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.

翻译：本文提出了一种基于神经网络的机器学习势能面，该势能面以矩阵乘积算子形式表达（NN-MPO）。MPO形式能够高效计算求解含时与不含时薛定谔方程时出现的高维积分，并有效克服所谓的“维度灾难”。这与其他基于神经网络的机器学习势能面方法（如多层感知机）形成鲜明对比——由于MLP主干架构采用全连接拓扑，其高维积分计算并不直接。尽管如此，NN-MPO仍保持了神经网络的高表达能力。对于完全耦合的六维从头算势能面，NN-MPO仅需625个在0至17,000 cm$^{-1}$能量范围内分布的训练点，即可达到光谱精度，测试平均绝对误差为3.03 cm$^{-1}$。我们的Python实现已发布于https://github.com/KenHino/Pompon。