Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Kentaro Hino, Yuki Kurashige

TL;DR

The paper presents NN-MPO, a neural-network-based PES model that leverages matrix product operator and tensor-train representations to enable accurate high-dimensional energy surfaces with a small training set. By mapping inputs to a latent space with a trainable Coordinator and expressing the weight tensor in TT form, NN-MPO achieves spectroscopic accuracy ($3.03\ \mathrm{cm}^{-1}$ MAE on a 6D H$_2$CO PES trained with 625 points) while permitting efficient evaluation of multi-dimensional integrals essential for quantum dynamics. The approach demonstrates compatibility with DMRG-based vibrational calculations, achieving $0.325\ \mathrm{cm}^{-1}$ MAE against exact MPO results for up to twenty excited states, underscoring its potential for quantum simulations beyond classical ML PES methods. The work also discusses practical considerations for scalability, symmetry, and coordinate choices, outlining pathways to extend NN-MPO to larger systems through localized coordinates and improved regularization. Overall, NN-MPO bridges traditional SOP/HDMR tensor representations and modern neural networks, offering a scalable, accurate framework for quantum molecular simulations and related high-dimensional integral tasks.

Abstract

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.

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

TL;DR

The paper presents NN-MPO, a neural-network-based PES model that leverages matrix product operator and tensor-train representations to enable accurate high-dimensional energy surfaces with a small training set. By mapping inputs to a latent space with a trainable Coordinator and expressing the weight tensor in TT form, NN-MPO achieves spectroscopic accuracy ( MAE on a 6D HCO PES trained with 625 points) while permitting efficient evaluation of multi-dimensional integrals essential for quantum dynamics. The approach demonstrates compatibility with DMRG-based vibrational calculations, achieving MAE against exact MPO results for up to twenty excited states, underscoring its potential for quantum simulations beyond classical ML PES methods. The work also discusses practical considerations for scalability, symmetry, and coordinate choices, outlining pathways to extend NN-MPO to larger systems through localized coordinates and improved regularization. Overall, NN-MPO bridges traditional SOP/HDMR tensor representations and modern neural networks, offering a scalable, accurate framework for quantum molecular simulations and related high-dimensional integral tasks.

Abstract

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 for a fully coupled six-dimensional ab initio PES, using only 625 training points distributed across a 0 to 17,000 cm energy range. Our Python implementation is available at https://github.com/KenHino/Pompon.

Paper Structure

This paper contains 12 sections, 43 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Model Architecture
  4. Optimization
  5. Multi-dimensional integral
  6. Results and Discussion
  7. Dataset sampling
  8. Training results
  9. DMRG calculations
  10. Conclusions
  11. Bond dimension of harmonic oscillator with and without the coordinate transformation
  12. Conjugate gradient method for TT core optimization

Figures (11)

  • Figure 1: The conventional diagram in the deep learning community. This diagram indicates NN-MPO for $n=f=6$, $N=3$, and $M=2$.
  • Figure 2: The conventional diagram in the tensor network community. This diagram indicates NN-MPO for $f=6$. Coordinator layer is omitted.
  • Figure 3: The diagram of two-dots and one-dot center core in TT, both of which are filled with navy color.
  • Figure 4: The diagram of basis concatenated force and energy.
  • Figure 5: The diagram of tensor network blocks used for the optimization.
  • ...and 6 more figures