App 17 — Invariance vs Equivariance

Rotate the molecule and watch which features stay fixed (invariant) and which rotate along with it (equivariant)

Rotating molecule (3 atoms)

Central atom Atom A Atom B

Force vector on atom A

Force vector — rotates with the crystal
Invariant feature — distance (SchNet sees this)
60.0
Distance from center to atom A. Never changes as the molecule rotates — this is all a purely invariant network like SchNet has access to.
Equivariant feature — direction (MACE keeps this)
0°
Bond direction angle. Changes exactly in step with the rotation — equivariant networks like MACE represent this directly via spherical harmonics.