Machine-learned potentials (MLPs) offer the promise of providing "first-principle" approaches for the full-span of molecular dynamics simulations. Among the panoply of machine-learned methods, 2 permutationally invariant polynomials (PIPs) have been used for nearly 20 years to develop precise, high-dimensional potential energy surfaces (PESs) for molecules, clusters, many-body terms for water potentials, and atom-centered many-body representations for materials and molecular force fields. The precision of a PIPs PES for "rMD17 ethanol" was shown to be as good as the best performing ML methods and to be substantially faster (factors of 10 or more) than all the ML methods considered, i.e., GAP-SOAP, ANI, DPMD, sGDML, PhysNet, KREG, and pKREG. Similar factors in speed differences and high precision for PIPs were recently reported for 21-atom aspirin. In addition, PIPs have been used to fit more extensive and complex datasets than are given in the rMD17 dataset for molecules with 10 or more atoms. Examples include formic acid dimer, glycine, N -methyl acetamide, acetyl acetone, and tropolone. PIPs have also been used effectively and extensively as inputs for Neural Network and Gaussian Process Regression 38 machine learning for potentials.