Fitting potential energy surfaces with fundamental invariant neural network. II. Generating fundamental invariants for molecular systems with up to ten atoms.

Symmetry adaptation is crucial in representing a permutationally invariant potential energy surface (PES). Due to the rapid increase in computational time with respect to the molecular size, as well as the reliance on the algebra software, the previous neural network (NN) fitting with inputs of fundamental invariants (FIs) has practical limits. Here, we report an improved and efficient generation scheme of FIs based on the computational invariant theory and parallel program, which can be readily used as the input vector of NNs in fitting high-dimensional PESs with permutation symmetry. The newly developed method significantly reduces the evaluation time of FIs, thereby extending the FI-NN method for constructing highly accurate PESs to larger systems beyond five atoms. Because of the minimum size of invariants used in the inputs of the NN, the NN structure can be very flexible for FI-NN, which leads to small fitting errors. The resulting FI-NN PES is much faster on evaluating than the corresponding permutationally invariant polynomial-NN PES.