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Spectral convergence of sum-of-Gaussians tensor neural networks for many-electron Schrödinger equation

Teng Wu, Qi Zhou, Huangjie Zheng, Hehu Xie, Zhenli Xu

Abstract

We present an improved version of the sum-of-Gaussians tensor neural network (SOG-TNN) architecture for solving many-electron Schrödinger 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.

Spectral convergence of sum-of-Gaussians tensor neural networks for many-electron Schrödinger equation

Abstract

We present an improved version of the sum-of-Gaussians tensor neural network (SOG-TNN) architecture for solving many-electron Schrödinger 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.
Paper Structure (9 sections, 17 equations, 3 figures, 3 tables)

This paper contains 9 sections, 17 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Tensor neural network representation
  4. Kernel tensor factorization
  5. Results
  6. Stable learning of SOG-TNN
  7. Convergence rate with basis size
  8. Comparison with the SG-CI results
  9. Conclusion

Figures (3)

  • Figure 1: Training curves of error $\mathcal{E}_r$ for (a) H and (b) He with basis size $P=12$ and $P=20$, respectively.
  • Figure 2: Convergence of relative error $\mathcal{E}_r$ as function of the basis size $P$ for 1D soft-Coulomb systems from H to O. The dashed lines are fitting lines by ansatz $\mathcal{E}_r=C\cdot P^{-\beta}\cdot e^{-\gamma P}$.
  • Figure 3: Log-log convergence of $\mathcal{E}_r$ against the basis size $P$ for the SOG-TNN (blue color) and SG-CI methods (red color). (a) H and He atoms; (b) Li and Be atoms; (c) B and C atoms; and (d) N and O atoms.