  Robustness against small perturbations is a crucial feature of topological
properties. This robustness is both a source of theoretical interest and a
drive for technological applications, but presents a challenge when looking for
new topological systems: Small perturbations cannot be used to identify the
global direction of change in the topological indices. Here, we overcome this
limitation by breaking the symmetries protecting the topology. The introduction
of symmetry-breaking terms causes the topological indices to become
non-quantized variables, which are amenable to efficient design algorithms
based on gradient methods. We demonstrate this capability by designing discrete
and continuous phononic systems realizing conventional and higher-order
topological insulators.
