Physical properties of a topological origin are known to be robust against small perturbations. This robustness is both a source of theoretical interest and a driver 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 smooth, nonquantized functions of the system parameters, 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.