Physics-Informed PRBPN for Tunnel Propagation

• The paper presents a novel deep learning framework that reconstructs fine-mesh RSS fields from coarse-mesh PWE outputs while integrating physics priors.
• It employs a Difference-Weighted Temporal Fusion module and a recurrent back-projection decoder to iteratively refine super-resolved predictions.
• PRBPN achieves high accuracy with R² > 0.85 and low error metrics across diverse tunnel geometries and real-world experimental validations.

The Physics-Informed Recurrent Back-Projection Propagation Network (PRBPN) is a deep learning framework developed for efficient, accurate modeling of radio wave propagation in large-scale railway tunnels, with direct application to communication-based train control (CBTC) systems. PRBPN addresses the prohibitive computational cost of fine-grid Parabolic Wave Equation (PWE) solvers while avoiding the fidelity loss inherent in coarse-grid approximations. The architecture reconstructs fine-mesh received-signal-strength (RSS) fields from a small sequence of coarse-mesh PWE outputs by integrating physics-based priors and data-driven temporal learning without any preliminary upsampling stage. It achieves data efficiency and generalization across diverse tunnel geometries and operational frequencies, as validated by both extensive synthetic experiments and real-world tunnel measurements (Wu et al., 5 Jan 2026).

1. Architectural Framework and Data Flow

PRBPN operates on a temporal window of $2n+1$ contiguous coarse-mesh RSS slices, $\{I_{t-n}, \dotsc, I_t, \dotsc, I_{t+n}\}$, where each $I_{\tau}$ is a matrix in $\mathbb{R}^{H \times W}$ representing coarse-resolution RSS derived from a PWE solver. The core pipeline comprises:

• Initial ConvBlock encoding: The central target slice $I_t$ is processed via a $3 \times 3$ convolution (stride 1, PReLU activation) yielding an initial feature map $L_t = \mathrm{ConvBlock}(I_t)$.
• Neighbor-frame encoding and difference computation: For each offset $k=1,\dotsc,n$, absolute differences $\mathrm{diff}_{t\pm k} = |I_t - I_{t\pm k}|$ are computed. Each neighbor pair $[I_t, I_{t\pm k}, \mathrm{diff}_{t\pm k}]$ is encoded via a further ConvBlock to form $M_{t\pm k}$.
• Difference-Weighted Temporal Fusion (DWTF): Temporal context is introduced by calculating mean-channel difference maps, convolving with $3 \times 3$ filters, and passing through sigmoid activation to obtain attention weights $\mathrm{attn}_{t\pm k}$. These weights modulate the mean differences, producing $$\mathrm{wdiff}_{t\pm k} = \mathrm{attn}_{t\pm k} \odot \mathrm{mean\_chan}(\mathrm{diff}_{t\pm k})$$. Summed and concatenated with $L_t$ and all $M_{t\pm k}$, this fusion guides the network’s focus on regions of significant wavefront variation.
• Recurrent Back-Projection Decoder: Fused features enter an iterative back-projection loop. At each iteration $i$:
  • A provisional high-resolution estimate is generated: $$H_t^{(i)} = \mathrm{NetE}(L_t^{(i-1)}, \{M_{t\pm k}^{(i-1)}\}; \theta_E)$$.
  • Coarse projection: $L_t^{(i)} = \mathrm{NetD}(H_t^{(i)}; \theta_D)$.
  • Residual computation: $e_t^{(i)} = H_t^{(i)} - L_t^{(i)}$.
  • HR estimate refinement: $$H_t^{(i+1)} = H_t^{(i)} + \mathrm{NetRes}(e_t^{(i)}; \theta_{\mathrm{res}})$$.

After a fixed number of iterations, the final SR slice is formed by a lightweight ConvBlock aggregating recent $H_t$ states. All convolutions support large upsampling factors (e.g., kernel size 12, stride 8, padding 2 for $\times8$ upsampling), eliminating the need for explicit pre-interpolation.

2. Integration of Physics-Based Knowledge

PRBPN incorporates physics priors essential for radio wave propagation:

• Propagation smoothness and local continuity: The short-range temporal window and DWTF module explicitly focus on slices with pronounced local variations (e.g., wall reflections), emulating the one-way parabolic wave equation

$$\frac{\partial E(x,y,z)}{\partial z} = \frac{j}{2k_0} \nabla_\perp^2 E(x,y,z) + j[k(x,y,z) - k_0]E(x,y,z)$$

This mechanism ensures that super-resolved predictions align with the expected physical evolution of wavefronts governed by transverse Laplacian variations.

