SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To show the effect of the number of symmetry groups of a PDE in the performance of the model trained with the symmetry loss, we analyzed the performance on (average MSE over the test dataset) as we increased the number of infinitesimal generators that were used to calculate  $\mathcal{L}_{\mathrm{sym}}$  for the Heat equation. The two tables below show this effect for two different models that are trained with  $N_{r} = 500$  and  $N_{r} = 10000$  unique points sampled from inside the grid, respectively. As it can be seen, including both of the useful symmetries of the Heat equation leads to the best performance compared to the models that are trained with 1 or 0 infinitesimal generators. Furthermore, as shown in previous results, we can see that the model's overall performance in all 3 cases increases as we
Figure 5: The effect of training the PDE solver for the Burgers' equation with and without the symmetry loss for one of the PDEs in the test dataset. (a) shows the ground truth solution and the predictions of the two models as the number of samples inside the domain increases from 5000 to 25000 and 100000. (b) shows the corresponding predictions and the ground truth solution at different time slices.
sample more points inside the grid and that the effect of the symmetry loss is more pronounced in the low-data regime. We also tried training models with and without a symmetry loss for the
Table 5: The average test set mean-squared error for the Heat equation as a function of increasing the number of infinitesimal generators used to compute the symmetry loss. The MSE is reported for models trained with different numbers of unique points sampled from inside the grid. It can be seen that including both of the useful symmetries of the Heat equation leads to the best performance compared to the models that are trained with 1 or 0 infinitesimal generators.
Number of Symmetries (K)	MSE when Nr = 500	MSE when Nr = 10000
0	0.73 ± 0.38	0.26 ± 0.21
1	0.61 ± 0.33	0.25 ± 0.17
2	0.44 ± 0.26	0.20 ± 0.12
Heat equation using a modified MLP architecture, as suggested in Wang et al. [2020b]. We found that, unexpectedly, this architecture was performing worse than a simple MLP architecture in our experiments. However, as it can be seen in Table 6, the model trained with the symmetry loss still performs better than the one trained with just the PINN loss.
Table 6: The average test set mean-squared error for the Heat equation as a function of increasing the number of unique points sampled inside the grid. The model architecture is a modified MLP, as suggested in Wang et al. [2020b]. It can be seen that the model trained with the symmetry loss performs betters than that trained without, especially in the low-data regime.
Number of Points (Nr)	No Symmetry	Symmetry
500	1.15 ± 0.754	0.859 ± 0.689
2000	0.974 ± 0.667	0.520 ± 0.488
10000	0.391 ± 0.577	0.357 ± 0.363