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 Schrödinger equation, and effectively overcomes the so-called curse of dimensionality. This starkly contrasts with other neural network-based machine learning PES methods, such as multilayer 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msup><a:mrow><a:mi>cm</a:mi></a:mrow><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:math> for a fully coupled six-dimensional PES, using only 625 training points distributed across a 0 to 17 000 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msup><b:mrow><b:mi>cm</b:mi></b:mrow><b:mrow><b:mo>−</b:mo><b:mn>1</b:mn></b:mrow></b:msup></b:math> energy range.