New Algorithms to Generate Permutationally Invariant Polynomials and Fundamental Invariants for Potential Energy Surface Fitting.

Symmetric functions, such as Permutationally Invariant Polynomials (PIPs) and Fundamental Invariants (FIs), are effective and concise descriptors for incorporating permutation symmetry into neural network (NN) potential energy surface (PES) fitting. The traditional algorithm for generating such symmetric polynomials has a factorial time complexity of N!, where N is the number of identical atoms, posing a significant challenge to applying symmetric polynomials as descriptors of NN PESs for larger systems, particularly with more than 10 atoms. Herein, we report a new algorithm which has only linear time complexity for identical atoms. It can tremendously accelerate generation process of symmetric polynomials for molecular systems. The proposed algorithm is based on graph connectivity analysis following the action of the generation set of molecular permutational group. For instance, in the case of calculating the invariant polynomials for a 15-atom molecule, such as tropolone, our algorithm is approximately 2 million times faster than the previous method. The efficiency of the new algorithm can be further enhanced with increasing molecular size and number of identical atoms, making the FI-NN approach feasible for systems with over 10 atoms and high symmetry demands.