Algorithm 1: Prediction algorithm process in TKDF—TKDF for Time-Series Prediction

Input: Historical timestamp $S$, future timestamp $T$, historical sequence $X$

Output: Future sequence $Y$

Hyperparameter: Learning rate $r$, weight $\alpha ,\beta$

1. Initialization: Network ${\theta }_1$ and Network ${\theta }_2$

2. [The pre-training stage]: utilize $S$ to predict $X$

3. for $iter<iters$ do:

4. Calculate self-distillation loss: $L_s=\alpha L_s^{(H)}+(1-\alpha )L_s^{(S)}$

5. Update parameters of the timestamp mapper: ${\theta }_s={\theta }_s+r\ast {\displaystyle \frac{\partial L_s}{\partial {\theta }_s}}$

6. end for

7. [The Multi-branch Prediction Stage]: utilize $T$ and $X$ to predict $Y$

8. for $iter<iters$ do:

9. Calculate mutual learning loss of network ${\theta }_1$: $L_{\theta 1}=\beta L_{\theta 1}^{(H)}+(1-\beta )L_{\theta 1}^{(S)}$

10. Calculate mutual learning loss of network ${\theta }_2$: $L_{\theta 2}=\beta L_{\theta 2}^{(H)}+(1-\beta )L_{\theta 2}^{(S)}$

11. Update parameters of network ${\theta }_1$: ${\theta }_1={\theta }_1+r\ast {\displaystyle \frac{\partial L_{\theta 1}}{\partial {\theta }_1}}$

12. Update parameters of network ${\theta }_2$: ${\theta }_2={\theta }_2+r\ast {\displaystyle \frac{\partial L_{\theta 2}}{\partial {\theta }_2}}$

13. Update the prediction result

14. End for