Source-linked AI summary
Fermi Sets: Universal and interpretable neural architectures for fermions
Liang Fu
TL;DR
Fermionic neural networks must satisfy antisymmetry, but the universality of determinant-based constructions has generally been unresolved beyond one dimension. Fermi Sets address this with physically interpretable antisymmetric cores combined with symmetric networks, and a shared model achieves competitive accuracy on three-dimensional metallic solid hydrogen across multiple geometries.
Problem
Fermionic wavefunctions obey a nontrivial exchange-antisymmetry constraint, while the universality of determinant-based neural architectures was unknown beyond d = 1.
Method
Fermi Sets combine permutation-invariant neural networks with antisymmetric cores including pairwise products, Slater determinants, or Jordan-Wigner factors, with learnable Slater bases providing a universal architecture.
Results
A single spinful Fermi Sets model reaches −0.49062(1) Ha/atom for metallic solid hydrogen and surpasses all three DMC benchmarks while training across four nuclear configurations.
Takeaways & Limitations
Fermi Sets provide universal, physically interpretable fermionic wavefunctions while supporting one-network multi-geometry learning in a challenging three-dimensional real-material system.
Takeaways & Limitations
A single fixed Slater core hard-wires its nodal set and generally excludes fermionic wavefunctions that are nonzero on the core's zeros.
Abstract
from arXiv · showhide
We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability. As a numerical illustration, a single Fermi Sets model applied to metallic solid hydrogen in three dimensions, trained simultaneously across multiple nuclear geometries, surpasses all diffusion Monte Carlo benchmarks.
NUMERICAL BENCHMARK: SOLID HYDROGEN IN THREE DIMENSIONS
The study benchmarks Fermi Sets on metallic solid hydrogen in three spatial dimensions, a challenging real-material setting combining electron correlation and three-dimensional geometry.
- Metallic solid hydrogen provides a three-dimensional, basis-free test of Fermi Sets for directly solving the many-electron Schrödinger equation in real space.
System and Hamiltonian
The benchmark studies hydrogen atoms arranged in a periodic BCC supercell and evaluates their electronic problem with a Born–Oppenheimer Hamiltonian using Ewald-summed Coulomb interactions.
- N = 16 hydrogen atoms form a BCC crystal at rs = 1.31 Bohr in a 2×2×2 supercell with periodic boundary conditions.
- Each hydrogen atom contributes one spin-1/2 electron, giving N = 16 electrons in total.
- The Hamiltonian uses fixed proton positions and includes proton–proton Ewald energy, with Coulomb interactions evaluated by standard Ewald summation.
Fermi Sets architecture for spinful electrons
For spinful electrons, Fermi Sets combine learnable Slater determinants as antisymmetric bases with permutation-invariant neural networks for symmetric factors, preserving universality and physical interpretability.
- The spinful ansatz uses generalized electron coordinates xi = (ri, si) with s = ↑, ↓.
- K learnable Slater determinants built from spin orbitals φk_i provide the antisymmetric bases.
- Deep Sets permutation-invariant networks implement the symmetric factors ϕk.
- Learnable spin orbitals guarantee universality because the ansatz contains a product of spin-up and spin-down Vandermonde determinants as a special case.
- The benchmark uses K = 8 Slater determinants and 238,024 network parameters, with periodicity represented through plane-wave envelope features.
Multi-geometry parameter sharing
A single Fermi Sets network is trained across equilibrium and displaced nuclear configurations, sharing parameters while taking nuclear positions as inputs to produce geometry-dependent wavefunctions and energies.
- Multi-geometry parameter sharing: One parameter set is trained simultaneously on four nuclear configurations: equilibrium BCC geometry and three randomly displaced geometries.
- Multi-geometry parameter sharing: Nuclear positions enter the network as inputs, allowing one wavefunction to describe the electronic ground state across all geometries without per-geometry re-optimization.
- Multi-geometry parameter sharing: This parameter sharing is presented as a step toward ab initio potential energy surfaces, with one trained network providing wavefunctions and energies at arbitrary nuclear configurations.
- Multi-geometry parameter sharing: Table S1 reports energy per atom for the shared model across equilibrium and three displaced configurations using 21,000 independent Monte Carlo blocks.
Results and comparison
Fermi Sets achieves competitive energy accuracy for equilibrium BCC solid hydrogen while using one parameter set across four nuclear geometries. Its result surpasses all DMC benchmarks and approaches a specialized equilibrium-only NQS result.
- −0.49062(1) Ha/atom surpasses all three DMC benchmarks for equilibrium BCC solid hydrogen at N = 16 and rs = 1.31.The comparison uses periodic boundary conditions and energies per atom.
- Fermi Sets comes within ∼1 mHa of the specialized NQS result while training one parameter set across equilibrium and three displaced geometries.The NQS reference is optimized for the equilibrium geometry alone, whereas Fermi Sets is trained simultaneously across all four.
- Energy varies smoothly and consistently across displaced geometries despite individual Cartesian displacements as large as 0.3 Bohr.This behavior is reported as evidence of transferability across nuclear configurations.
- The displaced geometries use independently drawn Gaussian nuclear displacements with σ = 0.1 Bohr per Cartesian direction.The system contains 16 atoms in the 2×2×2 BCC supercell, with atoms assigned to corner and body-centered sites.