Sum-of-Gaussians tensor neural networks for high-dimensional Schr\"odinger equation

TL;DR

This paper introduces a novel sum-of-Gaussians tensor neural network method that efficiently solves high-dimensional Schrödinger equations by overcoming the curse of dimensionality and accurately handling Coulomb interactions.

Contribution

The paper presents a new SOG-TNN algorithm with a dimensionally separable kernel approximation and a range-splitting scheme, improving efficiency and accuracy in high-dimensional quantum problems.

Findings

Outperforms existing methods in accuracy and efficiency

Effectively handles Coulomb singularities with SOG decomposition

Reduces computational complexity for high-dimensional Schrödinger equations

Abstract

We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schr\"{o}dinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor representation with rapid convergence. We further develop a range-splitting scheme that partitions the Gaussian terms into short-, long-, and mid-range components. They are treated with the asymptotic expansion, the low-rank Chebyshev expansion, and the model reduction with singular-value decomposition, respectively, significantly reducing the number of two-dimensional integrals in computing electron-electron interactions. The SOG decomposition well resolves the computational challenge due to the singularity of the Coulomb interaction, leading to an efficient algorithm for the high-dimensional problem under the TNN framework. Numerical results demonstrate the outstanding performance of the new method, revealing that the SOG-TNN is a promising way for accurately tackling quantum systems.