Prime-Attentive Neural Networks (PANNs)

A framework for interpretable decomposition of chaotic dynamical systems into fundamental "prime" components.

Overview

This repository implements Prime-Attentive Neural Networks (PANNs) – a novel neural architecture that decomposes chaotic systems into irreducible components called dynamical primes. Each prime represents a fundamental mode of the system with its own eigenvalue (growth/oscillation rate), eigenfunction (spatial pattern), and topological signature.

Key features:

• Mathematically grounded in spectral theory of dynamical systems
• Interpretable by design with symbolic equation extraction
• Topologically consistent using persistent homology regularization
• Validated on canonical chaotic systems (Lorenz, logistic map)

Installation

# Clone repository
git clone https://github.com/yourusername/pann.git
cd pann

# Install dependencies
pip install -r requirements.txt

# Optional: Install PyTorch with CUDA support
pip install torch torchvision --index-url https://download.pytorch.org/whl/cu118

Quick Start

Run Lorenz Experiment

python main.py --experiment lorenz --num_epochs 200 --visualize

Run Logistic Map Experiments

python main.py --experiment logistic --r_values 3.5 3.7 3.9 4.0 --visualize

Train Custom Model

from pann_architecture import PrimeAttentiveNN
from training import PANNTrainer

# Create model
model = PrimeAttentiveNN(
    input_dim=3,      # Lorenz system has 3 dimensions
    latent_dim=64,    # Size of latent space
    num_primes=16,    # Number of dynamical primes
    hidden_dim=128
)

# Create trainer and train
trainer = PANNTrainer(model, learning_rate=1e-3)
trainer.train(train_loader, val_loader, num_epochs=200)

Architecture

The PANN consists of four main components:

1. Prime Embedding Layer: Maps state space to structured latent space
2. Factorization Module: Decomposes latent representation into prime coefficients
3. Prime Evolution Module: Models dynamics in prime space
4. Symbolic Decoder: Extracts human-readable equations from prime representation

Mathematical Foundations

Dynamical Primes

A dynamical prime is defined as a triple:

• Invariant submanifold: Minimal invariant set
• Ergodic measure: Natural measure on the submanifold
• Koopman eigenfunction: Eigenfunction of the Koopman operator with eigenvalue λ

Prime Continuum

The set of all dynamical primes forms a prime continuum – a measurable space indexing the irreducible components of the system.

Key Theorems

1. Prime Decomposition: Chaotic systems admit decomposition into primes
2. Spectral Correspondence: Primes correspond to Koopman eigenmodes
3. Topological Consistency: Each prime has associated persistent homology barcode

Examples

Logistic Map Bifurcation

# Analyze Feigenbaum scaling
from experiments import analyze_feigenbaum_scaling
analysis = analyze_feigenbaum_scaling(logistic_results)
print(f"Estimated δ: {analysis['feigenbaum_estimate']:.3f}")
# Expected: ≈ 4.669

Symbolic Equation Extraction

# Get interpretable equations
outputs = model(x, return_symbolic=True)
for eq in outputs['equations'][:3]:
    print(eq)
# Example output:
# dx0/dt = 0.142*sin(1.832*x0) + 0.083*exp(0.512*x1)
# dx1/dt = -0.234*x0 + 0.167*x0*x1

Topological Validation

from topological_regularization import analyze_topological_invariants

# Analyze topology of learned attractor
topology = analyze_topological_invariants(trajectory, prime_coeffs)
print(f"Betti numbers: {topology['betti_numbers']}")
# Expected for Lorenz: β₀=1, β₁=2 (double loop)

Applications

1. Climate Science: Decompose ENSO into fundamental modes
2. Plasma Physics: Identify coherent structures in turbulence
3. Neuroscience: Decompose neural dynamics into functional modules
4. Financial Systems: Identify fundamental market regimes

Research References

Theoretical Foundations

• Koopman operator theory
• Ergodic decomposition theorems
• Persistent homology in dynamical systems
• Spectral theory of chaotic systems

Related Work

• SINDy: Sparse Identification of Nonlinear Dynamics
• Neural ODEs: Continuous-depth models
• Reservoir Computing: Echo state networks
• Dynamic Mode Decomposition (DMD)