Equivariant Continuous Normalizing Flows with Offline Sampling for Fermionic Ground State Estimation
Abstract
We introduce a framework for fermionic variational Monte Carlo in which a continuous normalizing flow refines a fixed antisymmetric base wavefunction. The flow is a permutation-equivariant neural ordinary differential equation (ODE), a smooth, topology-preserving map; equivariance preserves the antisymmetry of the base, so the flow can in principle improve any antisymmetric ansatz that can be sampled efficiently. The flow deforms the nodal surface of the base but cannot change its topology, so the base must supply the correct nodal topology. We use Slater and Jastrow-Slater bases in this work. Exact samples from the flow's Born distribution are obtained by pushing pre-cached base samples through the forward ODE, requiring no Markov chain Monte Carlo at training time. Base samples are generated offline and reused across batches and runs, enabling embarrassingly parallel training across multiple GPUs. We introduce three permutation-equivariant vector field architectures, Pairwise Deep Sets, FermiNet Vector Fields, and Pairwise Deep Sets Gradient, each balancing expressivity and computational cost differently. We further introduce an augmented dynamics formulation that co-evolves the derivatives required for the kinetic energy as ODE state variables, eliminating differentiation through the ODE trajectory and significantly reducing wall-clock time and memory. On harmonically trapped spinless electrons, the flow reaches ground-state energies below configuration interaction with single and double excitations, on par with a FermiNet baseline trained with the same optimizer. Training shows near-ideal strong scaling to 128 NVIDIA A100 GPUs on NERSC's Perlmutter for the largest systems tested, up to N = 48 particles in three dimensions.
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BibTeX
@misc{cochran2026equivariant,
title = {{Equivariant Continuous Normalizing Flows with Offline Sampling for Fermionic Ground State Estimation}},
author = {Sam Cochran and James B. Larsen and James Stokes and Andrew Wray and Michael J. Keiser and Chad Rigetti and Shravan Veerapaneni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.18486},
eprint = {2607.18486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18486},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Equivariant Continuous Normalizing Flows with Offline Sampling for Fermionic Ground State Estimation
AU - Sam Cochran
AU - James B. Larsen
AU - James Stokes
AU - Andrew Wray
AU - Michael J. Keiser
AU - Chad Rigetti
AU - Shravan Veerapaneni
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2607.18486
UR - https://arxiv.org/abs/2607.18486
AB - We introduce a framework for fermionic variational Monte Carlo in which a continuous normalizing flow refines a fixed antisymmetric base wavefunction. The flow is a permutation-equivariant neural ordinary differential equation (ODE), a smooth, topology-preserving map; equivariance preserves the antisymmetry of the base, so the flow can in principle improve any antisymmetric ansatz that can be sampled efficiently. The flow deforms the nodal surface of the base but cannot change its topology, so the base must supply the correct nodal topology. We use Slater and Jastrow-Slater bases in this work. Exact samples from the flow's Born distribution are obtained by pushing pre-cached base samples through the forward ODE, requiring no Markov chain Monte Carlo at training time. Base samples are generated offline and reused across batches and runs, enabling embarrassingly parallel training across multiple GPUs. We introduce three permutation-equivariant vector field architectures, Pairwise Deep Sets, FermiNet Vector Fields, and Pairwise Deep Sets Gradient, each balancing expressivity and computational cost differently. We further introduce an augmented dynamics formulation that co-evolves the derivatives required for the kinetic energy as ODE state variables, eliminating differentiation through the ODE trajectory and significantly reducing wall-clock time and memory. On harmonically trapped spinless electrons, the flow reaches ground-state energies below configuration interaction with single and double excitations, on par with a FermiNet baseline trained with the same optimizer. Training shows near-ideal strong scaling to 128 NVIDIA A100 GPUs on NERSC's Perlmutter for the largest systems tested, up to N = 48 particles in three dimensions.
ER -