相位即梯度：用振荡器网络实现频率学习，精度超耦合权重学习。
The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks
- 相位差在弱扰动下等于损失对固有频率的梯度。
- 频率学习在相同参数量下准确率达96.0%，优于耦合权重学习的83.3%。
- 谱初始化可消除50%收敛失败问题，适合复杂网络训练。
我们证明，在弱输出扰动下，稳定平衡状态的耦合Kuramoto振荡器网络中，物理相位差即为损失函数对固有频率的梯度，且在扰动强度β趋近于零时等号成立。以往振荡器平衡传播研究将固有频率视为不可学习参数；本文表明，在稀疏分层架构中，频率学习在收敛种子上的表现优于耦合权重学习（参数量匹配时准确率分别为96.0%与83.3%，p = 1.8e-12）。随机初始化下约50%的收敛失败是损失景观特性所致，非梯度误差；采用拓扑感知谱初始化可在所有测试场景中彻底消除该问题（主任务从46/100提升至100/100，次任务50/50，仅训练频率且更大架构下也成功）。
原文摘要 · Abstract (English)
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).
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