Oriented Simplicial Networks

Category: Phase 3 Core - Geometric Deep Learning Status: Skeleton Implementation Dependencies:

categorical-composition

,

persistent-homology

Overview

Implements directional simplicial neural networks (Dir-SNNs) with asymmetric message passing operators, E(n)-equivariance constraints, and persistent homology tracking for topological feature learning.

Capabilities

• Directional Message Passing: Asymmetric operators respecting simplex orientation
• E(n)-Equivariance: Rotation/translation invariant representations
• Persistent Homology: Track topological features during training
• Simplicial Attention: Higher-order attention mechanisms on simplicial complexes

Core Components

1. Simplicial Complex Builder (

   simplicial_complex.jl

   )
   • Construct oriented simplicial complexes from data
   • Boundary operator computation
   • Coboundary and Laplacian matrices

2. Dir-SNN Layers (

   dirsnn_layers.jl

   )
   • Asymmetric message passing on simplices
   • E(n)-equivariant convolutions
   • Higher-order pooling operators

3. Persistent Homology Tracker (

   persistent_homology.jl

   )
   • Compute persistence diagrams during forward pass
   • Track birth/death of topological features
   • Bottleneck/Wasserstein distance metrics

4. Training Loop (

   train_dirsnn.jl

   )
   • Integration with Flux.jl
   • Topologically-aware loss functions
   • Gradient flow on simplicial manifolds

Integration Points

• Input from:

  sheaf-theoretic-coordination

  (sheaf structures on simplicial complexes)
• Output to:

  categorical-composition

  (functorial network composition)
• Coordinates with:

  formal-verification-ai

  (verify topological invariants)

Usage

using OrientedSimplicialNetworks

# Build simplicial complex from point cloud
complex = SimplicialComplex(points, max_dimension=2)

# Create Dir-SNN model
model = DirSNN([
    SimplicialConv(in_features=3, out_features=16, dimension=0),
    SimplicialConv(in_features=16, out_features=32, dimension=1),
    SimplicialPooling(dimension=1)
])

# Train with persistent homology tracking
train!(model, complex, labels; track_topology=true)

References

• Bodnar et al. "Weisfeiler and Lehman Go Cellular" (NeurIPS 2021)
• Hajij et al. "Topological Deep Learning" (Nature 2023)
• Carlsson "Topology and Data" (AMS 2009)

Implementation Status

• Core data structures
• Basic message passing
• Full E(n)-equivariance
• Persistent homology integration
• Benchmark suite