Geometric Invariants of Neural Network Training: Certifying Quasi-Symplectic Dynamics

M. Axel Giebelhaus
CHISI Research
[email protected]

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Abstract

This repository contains the complete LaTeX source, experimental data, and figures for the paper "Geometric Invariants of Neural Network Training: Certifying Quasi-Symplectic Dynamics".

We characterize a quasi-symplectic operating regime in neural network training by formalizing and validating three geometric probes that certify structure at small step sizes:

1. Symmetric round-trip test (ΔNEGdt): measures reversibility
2. Paired-subspace Ω-invariance error (εΩ): tests local symplecticity
3. Tail-median energy drift (D): quantifies stability

Key Findings

• GroupNorm and calibrate-then-freeze BatchNorm satisfy all probes at small Δt
• Standard BatchNorm (train mode) systematically fails, breaking local reversibility and Ω-invariance
• Causal intervention: Sweeping BN running-statistics momentum produces a clean dose-response curve
• Mass scaling: SimpleCNN follows expected dt_zero ∝ √M relationship; deeper architectures deviate

These results provide operational probes that certify near-reversibility, local symplecticity, and low drift in neural network optimization—characterizing the anti-dissipative dynamics discovered in prior work.

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Repository Structure

.
├── paper/                    # LaTeX source files
│   ├── main.tex             # Main document
│   ├── macros.tex           # Custom macros and notation
│   ├── Makefile             # Build system
│   ├── sections/            # Individual paper sections
│   ├── tables/              # Generated tables (LaTeX + CSV)
│   └── biblio/              # Bibliography (references.bib)
├── figures/                 # All paper figures
│   └── figs/                # Generated plots and diagrams
├── data/                    # Experimental results
│   └── computed/            # Processed data files (CSV)
└── README.md                # This file

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Building the Paper

Requirements

• TeX Live 2020+ or equivalent LaTeX distribution
• pdflatex, bibtex, and make

Quick Build

cd paper/
make

This will compile main.tex and produce main.pdf.

Clean Build

cd paper/
make clean
make

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Experimental Data

All experimental results are provided in data/computed/:

• summary_*.csv: Per-architecture summary statistics
• merged_curves_*.csv: Training curves for different configurations
• accept_rtvol_*.csv: Acceptance rate analysis
• phase3_*.csv: BatchNorm momentum sweep results

Key Configurations

• SimpleCNN: Simple convolutional baseline (with/without BN)
• ResNet-18: Standard ResNet-18 architecture with:
  • BatchNorm (training mode)
  • BatchNorm (freeze after epoch 200)
  • GroupNorm (32 groups)

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Figures

All figures are provided as publication-ready PDFs/PNGs in figures/figs/:

• fig_abs_drift_*.png: Absolute energy drift traces
• fig_mass_slope_*.png: Mass scaling analysis
• fig_accept_probes*.png: Acceptance rate visualizations
• fig_omega_per_seed.png: Ω-invariance per random seed
• phase3_*.png: Phase 3 intervention results

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Citation

If you use this work, please cite:

@misc{giebelhaus2025geometric,
  title={Geometric Invariants of Neural Network Training: Certifying Quasi-Symplectic Dynamics},
  author={Giebelhaus, M. Axel},
  year={2025},
  publisher={Zenodo},
  doi={10.5281/zenodo.XXXXXXX},
  note={DOI will be assigned upon publication}
}

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Related Work

This paper builds on prior research investigating Hamiltonian structure in neural network optimization:

• [Paper A: Characterization work on anti-dissipative dynamics]
• [Paper C: Hamiltonian Memory - symplectic approaches to inference]

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License

This work is licensed under the MIT License - see LICENSE for details.

The LaTeX source, experimental data, and figures are provided for reproducibility and further research.

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Contact

M. Axel Giebelhaus
Independent Researcher, CHISI Research
Email: [email protected]
Location: Beech Mountain, North Carolina, USA

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Acknowledgments

This research was conducted independently with computational resources generously provided through academic GPU compute grants.

Special thanks to the open-source scientific computing community for developing the tools that made this work possible: PyTorch, NumPy, Matplotlib, and the entire Python scientific stack.

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Reproducibility

All experimental configurations, hyperparameters, and random seeds are documented in the paper's Methods section. The processed data in data/computed/ allows reproduction of all figures and tables without re-running experiments.

For questions about experimental methodology or data provenance, please contact the author.