Sum-of-Gaussians tensor neural networks for high-dimensional Schrödinger equation

TL;DR

An accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schr\"{o}dinger equation" and shows that the SOG-TNN is a promising way for accurately tackling quantum systems.

Abstract

We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger 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.

Figures (8)

• Figure 1: The TNN architecture for approximating solutions to high-dimensional equations. It consists of $d$ independent feedforward sub-neural networks mapping $\mathbb{R}$ to $\mathbb{R}^p$. The outputs are multiplied to form $p$ tensor-product basis functions that are linearly combined to approximate the solution.
• Figure 2: Singular value distribution of kernel matrices $\bm{G}_{\ell}$ consists of $600$ proxy points chosen as composite Gauss-Legendre nodes with respect to different bandwidth $s_\ell$.
• Figure 3: Computation results for the Hydrogen atom by the SOG-TNN algorithm, comparing performance without and with the piecewise learning rate decay schedule. (a,b) results without the decay schedule, and (c,d) results with the decay schedule. The training curves in (a,c) shows the relative error of the ground state energy $E_0$, and (b,d) shows the $L^{\infty}(\Omega)$ error of radial distribution of the ground state wavefunction $\Psi_0$.
• Figure 4: (a) The relative error training curve of the ground state energy and (b) The radial distribution of the ground state wavefunction of the helium atom by the SOG-TNN algorithm.
• Figure 5: (a) Relative error training curve of the ground state energy, and (b) radial distribution of the ground state wavefunction of the lithium atom by the SOG-TNN algorithm.

Theorems & Definitions (2)

• Lemma 5.1
• proof