# TGN: A Temporal Graph Network for Physics Prediction
## 4 Experiments
### Experimental Results  
**Qualitative results. ** We tested our model in two fluid systems, cylinder flow and airfoil flow. In Figure 4, we apply the proposed method to predict the velocity and pressure on the trajectories of the test set for the two systems and compare them with the ground truth (CFD). For cylindrical flow, our model can capture periodic and trending information from historical time series at different Reynolds numbers to accurately predict laminar flow and Karman vortex street behavior in trajectories. In the high-speed airfoil flow, our model can successfully learn the changing flow field at different Mach numbers and angles of attack, and accurately predict its flow separation phenomena as well as vortex-shedding behavior. In both systems, cylinder flow and airfoil flow, our model accurately captures the frequency and phase of vortex shedding and visually presents predictions that are largely consistent with the CFD reference. This long-term stable prediction demonstrates that our model can be applied to complex physical systems. **Model performance comparison. ** To quantitatively evaluate the performance of our model, model variants, and baseline, we calculate the relative mean square error (RMSE)  
Figure 4: Comparison of velocity and pressure fields predicted by our model with ground truth (CFD). Our model accurately predicts the flow field dynamics for different systems and different parameters. on single-step rollout (1-step), 50-step rollouts, and the rollouts of the entire trajectory (500-step). In Table 1, we compare the average prediction errors on the two systems. In most cases, our model based on spatiotemporal encoding outperforms the baseline and model variants. For both short-term forecasts (1-step, 50-step) and long-term forecasts (500-step), our model shows superior performance. For airfoil flow, due to its larger mesh size, it requires larger memory consumption, so we reduce its history time steps to three. Although this will lose part of the model performance, our model still performs the best under long-term prediction. The cylinder flow with a larger history time steps (12 time steps) validates the effectiveness of our method and the RMSE of our method is significantly lower than that of the next-step model and the model variants. Surprisingly, our model is able to maintain a stable rollout even beyond the training range, as discussed in Section 4.3. We also found an interesting experimental result. Although the simple Linear model shows poor results on the long history time steps of the cylinder flow, it achieves competitive prediction accuracy on the airfoil flow. In particular, the Linear model outperforms the complex transformer-based method on the airfoil flow. This is mainly due to the short history time steps of airfoil flow, resulting in its time information being well captured by simple linear models. The use of complex attention mechanisms in the transformer-based method may lead to overfitting, making its prediction accuracy lower than that of the simple Linear model instead. Figure 5 shows how error accumulation occurs in the different models. The error accumulation of our model is very slow in both systems. Especially in the cylinder flow based on long history time steps, the long-term rollout of our model has significantly lower error accumulation than the baseline and model variants. We attribute this slow error accumulation behavior to the ability of the sequential model to capture periodic and trending information in long time series, as discussed further later in this section. We also note that the Linear and Transformer models have opposite performance results on the two systems. This is due to the fact that the simple Linear model cannot handle the complex temporal information in long time steps (cylinder flow), and the complex Transformer model leads to overfitting when handling short time steps (airfoil flow). On the contrary, our model maintains superior performance for history time steps of different lengths.