SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Next, we integrate our method in EqMotion [55], a recent equivariant method with state-of-the-art performance on trajectory forecasting. We incorporate Aether in EqMotion by treating the predicted forces as geometric features similar to velocities. Namely, after computing the forces for each object at each timestep, we compute the magnitudes of the force vectors and the force angle sequence, i.e. the angles between forces in consecutive timesteps. We concatenate these quantities to the existing features for the feature initialization step.
We train and evaluate EqMotion and Aether with EqMotion on inD [4] following our experimental setup. We report the results in Figure 8. We see that Aether is beneficial even for a state-of-the-art trajectory forecasting method, which further strengthens our claims.
Figure 8: Ablation study on the choice of equivariant GNN backbone. Results on inD.
C.2 Non-equivariant network with neural field
Our ability to capture global components with a neural field stems from our overall architecture, which promotes disentanglement. Using a graph network that respects the underlying symmetries and has an inductive bias towards using local interactions, i.e. any equivariant graph network, allows
the neural field to "solve for" the global components, by "subtracting" the local interactions from the observable net effects.
The choice of an equivariant network is crucial here; a non-equivariant graph network like NRI [23] or dNRI [14] would merely gather "redundant" information from the neural field. We demonstrate this mathematically in the following equations for the static field, in which we first compute the force  $\mathbf{f}$  at a target position  $\mathbf{p}$ , and then compute the node embedding using equations from NRI/dNRI, including the forces.
$$
\mathbf {f} = f (\mathbf {p}) = \operatorname {M L P} _ {1} (\mathbf {p}) \tag {83}
$$
$$
\mathbf {h} = g (\mathbf {p}, \mathbf {u}, \mathbf {f}) = \operatorname {M L P} _ {2} ([ \mathbf {p}, \mathbf {u}, \mathbf {f} ]) \tag {84}
$$
We can see that the node embeddings depend on positions twice, one explicit and one through another MLP, in an architecture similar to a concatenated residual connection. In this case, we do not expect the neural field to isolate global forces, or to be helpful for future forecasting. We test this hypothesis with an ablation experiment on the electrostatic field setting, by combing our neural field with dNRI, instead of an equivariant network. In Table 5, we report the MSE at the final prediction timestep, i.e. MSE@20. We can see that adding a neural field to a non-equivariant network does not enhance performance, and in fact, it results in performance degradation, which enhances our hypothesis.
Table 5: Ablation study on the suitability of non-equivariant networks with Aether. Combining dNRI -a non-equivariant graph network- with a neural field does not enhance performance. Results on the electrostatic field setting.
Method	MSE@20 (↓)
dNRI [14]	1.20
dNRI+Aether	1.37
Aether	0.69
C.3 Choice of conditioning mechanism
FiLM [33] is used in the dynamic field setting to condition neural fields. To examine its influence on performance, we perform an ablation study on the 3D gravitational setting, where we replace FiLM layers with conditioning by concatenation, a very simple and successful conditioning mechanism. We term this model *Concat Aether*, and report the results in Table 6. We see that conditioning by concatenation underperforms, scoring almost on par with LoCS. This is perhaps expected, since concatenation is a rather weak form of conditioning, and our task is very challenging. On the other hand, FiLM is a powerful mechanism and is able to condition effectively.
Table 6: Ablation study on the choice of conditioning mechanism. Results on the 3D gravitational field setting.
Method	MSE@5 (↓)
LoCS [24]	0.1308
Aether	0.0660
Concat Aether	0.1474
D.1 Electrostatic field
Figure 9 shows qualitative results on the electrostatic field setting.
(a) Aether
(b) G-LoCS
(c) LoCS
Figure 9: Predictions on the electrostatic field setting. Lighter colors indicate predictions, and darker colors indicate the groundtruth. Predictions start where markers have black edges. Markers get bigger and more opaque as trajectories evolve in time. The background streamplots indicate the groundtruth field, and are not given as input to the networks. Similarly, the blue  $\oplus$  markers and the red  $\ominus$  markers, are merely shown for illustrative purposes, indicating the charges of the field sources, and are not given as input to the networks. Best viewed in color.
(d) dNRI
D.1.1 Discovered electrostatic field
In Figure 10 we visualize the discovered electrostatic field compared to the groundtruth one.
Predicted Field
Figure 10: Learned Field (left) in electrostatic field setting compared to groundtruth (right).
D.2 InD
Figure 11 shows qualitative results of our method on inD [4]. Figures 12, 13 and 14 show qualitative results for G-LoCS, LoCS, and dNRI, respectively.
Figure 11: Aether predictions (right) on inD, compared to groundtruth (left). Predictions start where markers are colored black. Best viewed in color.
Figure 12: G-LoCS predictions (right) on inD, compared to groundtruth (left). Predictions start where markers are colored black. Best viewed in color.
Figure 13: LoCS predictions (right) on inD, compared to groundtruth (left). Predictions start where markers are colored black. Best viewed in color.
Figure 14: dNRI predictions (right) on inD, compared to groundtruth (left). Predictions start where markers are colored black. Best viewed in color.
D.2.1 Discovered traffic force field
In Figure 15 we visualize the discovered traffic force field on inD. In the supplementary material, we provide video visualizations of the learned field, with input orientations evolving over time.
Figure 15: Discovered field on inD [4]. For simplicity, we only visualize the field for discrete input orientations in  $\mathrm{C}_4 = \left\{0,\frac{\pi}{2},\pi ,\frac{3\pi}{2}\right\}$ . Best viewed in color.
D.3 2D gravity
Figure 16 shows qualitative results on the 2D gravitational n-body problem.
(c) LoCS
Figure 16: Predictions on gravity. Lighter colors indicate predictions, and darker colors indicate the groundtruth. Predictions start where markers have black edges. Markers get bigger as trajectories evolve. Best viewed in color.
D.3.1 Discovered 2D gravitational fields
Figure 17 shows examples of discovered fields compared to the groundtruth ones.
Predicted Field
Groundtruth Field
Figure 17: Learned dynamic fields (left) in 2D gravitational field setting vs groundtruth (right).
E Quantitative results
In all settings, we report the total errors, i.e. the mean squared errors of positions and velocities over time,  $E(t) = \frac{1}{ND} \sum_{n=1}^{N} \| \mathbf{x}_n^t - \hat{\mathbf{x}}_n^t \|_2^2$ . Following Kofinas et al. [24], we also separately report the  $L_2$  norm position errors,  $E_p(t) = \frac{1}{N} \sum_{n=1}^{N} \| \mathbf{p}_n^t - \hat{\mathbf{p}}_n^t \|_2$ , and velocity errors,  $E_u(t) = \frac{1}{N} \sum_{n=1}^{N} \| \mathbf{u}_n^t - \hat{\mathbf{u}}_n^t \|_2$ .
E.1 Electrostatic field
Figure 18: Results in the electrostatic field setting.
E.2 InD
Figure 19: Results in inD.
E.3 Gravity
Figure 20: Results in the dynamic gravitational field setting.
E.4 2D gravity
Figure 21: Results in the dynamic 2D gravitational field setting.
E.5 Significance of the discovered field
Figure 22: Ablation study on the significance of the discovered field. Results in the electrostatic field setting.