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

arXiv: https://arxiv.org/abs/2410.23858 · HTML: https://mlsanity.com/paper/2410.23858/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 69 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [TeaNet: universal neural network interatomic potential inspired by iterative electronic relaxations](https://mlsanity.com/paper/1912.01398/) (2019)
- [On the Complexity of Learning with Kernels](https://mlsanity.com/paper/1411.1158/) (2014)

## Abstract (arXiv)

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.