  Interatomic potentials are essential to go beyond ab initio size limitations,
but simulation results depend sensitively on potential parameters. Forward
propagation of parameter variation is key for uncertainty quantification,
whilst backpropagation has found application for emerging inverse problems such
as fine-tuning or targeted design. Here, the implicit derivative of functions
defined as a fixed point is used to Taylor expand the energy and structure of
atomic minima in potential parameters, evaluating terms via automatic
differentiation, dense linear algebra or a novel sparse operator approach. The
latter allows efficient forward and backpropagation through relaxed structures
of arbitrarily large systems. The implicit expansion accurately predicts
lattice distortion and defect formation energies and volumes with classical and
machine-learning potentials, enabling high-dimensional uncertainty propagation
without prohibitive overhead. We then show how the implicit derivative can be
used to solve challenging inverse problems, minimizing an implicit loss to
fine-tune potentials and stabilize solute-induced structural rearrangements at
dislocations in tungsten.