• Low-Resolution–High-Resolution (LR–HR) consistency: The recurrent back-projection loop formalizes a coarse-to-fine residual correction, analogous to enforcing conservation laws and modal continuity. At each step, the residual between predicted HR and projected LR slices, $e_t^{(i)}$, is used to iteratively refine the reconstruction, maintaining physical fidelity to PWE-derived field dynamics.

3. Training Methodology and Loss Construction

PRBPN is trained end-to-end on matched pairs of coarse and fine PWE field slices. Training datasets include four canonical tunnel shapes (e.g., rectangular, arched), four frequencies (0.9, 2.4, 4.9, 5.8 GHz), and randomized wall permittivity $\varepsilon_r$, conductivity $\sigma$, and transmitter/receiver placements. The fine-mesh grid employs $\Delta x = \Delta y = 0.4\lambda$; the coarse mesh uses $\Delta x = \Delta y = 3.2\lambda$.

Data efficiency is pronounced: as few as 10–20 paired slices in synthetic domains, and four pairs per frequency in the real-world Massif Central tunnel, are sufficient for robust learning.

The composite loss function is: $$\mathcal{L} = \mathcal{L}_{\rm rec} + \beta\,\mathcal{L}_{\rm smooth}$$ with

$$\mathcal{L}_{\rm rec} = \| SR_t - HR_t \|_1,\quad \mathcal{L}_{\rm smooth} = \| \nabla SR_t \|_1$$

where $\beta$ is a regularization factor promoting smoothly varying, physically plausible solutions and suppressing spurious high-frequency artifacts. Standard augmentations (flips, rotations, random crops) and the direct usage of coarse-mesh input fields (bypassing any interpolation) further enhance data efficiency and consistency.

4. Quantitative Performance and Evaluation Metrics

PRBPN’s super-resolved predictions are evaluated against fine-grid PWE references using:

Metric	Formula	Benchmark Result Example
MAE	$\frac{1}{N} \sum |y_i - \hat{y}_i|$	0.8587 dB (rectangular, 0.9 GHz)
MAPE	$\frac{100\%}{N} \sum |(y_i - \hat{y}_i)/y_i|$	2.27% (rectangular, 0.9 GHz)
RMSE	$\sqrt{\frac{1}{N}\sum (y_i - \hat{y}_i)^2}$	1.8183 (rectangular, 0.9 GHz)
$R^2$	$$1 - \frac{\sum (\hat{y}_i - y_i)^2}{\sum (y_i - \bar{y})^2}$$	0.9753 (rectangular, 0.9 GHz)

Across all four tested shapes and frequencies, PRBPN consistently achieves $R^2 > 0.85$, MAE below a few dB, and MAPE $\lesssim$ 3.6%. For the arched tunnel at 5.8 GHz, MAE ≈ 1.0676, MAPE ≈ 2.57 %, $R^2$ = 0.8607.

5. Real-World Case Study: Massif Central Tunnel

Validation in the 2.5 km Massif Central tunnel (France) demonstrates PRBPN’s capacity to generalize under severe data scarcity. Only four paired coarse/fine slices per frequency (0.9 GHz and 2.1 GHz) were required for training. The model achieves:

• 0.9 GHz: MAE = 0.4192, MAPE = 1.32%, RMSE = 1.2544, $R^{2}$ = 0.9780
• 2.1 GHz: MAE = 0.4947, MAPE = 1.22%, RMSE = 1.4903, $R^{2}$ = 0.9520

Comparison against ground truth fine-mesh PWE fields at 1.25 km and 2.5 km demonstrates that PRBPN accurately recovers fine-scale reflection paths, higher-order mode structures, and attenuation gradients absent from coarse inputs.

6. Summary of Capabilities and Applications

PRBPN achieves high-fidelity, physically-consistent tunnel propagation modeling while minimizing reliance on computationally demanding fine-mesh PWE solvers. Its combination of compact encoding, physics-informed temporal fusion, and iterative projection/back-projection grants strong generalization capabilities across diverse tunnel cross-sections, operational frequencies, and material properties, even under data-scarce regimes. This enables engineers and researchers to efficiently obtain predictive RSS fields necessary for designing and validating reliable CBTC infrastructures (Wu et al., 5 Jan 2026).