phasegrad

Phase-Gradient Duality: backpropagation-free learning in coupled oscillator networks.

Paper (PDF)

In a Kuramoto oscillator network at equilibrium, the physical phase response to weak output clamping equals the gradient of the loss with respect to natural frequencies. This enables a learning rule that requires no separate backward pass — two forward passes of the same physics compute both the prediction and the gradient.

Quick start: verify the theorem

from phasegrad import run_verification

results = run_verification(sizes=[6, 10, 15, 20])
# All cosine similarities = 1.000000

Quick start: train a vowel classifier

from phasegrad import KuramotoNetwork, train, spectral_seed
from phasegrad.kuramoto import make_network
from phasegrad.data import load_hillenbrand

train_data, test_data, info = load_hillenbrand(vowels=['a', 'i'])
net = make_network(n_input=2, n_hidden=5, n_output=2)
spectral_seed(net)  # topology-aware init for sparse layered architectures
history = train(net, train_data, test_data, epochs=200)

Autograd cross-check (requires PyTorch)

from phasegrad.autograd_verify import verify_autograd_table

# Reproduces Table 1 with computationally independent PyTorch verification
results = verify_autograd_table(sizes=[6, 10, 15, 20])

Install with pip install -e ".[autograd]" for PyTorch support.

Installation

pip install -e .

Requires Python 3.10+, NumPy, SciPy. Optional: pip install -e ".[plots]" for matplotlib, pip install -e ".[dev]" for pytest + scikit-learn.

The theorem

For a network of coupled Kuramoto oscillators with dynamics

$$\frac{d\theta_i}{dt} = \omega_i + \sum_j K_{ij} \sin(\theta_j - \theta_i)$$

at a stable equilibrium $\theta^*$, the gradient of a quadratic output loss with respect to natural frequencies can be computed physically:

$$\lim_{\beta \to 0} \frac{\theta_k^\beta - \theta_k^*}{\beta} = -\frac{\partial L}{\partial \omega_k}$$

where $\theta^\beta$ is the equilibrium under weak output clamping with strength $\beta$.

The proof uses the implicit function theorem: the equilibrium defines $\theta^*(\omega)$ implicitly, and the Jacobian of the equilibrium equations (the negative of a coupling-weighted graph Laplacian) mediates the gradient flow.

Key results

Result	Value
Gradient identity (cosine similarity)	1.000000 at all N tested (6–200)
Equilibrium residuals	At or below machine epsilon
Natural freq vs coupling (converged, matched params)	96.0% vs 83.3% (p = 1.8 × 10⁻¹²)
Spectral seeding success rate	100/100 (vs 46/100 random init)

Repository structure

phasegrad/       Library: kuramoto, gradient, training, verification, seeding, autograd_verify
tests/           10 test files (gradient identity, coupling gradient, training, robustness, scaling, statistical, seeding, autograd)
experiments/     All experiment scripts + reproducible JSON results
paper/           LaTeX source, compiled PDF, figures
examples/        Quick-start scripts
data/            Hillenbrand vowel formant dataset

Running the tests

pip install -e ".[dev]"
pytest tests/

Generating paper figures

python paper/generate_figures.py
# Outputs to paper/figures/

Citation

Rashahmadi, M. (2026). Equilibrium Propagation with Natural Frequencies
in Coupled Oscillator Networks. Calibur Labs.

License

MIT