用矩阵乘积算子构建神经网络势能面，高效解决高维量子问题。
Neural Network Matrix Product Operator: A Multi-Dimensionally Integrable Machine Learning Potential
- 将神经网络与矩阵乘积算子结合，突破维度灾难限制。
- 仅用625个训练点即达3.03 cm⁻¹误差，精度接近光谱级。
- 适合需要高维量子模拟的分子动力学研究者使用。
提出一种基于神经网络的机器学习势能面（NN-MPO），以矩阵乘积算子（MPO）形式表达。MPO结构可高效计算求解时不变与时变薛定谔方程中出现的高维积分，有效克服维度灾难问题。这与传统多层感知机（MLP）等全连接结构形成鲜明对比，后者因拓扑限制难以高效处理高维积分。然而，NN-MPO仍保持神经网络的强表征能力。在全耦合六维从头算势能面上，仅用625个分布于0至17,000 cm⁻¹能量范围内的训练点，测试均绝对误差（MAE）可达3.03 cm⁻¹，达到光谱级精度。代码已开源：https://github.com/KenHino/Pompon。
原文摘要 · Abstract (English)
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 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.
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