PROTOCOL.md

EMVal evaluation protocol

Protocol version: 1.0.0.

EMVal evaluates models of time-harmonic electromagnetic behavior. EMVal-NF measures complex near-field predictions, and EMVal-SP measures complex S-parameter predictions.

Evaluation subsets and aggregation

EMVal separates in-distribution (ID) boards from out-of-distribution (OOD) boards by geometry family, where an ID board belongs to a design family that is included in model training, and an OOD board belongs to a design family that is excluded from model training. Regardless of whether a board geometry in our evaluation sets is ID or OOD, it is NOT included in the model training set. The public and private_standard groups are ID. The private_challenge group is OOD.

Each metric is computed for one board. The dataset-level value is the equal-weight mean of the board values. Thus, a board with more ports or probes does not receive more weight. Near-field metrics pool all excited ports, frequencies, and probes within each board before the board-level reduction.

Metrics

EMVal-NF

For a board excited at port $jj$ and frequency $ff$, let ${\mathbf{F}}_{jfp}\in {\mathbb{C}}^3\backslash mathbf\{F\}\_\{jfp\}\backslash in\backslash mathbb\{C\}^3$ be the simulated field vector at probe $pp$, and let ${\widehat{\mathbf{F}}}_{jfp}\backslash hat\{\backslash mathbf\{F\}\}\_\{jfp\}$ be the prediction. Here, $\mathbf{F}\backslash mathbf\{F\}$ is either the electric field $\mathbf{E}\backslash mathbf\{E\}$ or the magnetic field $\mathbf{H}\backslash mathbf\{H\}$.

The global relative $L_{2}L\_2$ error compares total error energy with total simulated field energy across every excited port, frequency, and probe:

$$\mathrm{rel}{L}_2(\mathbf{F})=\frac{\sqrt{\sum _{jfp}\parallel {\widehat{\mathbf{F}}}_{jfp}-{\mathbf{F}}_{jfp}{\parallel }_2^2}}{\sqrt{\sum _{jfp}\parallel {\mathbf{F}}_{jfp}{\parallel }_2^2}}\mathrm{.}\backslash operatorname\{rel\}L\_2(\backslash mathbf\{F\})\; =\; \backslash frac\{\backslash sqrt\{\backslash sum\_\{jfp\}\backslash lVert\backslash hat\{\backslash mathbf\{F\}\}\_\{jfp\}-\backslash mathbf\{F\}\_\{jfp\}\backslash rVert\_2^2\}\}\{\backslash sqrt\{\backslash sum\_\{jfp\}\backslash lVert\backslash mathbf\{F\}\_\{jfp\}\backslash rVert\_2^2\}\}.$$

The scorer reports rel_l2_E and rel_l2_H. Because the metric sums squared absolute errors before normalization, it gives more weight to regions where the simulated field is strong. Lower values are better, and 0 is the optimum.

The median local error normalizes the error at each probe by the simulated field magnitude and then takes the median:

$$\mathrm{}(\mathbf{F})={\mathrm{median}}_{jfp}(e_{jfp}),e_{jfp}=\frac{\parallel {\widehat{\mathbf{F}}}_{jfp}-{\mathbf{F}}_{jfp}{\parallel }_2}{\parallel {\mathbf{F}}_{jfp}{\parallel }_2}\mathrm{.}\backslash operatorname\{local\backslash text\{-\}q\_\{50\}\}(\backslash mathbf\{F\})\; =\; \backslash operatorname\{median\}\_\{jfp\}(e\_\{jfp\}),\; \backslash qquad\; e\_\{jfp\}\; =\; \backslash frac\{\backslash lVert\backslash hat\{\backslash mathbf\{F\}\}\_\{jfp\}-\backslash mathbf\{F\}\_\{jfp\}\backslash rVert\_2\}\{\backslash lVert\backslash mathbf\{F\}\_\{jfp\}\backslash rVert\_2\}.$$

The scorer reports local_q50_E and local_q50_H. The local normalization gives low- and high-magnitude probes equal influence. The median limits the effect of a small number of large errors. Lower values are better, and 0 is the optimum.

The median local alignment measures the alignment of the predicted and simulated complex vectors at each probe:

$$\mathrm{}(\mathbf{F})={\mathrm{median}}_{jfp}(c_{jfp}),c_{jfp}=\frac{\mathrm{Re}\text{\hspace{0.17em}⁣}\left(\sum _{k=1}^3{\widehat{F}}_{jfp,k}{F}_{jfp,k}^{\ast }\right)}{\parallel {\widehat{\mathbf{F}}}_{jfp}{\parallel }_2\parallel {\mathbf{F}}_{jfp}{\parallel }_2}\mathrm{.}\backslash operatorname\{local\backslash text\{-\}cos\backslash text\{-\}q\_\{50\}\}(\backslash mathbf\{F\})\; =\; \backslash operatorname\{median\}\_\{jfp\}(c\_\{jfp\}),\; \backslash qquad\; c\_\{jfp\}\; =\; \backslash frac\{\backslash operatorname\{Re\}\backslash !\backslash left(\backslash sum\_\{k=1\}^\{3\}\backslash hat\; F\_\{jfp,k\}F\_\{jfp,k\}^\{*\}\backslash right)\}\{\backslash lVert\backslash hat\{\backslash mathbf\{F\}\}\_\{jfp\}\backslash rVert\_2\backslash lVert\backslash mathbf\{F\}\_\{jfp\}\backslash rVert\_2\}.$$

The scorer reports local_cos_q50_E and local_cos_q50_H. A value of 1 means that the prediction is aligned with the simulated vector in direction and complex phase. Higher values are better, and 1 is the optimum.

