Thank you for the new feedback!
We would first like to clarify potential ambiguities surrounding the term "error" by distinguishing between "performance error" and "numerical error":
> Performance error refers to the efficacy of GNNs in tasks such as node classification, where the focus is on the model's ability to correctly predict outcomes.
> Numerical error, on the other hand, concerns the accuracy of numerical solutions to fractional differential equations compared to the "true" solution trajectory, which is a key issue in computational mathematics.
The two errors are related, but not fully equivalent to each other. The approximation error in (S2) and (S3) refers specifically to the "numerical error."
For further clarity, we then conducted ablation studies on the Cora dataset to observe how errors change with respect to the step size $h$.
> (1). In the first ablation study, we fixed all other parameters and varied the step size $h$ used in each experiment. We trained the model using different step sizes $h$ during the training phase and tested it with the corresponding step sizes during the test phase. The results are presented in **Table R5**. According to (S2) and (S3), the numerical approximation error should be large when $h$ is large. However, the results from Table R5 indicate that while classification performance deteriorates with larger step size $h$, it does not degrade to an unreasonable level and still maintains adequate classification performance. This occurs because both the training and testing phases follow the same discretization procedure, and although both are far from the true FDE solution, the loss function is designed to minimize the final classification error, resulting in satisfactory performance.
> (2). In the second ablation study, we maintained fixed parameters as in the first study but changed our approach by training the model exclusively with a small step size $h=0.1$. During the testing phase, we varied the step size $h$. The results are presented in **Table R6**. According to (S2) and (S3), the numerical approximation error should be minimal when $h$ is small, and the solution with $h=0.1$ can be presumed close to the "true" solution trajectory of the FDE.
We noted that when $h < 1$ and remains relatively small, the model still achieves good classification performance. This occurs because the "numerical error" is still minimal, keeping the numerical solution close to the "true" solution trajectory.
However, as $h$ increases beyond this range, the "numerical error" grows significantly, diverging from both the "true" solution trajectory and the approximate solution with $h=0.1$. Consequently, classification performance deteriorates substantially.
**Table R5: Step size and classification accuracy(\%) (training and test)**
| Step size | 5 | 2 | 1 | 0.5 | 0.2 | 0.1 |
|-------------|-------|-------|-------|-------|-------|-------|
| Solver Eq.(28) | 80.24 | 82.98 | 83.11 | 83.18 | 83.15 | 83.11 |
| Solver Eq.(36) | 80.03 | 82.33 | 82.57 | 82.91 | 83.11 | 83.19 |
**Table R6: Step size and classification accuracy(\%) (test phase)**
| Step size | 5 | 2 | 1 | 0.5 | 0.2 | 0.1 |
|------------|-------|-------|-------|-------|-------|-------|
| Solver Eq.(28) | 35.23 | 65.48 | 74.42 | 80.91 | 81.02 | 83.11 |
| Solver Eq.(36) | 39.80 | 61.62 | 73.91 | 76.14 | 80.41 | 83.19 |