The charge that stays where it started.

A symmetry fixes a set of conserved charges, and a model that respects the symmetry does not move them. Each model is rolled forward twenty steps and the largest relative change of its charge is recorded. The white arrow is the charge at the start. Each ring is one seed, drawn at the distance its charge had moved, on a log scale marked from a tenth of a per cent to ten times the charge's own size. Only the size of each drift was recorded, so it is drawn as a ring and no direction is implied. The bands on the right are the same seeds, read off along one axis.

Blue is ARCHON, over 100 seeds. Amber is a physics-informed network, the method that adds the physical law to the training loss, over 100 seeds. Rose is a Lie group integrator network, which builds the group into its own integrator, over 50 seeds. Readouts are each network's drift as a multiple of ARCHON's: 349 to 2,772 times for the physics-informed network and 5,018 to 6,443 times for the Lie group integrator network, across the training range, 3 times and 11 times that range, and pure noise.