【Ray tracing with NeRF】Learnable Wireless Digital Twins: Reconstructing Electromagnetic Field with Neural Representations

Learnable Wireless Digital Twins: Reconstructing Electromagnetic Field with Neural Representations

### 1. Overview

2.ML model for the EM property and the interaction behaviour

2.1 Neural Object

\[\mathbf{e}=\widetilde{g}_{\mathscr{E}, o}\left(\operatorname{PosEnc}(\overline{\mathbf{p}}) ; \boldsymbol{\Theta}_{\mathscr{E}, o}\right) \]

where \(\mathbf{p} \in \mathbb{R}^{3 \times 1}\) denote a position in the local coordinate of the o-th object, \(\mathbf{e}\) is high dim representation of EM property.

2.2 Neural Interaction

\[\mathbf{E}_{l, i+1}=\widetilde{g}_{T, \kappa}\left(\mathbf{e}, \mathbf{d}_{l, i}^{\mathrm{AoA}}, \mathbf{d}_{l, i}^{\mathrm{AoD}}, \mathcal{I}_{l, i}, \mathbf{E}_{l, i} ; \boldsymbol{\Theta}_{T, \kappa}\right) \]

The same neural interaction model can be shared across different objects.
Polarization Direction Normalization
The polarization decomposition of incoming and outcoming E-field is pre-defined.
Scattering

\[\begin{aligned} \hat{\mathbf{e}}_{i, \perp} & =\frac{\mathbf{d}^{\mathrm{AoA}} \times \mathbf{n}}{\left\|\mathbf{d}^{\mathrm{AoA}} \times \mathbf{n}\right\|} \\ \hat{\mathbf{e}}_{i, \|} & =\hat{\mathbf{e}}_{i, \perp} \times \mathbf{d}^{\mathrm{AoA}}\\ \hat{\mathbf{e}}_{\mathrm{o}, s} & =\left[\cos \left(\theta_{s}\right) \cos \left(\varphi_{s}\right), \cos \left(\theta_{s}\right) \sin \left(\varphi_{s}\right),-\sin \left(\theta_{s}\right)\right]^{T} \\ \hat{\mathbf{e}}_{\mathrm{o}, p} & =\left[\sin \left(\varphi_{s}\right), \cos \left(\varphi_{s}\right), 0\right]^{T}, \end{aligned} \]

The predicted outgoing E-field is given by

\[\left[E_{\mathrm{s}, \theta}, E_{\mathrm{s}, \varphi}\right]^{T}=\sqrt{\left|E_{\mathrm{i}, \perp}\right|^{2}+\left|E_{\mathrm{i}, \perp}\right|^{2}}\left[t_{1} e^{j t_{2}}, t_{3} e^{j t_{4}}\right]^{T}\\ \left[t_{1}, t_{2}, t_{3}, t_{4}\right]^{T}=\widetilde{g}_{T, \mathrm{reff}}\left(\mathbf{v}_{\mathrm{reff}} ; \boldsymbol{\Theta}_{T, \mathrm{reff}}\right) \]

where \(\mathbf{v}_{\text {scat }}=\left[\mathbf{e}^{T}, \overline{\mathbf{v}}_{\text {scat }}^{T}, \text { PosEnc }^{T}\left(\overline{\mathbf{v}}_{\text {scat }}\right), \Re\left\{\bar{E}_{\mathbf{i}, \perp}\right\}, \Im\left\{\bar{E}_{\mathbf{i}, \perp}\right\}, \Re\left\{\bar{E}_{\mathbf{i}, \|}\right\}, \Im\left\{\bar{E}_{\mathbf{i}, \|}\right\}\right]^{T}\) and \(\overline{\mathbf{v}}_{\text {scat }}=\left[\left(\mathbf{d}^{\mathrm{AoA}}\right)^{T},\left(\mathbf{d}^{\mathrm{AoD}}\right)^{T}, \mathbf{n}\right]^{T}\).

3. Summary

相比WINERT，主要做了以下改进：

1. 考虑散射、衍射
2. 场景更复杂
3. 考虑了极化
4. 考虑了物体表面的不同位置材质的变化，更精细
5. 使用positional encoding

缺点：

1. 每个物体都要单独训练一个网络
2. 泛化能力弱和仅使用仿真软件产生数据的问题仍未解决