Algorithm 2 Implementation of the CNPs–SDE or ANPs–SDE models

Inputs: ID dataset $p_{\mathit{ID}}\left(x,y\right)$; MR is the missing rate of the ID dataset; the downsampling layer $h_1$ is the encoder of the CNPs or ANPs models; f and g are the drift net and diffusion net, respectively;${ L}_1$ is the negative log-likelihood loss function for the CNPs model or the ELBO for the ANPs model; $L_2$ is the binary cross-entropy loss function; the fully collected layer $h_2$ is the decoder of the CNPs or ANPs models to produce Means and Vars. 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. Forward through the downsampling net: d_mean_z = $h_1\left(X^m\right)$ and $X_0{}^m=\left(X^m, d\_\mathit{mean}\_z \right)$;

3. Forward through the SDE-Net block:

4. for k = 0 to $n-1$ do

5. Sample $Z_k{}^{{ N}_M}~N \left(0,1\right)$;

6. $X_{k+1}{}^m=X_k{}^m+f \left(X_0{}^m,t\right)∆t+g\left(X_0{}^m\right)\sqrt{∆t}{Z}_k$;

7. end for

8. Means, Vars $=$ $h_2$($X_{k+1}{}^m$)

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

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

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

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

13. Update g by ${\nabla }_g\frac{1}{m}{L}_2(g({X_0}^m),0)-{\nabla }_g{L}_2(g({{\tilde{X}}_0}^m),1)$;

for #testing iterations do

14. Evaluate the CNPs–SDE or ANPs–SDE models;

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

16. mask = Bernoulli(1-MR);

17. masked_$X^m$ = mask ∗ $X^m$;

18. Means, Vars = CNPs_SDE (masked_$X^m$) or ANPs_SDE (masked_$X^m$).