  We introduce and analyze numerical companion matrix methods for the
reconstruction of hypersurfaces with crossings from smooth interpolants given
unordered or, without loss of generality, value-sorted data. The problem is
motivated by the desire to machine learn potential energy surfaces arising in
molecular excited state computational chemistry applications. We present
simplified models which reproduce the analytically predicted convergence and
stability behaviors as well as two application-oriented numerical experiments:
the electronic excited states of Graphene featuring Dirac conical cusps and
energy surfaces corresponding to a sulfur dioxide ($SO_2$) molecule in
different configurations.
