Table 2.

The Hopf index calculated by using the conventional discretization approach

Input	h = 0.5	h = 2	h = 3.2
Theory (N → ∞)	−2	1	0
Theory (N = 10)	−2.045	1.041	0.011
Leg. samples	−2.056	1.039	0.009
Adv. examples (C.)	−2.039	0.952	0.026
Adv. examples (D.)	−	−	0.073

For a 10 × 10 × 10 grid, the theoretical value for the Hopf index is shown in the third row, where the small deviations from the corresponding integers are caused by the discretization error in the 3D momentum integration. The Hopf indices of experimentally implemented legitimate (Leg.) samples, and adversarial (Adv.) examples with continuous (C.) and discrete (D.) attacks are close to the theoretical values.