# The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

arXiv: https://arxiv.org/abs/2604.10272 · HTML: https://mlsanity.com/paper/2604.10272/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 11 Apr 2026.

## Builds on (most cited first, 18 of 18)

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- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Learning at the Speed of Physics: Equilibrium Propagation on Oscillator Ising Machines](https://mlsanity.com/paper/2510.12934/) (2025)

## Abstract (arXiv)

We prove that in a coupled Kuramoto oscillator network at stable equilibrium, the physical phase displacement under weak output nudging is the gradient of the loss with respect to natural frequencies, with equality as the nudging strength beta tends to zero. Prior oscillator equilibrium propagation work explicitly set aside natural frequency as a learnable parameter; we show that on sparse layered architectures, frequency learning outperforms coupling-weight learning among converged seeds (96.0% vs. 83.3% at matched parameter counts, p = 1.8e-12). The approximately 50% convergence failure rate under random initialization is a loss-landscape property, not a gradient error; topology-aware spectral seeding eliminates it in all settings tested (46/100 to 100/100 seeds on the primary task; 50/50 on a second task, K-only training, and a larger architecture).