Spectral convergence of sum-of-Gaussians tensor neural networks for many-electron Schr\"odinger equation

TL;DR

This paper introduces an improved sum-of-Gaussians tensor neural network architecture for solving the many-electron Schrödinger equation, demonstrating high accuracy, spectral convergence, and efficiency in one-dimensional soft-Coulomb systems.

Contribution

The paper develops an enhanced SOG-TNN model with model reduction and anti-symmetry preservation, achieving efficient, high-accuracy solutions for multi-electron wave functions.

Findings

High accuracy with small basis sizes

Robust spectral convergence observed

Efficient low-rank representation of wave functions

Abstract

We present an improved version of the sum-of-Gaussians tensor neural network (SOG-TNN) architecture for solving many-electron Schr\"{o}dinger equation for one-dimensional soft-Coulomb systems. Model reduction techniques are introduced to reduce the number of tensor-factorized bases under the SOG approximation of the kernel. The Slater determinant ansatz is employed so that the anti-symmetric property of the wave function can be strictly preserved. Numerical results show that the SOG-TNN achieves high accuracy with remarkably small basis sizes. Robust spectral convergence with respect to the basis size is also observed, consistently characterized by a mixed algebraic-exponential model for the error decay. These findings validate that the SOG-TNN architecture provides an ultra-efficient and low-rank representation of complex multi-electron wave functions, shedding light on high-fidelity quantum calculations in larger-scale many-electron systems.