# Tensor Hypercontraction Error Correction Using Regression

## TL;DR

This paper uses machine learning to correct errors in a computationally efficient quantum chemistry method, improving accuracy for molecular energy predictions.

## Contribution

The novel contribution is applying regression models to correct errors in tensor hypercontraction approximations of quantum methods.

## Key findings

- Nonlinear regression models reduced THC-MP3 energy errors by up to 9× compared to canonical MP3.
- Error corrections improved both molecule and reaction energy predictions significantly.
- Absolute and relative correction approaches were evaluated for their effectiveness.

## Abstract

Wavefunction‐based quantum methods are some of the most accurate tools for predicting and analyzing the electronic structure of molecules, in particular for accounting for dynamical electron correlation. However, most methods of including dynamical correlation beyond the simple second‐order Møller–Plesset perturbation theory (MP2) level are too computationally expensive to apply to large molecules. Approximations which reduce scaling with system size are a potential remedy, such as the tensor hyper‐contraction (THC) technique of Hohenstein et al., but also result in additional sources of error. In this work, we correct errors in THC‐approximated methods using machine learning. Specifically, we apply THC to third‐order Møller–Plesset theory (MP3) as a simplified model for coupled cluster with single and double excitations (CCSD), and train several regression models on observed THC errors from the Main Group Chemistry Database (MGCDB84). We compare performance of multiple linear regression models and nonlinear Kernel Ridge regression models. We also investigate correlation procedures using absolute and relative corrections and evaluate the corrections for both molecule and reaction energies. We discuss the potential for using regression techniques to correct THC‐MP3 errors by comparing it to the “canonical” MP3 reference values and find the optimum technique based on accuracy. We find that nonlinear regression models reduced root mean squared errors between THC‐ and canonical MP3 by a factor of 6–9× for total molecular energies and 2–3× for reaction energies.

The combination of tensor hypercontraction with machine learning bridges the gap between accuracy and computational efficiency through scaling reduction.