Algorithm 1 C-T-GPR: penalised maximum likelihood estimation.

1:Input: Training data $\mathcal{D}={\{X_i,Z_i,Y_i\}}_{i=1}^n$, test point $(X^{*},Z^{*})$, regularisation $\lambda$, tolerance ${\epsilon }_{\mathrm{tol}}$. 2:Initialise $\eta$ (Xavier), ${\sigma }^2$ (sample variance), $\mathit{\theta }$ (median heuristic for length scales). 3:repeat 4:   (Block 1: NN update) Fix $({\sigma }^2,\mathit{\theta })$; compute $\mathbf{\alpha }={\mathbf{C}}^{-1}(\mathbf{Y}-{\mathbf{f}}_{\eta })$; update $\eta$ via Adam using Equation (22). 5:   (Block 2: GP update) Fix $\eta$; update $({\sigma }^2,\mathit{\theta })$ via L-BFGS-B using Equations (21)–(23). 6:until$|\Delta \mathcal{L}|<{\epsilon }_{\mathrm{tol}}$ 7:Compute predictive $({\widehat{\mu }}^{*},{\widehat{v}}^{*})$ via Equations (25) and (26). 8:Compute predictive entropy ${\mathcal{H}}^{*}={\displaystyle \frac{1}{2}}ln\left(2\pi e{\widehat{v}}^{*}\right)$. 9:Output: Point prediction ${\widehat{\mu }}^{*}$, predictive variance ${\widehat{v}}^{*}$, predictive entropy ${\mathcal{H}}^{*}$.