  Background: The generation of potential energy surfaces is a critical step in
theoretical models aiming to understand and predict nuclear fission.
Discontinuities frequently arise in these surfaces in unconstrained collective
coordinates, leading to missing or incorrect results.
  Purpose: This work aims to produce efficient and physically-motivated
computational algorithms to refine potential energy surfaces by removing
discontinuities.
  Method: Procedures based on tree-search algorithms are developed which are
capable of smoothing discontinuities in one and two-dimensional potential
energy surfaces while minimising their overall energy.
  Results: Each of the new methods is applied to smooth candidate
discontinuities in ${}^{252}\mathrm{Cf}$, ${}^{222}\mathrm{Th}$ and
${}^{218}\mathrm{Ra}$. The effectiveness of each case is analysed both
qualitatively and quantitatively. The one-dimensional method is also compared
to the adiabatic and linear interpolation approaches which are commonly used to
remove discontinuities.
  Conclusions: The smoothing methods presented in this work are
resource-efficient and successful for one- and two-dimensional discontinuities;
they will improve the fidelity of potential energy surfaces as well as their
subsequent uses in beyond mean-field applications. Complex discontinuities
occurring in higher dimensions may require alternative approaches which better
utilise prior knowledge of the potential energy surface to narrow their
searches.
