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On Representing Electronic Wave Functions with Sign Equivariant Neural Networks

Nicholas Gao, Stephan Günnemann

TL;DR

This work explores the flipped approach, where it is demonstrated that it reduces to a Jastrow factor, a commonly used permutation-invariant multiplicative factor in the wave function, and concludes with neither theoretical nor empirical advantages of sign equivariant functions for representing electronic wave functions.

Abstract

Recent neural networks demonstrated impressively accurate approximations of electronic ground-state wave functions. Such neural networks typically consist of a permutation-equivariant neural network followed by a permutation-antisymmetric operation to enforce the electronic exchange symmetry. While accurate, such neural networks are computationally expensive. In this work, we explore the flipped approach, where we first compute antisymmetric quantities based on the electronic coordinates and then apply sign equivariant neural networks to preserve the antisymmetry. While this approach promises acceleration thanks to the lower-dimensional representation, we demonstrate that it reduces to a Jastrow factor, a commonly used permutation-invariant multiplicative factor in the wave function. Our empirical results support this further, finding little to no improvements over baselines. We conclude with neither theoretical nor empirical advantages of sign equivariant functions for representing electronic wave functions within the evaluation of this work.

On Representing Electronic Wave Functions with Sign Equivariant Neural Networks

TL;DR

This work explores the flipped approach, where it is demonstrated that it reduces to a Jastrow factor, a commonly used permutation-invariant multiplicative factor in the wave function, and concludes with neither theoretical nor empirical advantages of sign equivariant functions for representing electronic wave functions.

Abstract

Recent neural networks demonstrated impressively accurate approximations of electronic ground-state wave functions. Such neural networks typically consist of a permutation-equivariant neural network followed by a permutation-antisymmetric operation to enforce the electronic exchange symmetry. While accurate, such neural networks are computationally expensive. In this work, we explore the flipped approach, where we first compute antisymmetric quantities based on the electronic coordinates and then apply sign equivariant neural networks to preserve the antisymmetry. While this approach promises acceleration thanks to the lower-dimensional representation, we demonstrate that it reduces to a Jastrow factor, a commonly used permutation-invariant multiplicative factor in the wave function. Our empirical results support this further, finding little to no improvements over baselines. We conclude with neither theoretical nor empirical advantages of sign equivariant functions for representing electronic wave functions within the evaluation of this work.
Paper Structure (14 sections, 3 theorems, 21 equations, 7 figures, 2 tables)

This paper contains 14 sections, 3 theorems, 21 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Sign Equivariant Functions
  4. Theoretical Results
  5. Empirical Results
  6. Conclusion
  7. Proof of Theorem 1
  8. Proof of Theorem 2
  9. Distribution of Wave Function Amplitudes
  10. LinLog Transformation
  11. Used structures
  12. Variational Monte Carlo
  13. Setup
  14. Additional Experiments

Key Result

Theorem 1

Any odd function $f:{\mathcal{X}} \to {\mathcal{Y}}$ on real vector spaces ${\mathcal{X}}, {\mathcal{Y}}$ can be represented as $f({\bm{x}})=g({\bm{x}})-g(-{\bm{x}})$ where $g:{\mathcal{X}} \to {\mathcal{Y}}, {\bm{x}}\in{\mathcal{X}}$.

Figures (7)

  • Figure 1: CASSCF for different values of $\alpha$ without symmetric functions.
  • Figure 2: FermiNet with different choices of symmetric functions and $\alpha=-2$.
  • Figure 3: Distribution of wave function amplitudes for LiH in logarithmic (left), linear (center left), and lin-log (right figures) domain. Magnitudes vary between $10^{-2}$ and $10^{-20}$.
  • Figure 4: Illustration of different domains relative to linear data.
  • Figure 5: Final energies of CASSCF+odd optimized with Prodigy. Missing entries were numerically unstable and encountered NaNs during training.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:odd_construction']}
  • Lemma 1
  • proof : Proof of Lemma \ref{['le:linear']}
  • ...and 1 more