gfn 2.5.0

Manifold: Geometric Intelligence via Symplectic Geodesic Flows | GFN: Geometric Flow Networks

GFN | Manifold

Geodesic Flow Networks: Geometric Intelligence via Symplectic Flows

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What's New in v2.5.0 (Riemannian Stability)

• Riemannian Optimization: RiemannianAdam optimizer ensures parameter updates respect manifold geometry
• Adaptive Curvature Gating: Learnable valve mechanism enables inertial coasting when optimal
• Zero-Force Inductive Bias: Architectural enforcement of E(0) = 0 for perfect state preservation
• Velocity Normalization: Automatic stabilization preserving memory direction while controlling magnitude

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Overview

GFN (Geodesic Flow Networks), publicly known as Manifold, reformulates sequence modeling as geodesic flow on a learned Riemannian manifold. Instead of attention matrices (O(N²)) or fixed-state compression, GFN models the latent state as a physical particle governed by symplectic integrators, enabling O(1) memory with infinite horizon stability.

Core Innovation: State transitions follow Einstein's geodesic equation with learned curvature, ensuring information conservation via Hamiltonian dynamics.

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Installation

pip install gfn

Or install from source:

git clone https://github.com/Manifold-Laboratory/manifold.git
cd manifold
pip install -e "."

Requirements: Python 3.10+, PyTorch 2.0+, CUDA (optional)

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Quick Start

from src.model import Manifold
from src.optim import RiemannianAdam

# Model
model = Manifold(
    vocab_size=50257,
    dim=512,
    depth=12,
    heads=8,
    integrator_type='leapfrog'
).cuda()

# Optimizer (REQUIRED: standard Adam will fail)
optimizer = RiemannianAdam(
    model.parameters(),
    lr=1e-4,
    retraction='normalize',
    max_norm=10.0
)

# Training
for x, y in dataloader:
    optimizer.zero_grad()
    logits, _, _ = model(x)
    loss = criterion(logits.view(-1, vocab_size), y.view(-1))
    loss.backward()
    torch.nn.utils.clip_grad_norm_(model.parameters(), 0.05)
    optimizer.step()

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Verified Performance

Binary Parity Task (Cumulative XOR)

Challenge: Predict cumulative XOR over arbitrarily long sequences (requires infinite-precision state tracking)

Training Performance

Model	Steps to Convergence	Final Loss	Training Time	Final Accuracy
GFN	728	0.00494	47 min (L=20)	99.9%
MicroGPT	4,000	0.0254	1m 27s (L=20)	99.0%

GFN achieves lower loss (0.00494 vs 0.0254) and higher accuracy despite longer training time

Zero-Shot Generalization Results

Trained on L=20 only, tested on sequences up to L=1000 (50× longer):

Figure: Left plot shows perfect accuracy generalization. Right plot demonstrates O(1) memory scaling (flat line) vs theoretical O(N) baseline.

Detailed Results:

Test Length	GFN Accuracy	GFN VRAM	MicroGPT Accuracy	MicroGPT VRAM
20 (seen)	100.0%	28.3 MB	98.0%	44.7 MB
50	100.0%	28.4 MB	49.5% (collapsed)	73.7 MB
100	100.0%	28.6 MB	50.1% (random)	156.0 MB
200	100.0%	29.0 MB	51.8% (random)	420.9 MB
400	100.0%	29.8 MB	49.9% (random)	1,363 MB (1.3GB)
500	100.0%	30.4 MB	49.1% (random)	2,040 MB (2.0GB)
1000	100.0%	32.1 MB	50.7% (random)	7,488 MB (7.3GB)
10000	100.0%	60.3 MB	FAILED (OOM)	> 8GB

Key Findings:

• ✅ Perfect Generalization: 100% accuracy on all lengths including L=10,000 (500× training length)
• ✅ O(1) Memory Verified: VRAM growth of only 32 MB (113%) from L=20→10,000
• ✅ Transformer Collapse: MicroGPT accuracy drops to random chance (50%) immediately at L=50
• ✅ Memory Advantage: At L=1000, GFN uses 32MB vs Transformer's 7.5GB (234× less memory)

