Authors response for follow-up questions
Thanks for raising these fair concerns. Let us clarify!
The 5-second rollout is, in fact, our mistake and only refers to the heat equation. We use different time stepping depending on each specific dataset - for instance, the Black Sea data has a time resolution of one full day. We summarize in the following table a more detailed overview of the experiments, such as total rollout time, space, and time resolution for both OSC space and time.
| Experiment | Total Rollout Time | Ground Truth Step $\Delta t$ | Collocation Step $\Delta t$ | Ground Truth Resolution | Collocation Points Resolution |
| --- | --- | --- | --- | --- | --- |
| Heat Equation | 5 Seconds | 0.1 Seconds | 0.2 Seconds | 64 * 64 | 12 * 12 |
| Damped Wave | 10 Seconds | 1 Seconds | 2 Seconds | 64 * 64 | 32 * 32 |
| 2D Navier Stokes | 10 Seconds | 1 Seconds | 2 Seconds | 64 * 64 | 32 * 32 |
| Ocean Currents | 10 Hours | 1 Hour | 2 Hours | 73 * 73 (3 features) | 36 * 36 (3 features) |
| Black Sea | 10 Days | 1 Day | 2 Days | 5000 nodes (3 features) | 1000 Nodes (3 features) |
| 3D Navier Stokes | 0.5 Seconds | 0.05 Seconds | 0.1 Seconds | 262,144 nodes (5 features) | 32,768 nodes (5 features) |
> For each of the experiment, how many time steps do the ground truth trajectories consist of and what is the cardinality of the in-time collocation points $\{t_0, t_1, \cdots, t_K\}$? […] For experiment of Navier-Stokes equations, the time horizon of the dataset is t=10 (in Appendix B.3), as opposed to t=5 reported in Table 5.1. Why did you choose t=5 as the rollout length? Are there any difficulties that prevent the proposed model from predicting 10 steps?
>
Let us take the previous table. For example, in the 2D Navier-Stokes experiment, the ground truth trajectories consist of 10-time steps (10 seconds), and step $\Delta t$ is 1 second; while in our model, the neural network autoregressively performs the rollout 5 times with $\Delta t = 2$ seconds and the OSC will interpolate through the skipped steps. In other words, the ground truth consists of $\mathbf{Y}^{t}, t\in\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\rbrace$ and the neural networks in our model gives $\mathbf{\tilde{Y}}^{t_{\mathrm{interpolate}}}, t_{\mathrm{interpolate}}\in\lbrace2, 4, 6, 8, 10\rbrace$ and interpolate to $\mathbf{\hat{Y}}^{t} = \mathrm{TimeOrientedOSC}(\mathbf{\tilde{Y}}^{t_{\mathrm{interpolate}}})$.
This is similar to the space resolution, in which GraphSplineNets predict only a subset of the domain and interpolate through the rest. We will add this information to our manuscript.
> On which data points is the MSE defined? Does it take the mean for all spatial points and all steps in each trajectory, and all the ground-truth trajectories in the test dataset?
>
The MSE is defined based on the ground truth resolution and time steps, i.e., all spacial points and all time steps in each trajectory.
> Is the MSE error reported in Table 5.1 comparable to the MSE error defined on only in-space collocation points over $\{t_0, t_1, \cdots, t_K\}$? I’m curious if the relatively high number of data points in the ground truth dataset would have an impact to reduce the MSE defined on only in-time and in-space collocation points.
>
As we mentioned in the previous question, the MSE reported in Table 5.1 is defined on all spacial points and all time steps in each trajectory. We report this small experiment decoupling the effect of space and time collocation separately.
| | MGN | MGN+SpaceOSC only | MGN+TimeOSC only | MGN+SpaceOSC+TimeOSC (GraphSplineNets) |
| --- | --- | --- | --- | --- |
| MSE(x10^{-3}) | 3.01±0.38 | 2.87±0.54 | 1.27±0.08 | 1.14±0.11 |
| Runtime [s] | 6.99±0.12 | 1.88±0.11 | 4.63±0.15 | 1.38±0.10 |
The insight is that space collocation is mostly responsible for efficiency, while time collocation yields comparatively more accuracy.
> Is the inference time reported in Figure 5.1 evaluated by the same models that produce the results for one of the experiments in Table 5.1?
>
Correct, we use the heat equation experiment to measure the times reported in Figure 5.1, while the model structure is the same across experiments. We noticed that these trends hold in general for the experiments.
Please let us know if you have further questions or concerns; we will be happy to address them!