Algorithm 1 Implementation of the ConvCNPs–SDE model

Inputs: ID dataset $p_{\mathit{ID}}\left(x,y\right)$; CR and MR are the context rate and missing rate, respectively; ccnps represents the ConvCNPs model of vNPs for completing the ID dataset; $h_1$ is the downsampling net for 2D image classification tasks or the upsampling net for 1D regression tasks; $h_2$ is the fully connected net; f represents the drift net and g represents the diffusion net; t is the layer depth; $L_1$ is the cross-entropy loss function, $L_2$ is the log-likelihood loss function, and $L_3$ is the binary cross-entropy loss function. Outputs: Means and Vars for #training iterations do

1. Sample a minibatch of m data: $\left(X^m,Y^m\right){ ~ p}_{\mathit{ID}}\left(x,y\right)$;

2. if for 1D regression task:

3. Context points $\left(X_c^m,Y_c^m\right)$ are generated from sampled target points $\left(X_t^m,Y_t^m\right)$ based on CR, where $\left(X_t^m,Y_t^m\right)$ equals $\left(X^m,Y^m\right)$;

4. Forward through the ConvCNPs model: Y_dist = ccnps$\left(X_c^m,Y_c^m,X_t^m\right)$;

5. Forward through the upsampling net of the SDE-Net block: $X_0{}^m=h_1\left(X^m,Y^m\right)$;

6. else for 2D image classification task:

7. Forward through the ConvCNPs model: Y_dist = ccnps$\left(X^m,Y^m\right)$;

8. Forward through the downsampling net of the SDE-Net block: $X_0{}^m=h_1\left(X^m,Y^m\right)$;

9. for k = 0 to t − 1 do

10. Sample $Z_k{}^m~\mathcal{N}\left(0,1\right)$;   $X_{k+1}{}^m=X_k{}^m+f \left(X_k{}^m,t\right)∆t+g\left(X_0{}^m\right)\sqrt{∆t}{Z}_k$;

11. end for

12. Forward through the fully connected layer of the SDE-Net block: $Y_f{}^m=h_2\left(X_k{}^m\right)$;

13. Update $h_1{, h}_2$ and f by ${\nabla }_{{ h}_1{, h}_2} \mathrm{and} f \frac{1}{m}{L}_1\left(Y_f{}^m,Y^m\right)$;

14. Update ccnps by ${\nabla }_{\mathit{ccnps} }\frac{1}{m}{L}_2\left(Y\_\mathit{dist}.\mathrm{means},Y_t^m\right)$;

15. Sample a minibatch of $m$ data from ID: $\left(X^m,0\right)~p_{\mathit{ID}}\left(x,y\right)$;

16. Sample a minibatch of $m$ data from OOD: $({\tilde{X}}^m,1)~p_{\mathit{OOD}}(x,y)$;

17. Forward through the downsampling or upsampling nets of the SDE-Net block: ${X_0}^m,{{\tilde{X}}_0}^m=h_1(X^m),h_1({\tilde{X}}^m)$;

18. Update g by ${\nabla }_g\frac{1}{m}{L}_3(g({X_0}^m),0)-{\nabla }_g{L}_3(g({{\tilde{X}}_0}^m),1)$;

for #testing iterations do

19. Evaluate the of ConvCNPs–SDE model;

20. Sample a minibatch of m data from ID: $\left(X^m,Y^m\right)~{ p}_{\mathit{ID}}\left(x,y\right)$;

21. mask = Bernoulli (1-MR)

22. masked_$X^m$ = mask ∗ $X^m$;

23. completed_$X^m$= ccnps$\left({\mathit{masked}\_X}^m\right)$;

24. Means, Vars = SDE-Net(completed_$X^m$);