orders K = 1 to 3 and varying sampling budgets M.Both methods use just-in-time compilation and run on the same CPU and GPU hardware.
  • Efficiency evaluation: For M = 10, the learned approach achieves roughly two orders of magnitude speedup on CPU and one order of magnitude on GPU over exhaustive ray tracing.The comparison measures total computation time as scene object count increases.
  • Coverage evaluation: Coverage evaluation compares exhaustive-search ground-truth maps with sampled predictions using relative residual and relative dB residual metrics.The model is trained for first-, second-, and third-order specular reflections, while line-of-sight paths require no model.

Idealized Street-Canyon Scenario.

The idealized street-canyon evaluation tests coverage prediction with sampled reflection paths, then examines whether the model generalizes to substantially different Manhattan geometries.

  • Idealized evaluation: Coverage maps use inverse squared path length and unit reflection coefficients to isolate the effect of correctly sampling reflection paths.The comparison uses a transmitter at x_TX = [0 m 0 m 32 m]^T.
  • Idealized evaluation: With M = 20, relative and log-relative residual maps show where sampled paths omit valid contributions and visualize corresponding errors in decibels.An uncolored log-relative map does not necessarily indicate that all solutions were found.
  • Generalization: The idealized training and validation data remain procedural variations of one canonical street-canyon geometry, limiting the evaluation’s distributional breadth.Variations include building removal, transmitter-receiver placement, and optional ground-plane inclusion.
  • Generalization: Manhattan evaluation introduces real-world differences in building proportions, facade complexity, street widths, and block organization, with scenes obtained from OpenStreetMap.The scenes contain N = 400 and N = 805 objects, compared with up to N = 74 in the idealized setting; only second-order reflections and M = 100 are used.

Practical note on coverage-map generation.

Exhaustive point-to-point ray tracing is poorly suited to practical coverage-map generation because it recomputes paths independently for each receiver location.

  • Practical deployment: Exhaustive enumeration scales poorly across many receiver positions, whereas SBR and ray-launching methods can reuse launched rays for coverage maps.The practical deployment boundary differs from the point-to-point path-enumeration problem evaluated here.

5.4 Discussion

The framework substantially improves path discovery and computational efficiency in idealized street-canyon environments, but its benefits and generalization remain strongly dependent on scene similarity and architectural choices.

  • Component contributions: The replay buffer is crucial for early convergence and high hit rates at higher interaction orders, while exploration and masking provide smaller or scenario-dependent gains.The ablation study reports trade-offs among components, emphasizing careful design and tuning.
  • Effectiveness across interaction orders: For K = 1 and K = 2, the model rapidly exceeds 90 % hit rate; for K = 3, it reaches almost 45 % accuracy and a 65 % hit rate.The K = 3 result remains below 90 %, and evaluation uses only M = 10 samples per scene.
  • Coverage prediction: The model approximates exhaustive-search coverage with RMSE of 3.34 dB for the full scene and 1.51 dB for the main canyon, while missing some higher-order reflection regions.Visual inspection identifies missing second- and especially third-order reflection regions despite clear reflection patterns.
  • Practical deployment: The hit rate exceeding 90 % contrasts with ≪1 % from naive random sampling, but training requires 45 minutes to 1.5 hours on an RTX™3070 GPU.The method is therefore described as an amortized accelerator rather than a zero-shot solution for arbitrary unseen scenes.
  • Computational advantages: On CPUs, reducing candidate paths is particularly valuable, and increasing scene objects or interaction order can make the model faster than exhaustive search even on GPUs.GPU speedups are less pronounced because parallelism already mitigates candidate exploration costs.
  • Computational advantages: Benefits are largest when valid paths are rare, whereas open environments with many valid candidates may make exhaustive enumeration competitive.The framework is consequently most beneficial for challenging geometric scenarios.
  • Generalization and robustness: Generalization is reasonable within randomized street-canyon geometries but remains imperfect for Manhattan scenes with different layouts, widths, facades, and block organization.Some facades and spatial regions show missing or discrepant reflection paths, indicating adaptation to training-specific structural patterns.
  • Scope and limitations: The model focuses on geometric path finding, while antenna patterns, polarization, and frequency-selective reflection coefficients remain handled by the ray-tracing engine.Sampling strategies optimized for geometric validity could therefore be suboptimal for received-signal contribution.

6 Conclusion

The paper integrates GFlowNets and architectural refinements into ray tracing to replace exhaustive path enumeration with sequential intelligent sampling. Experiments show physical interpretability, accuracy, linear per-sample inference complexity, and substantial speedups, while motivating further work on reciprocity.

  • Conclusion: The framework transforms point-to-point ray-path enumeration into sequential sampling with GFlowNets, using replay, exploration, and physics-based masking to address sparse rewards and overfitting.These refinements collectively target issues identified in previous framework iterations.
  • Conclusion: In urban street-canyon environments, the model preserves ray-tracing interpretability and accuracy while achieving linear inference complexity per sample with respect to scene size.Coverage maps align strongly with ground truth, although some higher-order paths and near-field stability remain imperfect.
  • Conclusion: For K ≥3, the invariant architecture provides speedups and memory savings by avoiding storage of millions of physically invalid candidates.The paper identifies large-scale digital twins and real-time wireless network optimization as promising application contexts.
  • Future work: Future work should investigate explicitly embedding multipath reciprocity through architectural mechanisms or loss functions comparing forward and reversed path predictions.Reversed TX-RX data augmentation is presented as a baseline for this direction.

Declarations

The paper reports no funding, provides its code and supporting materials publicly, and declares no conflicts of interest.

  • Declarations: The work was not supported by any funding.
  • Declarations: Code, benchmarks, tests, and a tutorial are available in a public GitHub repository.
  • Declarations: The authors declare that they have no conflicts of interest.

Appendix A Invariance Properties

The appendix establishes that the coordinate transformation remains invariant under global translation, scaling, and azimuthal rotation around the vertical axis.

  • Transformation setup: The transformation is evaluated under x_i 7→ αQx_i + b, with positive scaling α, translation b, and azimuthal rotation Q satisfying Qe_z = e_z.
  • Translation invariance: Translation invariance follows because the translation vector b cancels from all position-difference vectors.
  • Scaling invariance: Scaling invariance follows because the transformed distance becomes αs and the same α factor cancels in the normalized transformation.
  • Rotation invariance: Azimuthal rotation preserves the relevant vector operations because Q is a rotation matrix and fixes the vertical axis e_z.
  • Basis transformation: The transformed local vertical axis satisfies v′ = w′ × u′ = Qv, and the new basis matrix is R′ = RQ⊤.
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