Based on this idea, several promising protocols have been proposed, such as the dual-level Shepard interpolation method, the hierarchical construction scheme, the Δ-machine learning (Δ-ML) method. As ∆VHL-LL, with a small energy scope, features a very flat characterization, only a much smaller portion of the low-level dataset needs be calculated at the target high-level electronic structure method to fit the correction PES, ∆VHL-LL. In the dual-level Shepard interpolation and the hierarchical construction method, the interpolation was used to represent VLL and ∆VHL-LL, and data points were selected from reduced grids of data points for VLL. In the ∆-ML approach, both VLL and ∆VHL-LL were fitted by PIP or fragment-PIP, and the method was successfully applied to developing PESs of several systems, including CH4, H3O + , trans and cis-N-methyl acetamide (NMA), CH3CONHCH3, and acetylacetone.