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.

翻译：本文引入并分析了数值伴随矩阵方法，用于从无序（或经值排序）数据的光滑插值函数中重建具有交叉结构的超曲面。该问题的研究动机源于机器学习在分子激发态计算化学应用中势能曲面构建的需求。我们提出了简化模型，这些模型再现了理论预测的收敛性与稳定性行为，并进行了两项面向应用的数值实验：呈现狄拉克锥形奇点的石墨烯电子激发态，以及不同构型下二氧化硫（$SO_2$）分子对应的能量曲面。