Full benchmark results and plots available in tests/benchmarks/results/gfn_superiority/

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Core Architecture

Geodesic Equation

d²x/dτ² + Γ(v, x) = F_ext(token)

• x: Position in semantic manifold
• v: Velocity (momentum/memory)
• Γ: Christoffel symbols (learned curvature)
• F: Input token force

Symplectic Integration (Leapfrog)

# Half-step velocity
v_half = v + 0.5 * dt * (F - Γ(v, x))

# Full-step position
x_next = x + dt * v_half

# Half-step velocity finalization
v_next = v_half + 0.5 * dt * (F - Γ(v_half, x_next))

# Stabilization
v_next = v_next / (||v_next|| + ε)

Properties:

• Time-reversible
• Volume-preserving (det(J) = 1)
• Energy-conserving (|ΔH| ≈ O(dt²))

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Comparison with Baselines

Architecture	Memory (Inference)	Compute (per token)	Gradient Stability	Verified
Transformer	O(N) KV cache	O(N·d) attention	Good	—
LSTM/GRU	O(1)	O(d²) gates	Poor	—
Mamba (SSM)	O(1)	O(d²) state update	Medium	—
GFN	O(1) state	O(d²·R) Christoffel	Excellent	✓

Where N = sequence length, d = hidden dim, R = Christoffel rank (typically 16-32)

Note: GFN's O(d²·R) per-token cost is comparable to LSTMs/Mamba. For training full sequences, all architectures are O(N·...) in sequence length.

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Documentation

• SCIENTIFIC_PAPER.md - Complete research paper with mathematical derivations
• API.md - Python API reference
• TRAINING.md - Training guide and best practices
• ARCHITECTURE.md - System design and components
• BENCHMARKS.md - Empirical performance validation
• PHYSICS.md - Mathematical foundations

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Use Cases

• Long-Context Reasoning: Process sequences >10K tokens with constant memory
• Algorithmic Tasks: Perfect extrapolation on logical reasoning (XOR, sorting, arithmetic)
• Edge Deployment: Run large models on memory-constrained devices (<4GB RAM)
• Scientific Computing: Model systems requiring conservation laws (physics simulations)

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Repository Structure

/
├── src/                # Core Implementation
│   ├── model.py        # Main Manifold Architecture
│   ├── geometry.py     # Christoffel Symbols & Integrators
│   ├── layers.py       # M-Layer (Manifold Layer)
│   ├── embeddings.py   # Functional Embeddings
│   └── optim.py        # RiemannianAdam Optimizer
├── docs/               # Technical Documentation
│   ├── SCIENTIFIC_PAPER.md
│   ├── API.md
│   ├── TRAINING.md
│   └── BENCHMARKS.md
├── tests/              # Verification Suite
│   └── benchmarks/     # Reproducible Benchmarks
└── LICENSE             # Apache 2.0

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Development Status

Version 2.5.0 is production-ready for research and experimentation.

Verified:

• ✅ O(1) memory scaling (empirically confirmed)
• ✅ Perfect generalization on Parity task
• ✅ Stable training with RiemannianAdam
• ✅ Symplectic gradient flow

In Development:

• CUDA kernel acceleration (10-50× speedup expected)
• Mixed precision training (FP16/BF16)
• Language modeling benchmarks (WikiText)
• Mixture of Manifolds (MoM) architecture

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Citation

If you use GFN in your research, please cite:

@software{gfn2026,
  title={GFN: Geodesic Flow Networks},
  author={Stürtz Joaquín},
  year={2026},
  version={2.5.0},
  url={https://github.com/Manifold-Laboratory/manifold},
  license={Apache-2.0}
}

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Contributing

See CONTRIBUTING.md for guidelines.

Quick Links:

• Report Issues
• Request Features
• View Roadmap

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License

Apache License 2.0 - See LICENSE for details.

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Manifold Laboratory
Geometric intelligence through physical principles.