Neural Network Matrix Product Operator: A Multi-Dimensionally Integrable Machine Learning Potential

A neural network-based machine learning potential energy surface (PES) expressed in a matrix product operator (NN-MPO) is proposed. The MPO form enables efficient evaluation of high-dimensional integrals that arise in solving the time-dependent and time-independent Schr\"odinger equation and effectively overcomes the so-called curse of dimensionality. This starkly contrasts with other neural network-based machine learning PES methods, such as multi-layer perceptrons (MLPs), where evaluating high-dimensional integrals is not straightforward due to the fully connected topology in their backbone architecture. Nevertheless, the NN-MPO retains the high representational capacity of neural networks. NN-MPO can achieve spectroscopic accuracy with a test mean absolute error (MAE) of 3.03 cm$^{-1}$ for a fully coupled six-dimensional ab initio PES, using only 625 training points distributed across a 0 to 17,000 cm$^{-1}$ energy range. Our Python implementation is available at https://github.com/KenHino/Pompon.