Algorithm 1 ViT Variant Training for Radar Error Correction

1:Input: Radar measurements ${\left\{{\mathbf{r}}_i\right\}}_{i=1}^N$, AIS ground-truth positions ${\left\{{\mathbf{a}}_i\right\}}_{i=1}^N$   2:Output: Trained model parameters ${\mathsf{\Theta }}^{*}$   3: Preprocessing:   4: Convert AIS $({\varphi }_i,{\lambda }_i)$ to local Cartesian $(x_i,y_i)$ via Equations (4) and (5)   5: Temporal synchronisation: interpolate AIS positions to radar timestamps via Equations (6) and (7)   6: Compute target errors: $\Delta r_i=r_i-r_i^{\mathrm{AIS}}$, $\Delta {\theta }_i={\theta }_i-{\theta }_i^{\mathrm{AIS}}$ via Equation (1)   7: Construct 12-D feature vectors ${\mathbf{x}}_i\in {\mathbb{R}}^{12}$ via Section 2.3.3   8: Vessel-level split: ${\mathcal{D}}_{\mathrm{train}}\cup {\mathcal{D}}_{\mathrm{val}}\cup {\mathcal{D}}_{\mathrm{test}}$ (70:15:15)   9:Initialise: $\mathsf{\Theta }=\{\mathbf{E},{\mathbf{E}}_{\mathrm{pos}},{\mathbf{z}}_{\mathrm{cls}},{\theta }_{\mathrm{enc}},{\mathbf{W}}_r,{\mathbf{W}}_{\theta },b_r,b_{\theta },s_r,s_{\theta }\}$ via Xavier 10:Hyperparameters: $B=512$, ${\eta }_0={10}^{-3}$, ${\eta }_{\min }={10}^{-6}$, $\lambda ={10}^{-4}$, $P=25$ 11:$t\leftarrow 0$, $\mathrm{patience}\_\mathrm{counter}\leftarrow 0$, ${\mathcal{L}}_{\mathrm{best}}\leftarrow \infty$ 12:while $t<150$ and $\mathrm{patience}\_\mathrm{counter}<P$ do 13: Sample mini-batch $\mathcal{B}\subset {\mathcal{D}}_{\mathrm{train}}$, $\left|\mathcal{B}\right|=B$ 14: for each $\mathbf{x}\in \mathcal{B}$ do 15:  Patch embedding: ${\mathbf{z}}_0=[{\mathbf{z}}_{\mathrm{cls}};\mathbf{E}·{\mathbf{x}}^{\left(1\right)}+{\mathbf{E}}_{\mathrm{pos}}^{\left(1\right)};\dots ;\mathbf{E}·{\mathbf{x}}^{\left(4\right)}+{\mathbf{E}}_{\mathrm{pos}}^{\left(4\right)}]$   (Equation (9)) 16:  Transformer: ${\mathbf{z}}_N=\mathrm{TransformerEncoder}\left({\mathbf{z}}_0\right)$      (Equations (10) and (11)) 17:  Predict: $\widehat{\Delta r}={\mathbf{W}}_r·{\mathbf{z}}_N^{\left[\mathrm{CLS}\right]}+b_r$, $\widehat{\Delta \theta }={\mathbf{W}}_{\theta }·{\mathbf{z}}_N^{\left[\mathrm{CLS}\right]}+b_{\theta }$  (Equations ((12) and (13))) 18: end for 19: ${\mathcal{L}}_{\mathrm{total}}\leftarrow {\displaystyle \frac{1}{B}}\sum (w_r|\Delta r-\widehat{\Delta r}| + w_{\theta }|\Delta \theta -\widehat{\Delta \theta }\left|\right)+{\lambda \parallel \mathsf{\Theta }\parallel }_2^2$   (Equation (18)) 20: $w_r=\mathrm{softplus}(-s_r)$, $w_{\theta }=\mathrm{softplus}(-s_{\theta })$             (Equation (19)) 21: $\mathsf{\Theta }\leftarrow \mathsf{\Theta }-{\eta }_t·{\nabla }_{\mathsf{\Theta }}{\mathcal{L}}_{\mathrm{total}}$ (AdamW, clipped at ${\parallel \nabla \parallel }_{\infty }=1.0$) 22: ${\eta }_t\leftarrow {\eta }_{\min }+{\displaystyle \frac{1}{2}}({\eta }_0-{\eta }_{\min })(1+\cos \left({\displaystyle \frac{t}{150}}\pi \right))$          (cosine annealing) 23: if ${\mathcal{L}}_{\mathrm{val}}<{\mathcal{L}}_{\mathrm{best}}$ then 24:  ${\mathsf{\Theta }}^{*}\leftarrow \mathsf{\Theta }$, ${\mathcal{L}}_{\mathrm{best}}\leftarrow {\mathcal{L}}_{\mathrm{val}}$, $\mathrm{patience}\_\mathrm{counter}\leftarrow 0$ 25: else 26:  $\mathrm{patience}\_\mathrm{counter}\leftarrow \mathrm{patience}\_\mathrm{counter}+1$ 27: end if 28: $t\leftarrow t+1$ 29: end while