The scorer also reports two field-average values. rel_l2_avg is the arithmetic mean of the dataset-level rel_l2_E and rel_l2_H values. local_q50_avg is the corresponding mean of local_q50_E and local_q50_H.

EMVal-SP

For one board, let $S_{ijf}\in \mathbb{C}S\_\{ijf\}\backslash in\backslash mathbb\{C\}$ be the true S-parameter for receiving port $ii$, excited port $jj$, and frequency $ff$. Let ${\widehat{S}}_{ijf}\backslash hat\; S\_\{ijf\}$ be the prediction, and let $NN$ be the number of $(i,j,f)(i,j,f)$ entries.

The real-and-imaginary RMSE is

$$\mathrm{riRMSE}=\sqrt{\frac{1}{2N}\sum _{ijf}[{(\mathrm{Re}({\widehat{S}}_{ijf}-S_{ijf}))}^2+{(\mathrm{Im}({\widehat{S}}_{ijf}-S_{ijf}))}^2]}\mathrm{.}\backslash operatorname\{riRMSE\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{2N\}\backslash sum\_\{ijf\}\backslash left[\backslash left(\backslash operatorname\{Re\}(\backslash hat\; S\_\{ijf\}-S\_\{ijf\})\backslash right)^2+\backslash left(\backslash operatorname\{Im\}(\backslash hat\; S\_\{ijf\}-S\_\{ijf\})\backslash right)^2\backslash right]\}.$$

The scorer reports this value as real_imag_rmse. The factor $2N2N$ treats the real and imaginary parts as separate dimensions. Lower values are better, and the mathematical optimum is 0.

For the magnitude weighted-MAE, define $M=20{\log }_{10}\mathrm{\mid }S\mathrm{\mid }M=20\backslash log\_\{10\}|S|$, $\widehat{M}=20{\log }_{10}\mathrm{\mid }\widehat{S}\mathrm{\mid }\backslash hat\; M=20\backslash log\_\{10\}|\backslash hat\; S|$, and

$$w_{ijf}=\sigma \text{\hspace{0.17em}⁣}\left(\frac{M_{ijf}-N_0}{\tau }\right),N_0=-20\mathrm{d}\mathrm{B},\tau =2\mathrm{d}\mathrm{B}\mathrm{.}w\_\{ijf\}\; =\; \backslash sigma\backslash !\backslash left(\backslash frac\{M\_\{ijf\}-N\_0\}\{\backslash tau\}\backslash right),\; \backslash qquad\; N\_0=-20\backslash \; \backslash mathrm\{dB\},\; \backslash qquad\; \backslash tau=2\backslash \; \backslash mathrm\{dB\}.$$

Then

$$\mathrm{mag-wMAE}=\frac{\sum _{ijf}{w}_{ijf}\mathrm{\mid }{\widehat{M}}_{ijf}-M_{ijf}\mathrm{\mid }}{\sum _{ijf}{w}_{ijf}}\mathrm{.}\backslash operatorname\{mag\backslash text\{-\}wMAE\}\; =\; \backslash frac\{\backslash sum\_\{ijf\}w\_\{ijf\}|\backslash hat\; M\_\{ijf\}-M\_\{ijf\}|\}\{\backslash sum\_\{ijf\}w\_\{ijf\}\}.$$

The scorer reports this value as mag_wmae in dB. The sigmoid weight uses the simulated magnitude and reduces the contribution of deep nulls, where magnitude error in dB has limited engineering significance. Lower values are better, and 0 is the optimum.

The phase weighted-MAE applies the same magnitude-derived weights to the shortest angular distance between the predicted and simulated phases:

$$\mathrm{phase-wMAE}=\frac{\sum _{ijf}{w}_{ijf}\mid {\mathrm{wrap}}_{[-\pi ,\pi ]}({\widehat{\varphi }}_{ijf}-{\varphi }_{ijf})\mid }{\sum _{ijf}{w}_{ijf}}\mathrm{.}\backslash operatorname\{phase\backslash text\{-\}wMAE\}\; =\; \backslash frac\{\backslash sum\_\{ijf\}w\_\{ijf\}\backslash left|\backslash operatorname\{wrap\}\_\{[-\backslash pi,\backslash pi]\}(\backslash hat\backslash phi\_\{ijf\}-\backslash phi\_\{ijf\})\backslash right|\}\{\backslash sum\_\{ijf\}w\_\{ijf\}\}.$$

The scorer reports this value as ph_wmae in radians. The magnitude weights prevent phase errors in deep nulls from dominating the metric. Lower values are better, and 0 is the optimum.