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Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schrödinger Equation

ICML 2025 · Probabilistic and scientific modeling › Molecules and proteins · approach: Machine learning for physical simulation · 0 citations (OpenAlex)

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Studies symmetrization (data augmentation, group averaging, smoothed canonicalization, post hoc averaging) of DeepSolid VMC neural network solvers under diagonal space-group symmetries, finding in-training symmetrization can hurt while post hoc averaging helps.

Problem

Neural network variational Monte Carlo solvers for the many-electron Schrödinger equation with diagonal group symmetries (space groups in crystals), where invariant maps are hard to construct and it is unclear whether symmetrization helps.

Approach

Compares in-training symmetrization (data augmentation, group averaging, smoothed canonicalization) against post hoc averaging of a trained wavefunction, showing that the computational-statistical tradeoff in VMC (sampling cheap, gradients expensive) makes in-training symmetrization harmful while post hoc averaging is simple and effective.

Key results

• Post hoc averaging on graphene with P6mm (k=12) reaches -76.050(3) Ha with local-energy variance 0.33 Ha^2, versus -76.039(6) Ha and 2.02 Ha^2 for the unsymmetrized OG at similar GPU hours (281).
• Group averaging with same batch size and 12x cost gives -76.049(3) Ha but variance inflated to 0.58 Ha^2 at 304 GPU hours; data augmentation gives 2.48 Ha^2 variance, showing in-training symmetrization destabilizes training.
• For bcc-Li, post hoc averaging over P4/mmm (k=16) improves energy from -15.011(1) to -15.021(2) Ha with variance 0.033 vs 0.059 Ha^2.

Measured effects reported in the paper

Surprising findings

• In-training data augmentation and group averaging increase gradient variance and can lead to worse energy despite the symmetry, contrary to standard ML expectations.
• Post hoc smoothed canonicalization performs significantly worse than other methods, likely due to second-derivative blowup at boundaries.

Limitations

• Theoretical analysis restricted to first-order updates; second-order (KFAC) extension only numerical.
• Experiments focus on a single architecture (DeepSolid) and three small crystal systems.
• Averaging over subsets (generators) gives inconclusive results and inflated variance for graphene.

Experimental setup

Datasets: graphene 1x1 supercell LiH 2x2x2 supercell bcc-Li 2x2x2 supercell

Benchmarks: graphene LiH bcc-Li

Metrics: energy (Ha) variance of local energy Var[PA/OG] symmetry measure GPU hours

Baselines: DeepSolid OG data augmentation group averaging group averaging with subsampling post hoc canonicalization

Builds on: DeepSolid FermiNet KFAC

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