Physical learning methods train physical networks to perform computational tasks using only local update rules, exploiting the physics of the system to handle the global transfer of information. We provide the first local convergence analysis of three such methods -- Equilibrium Propagation (EP), Coupled Learning (CL), and a new method we call Adjoint Coupled Learning (AL) -- for linear circuits, in the limit of small-nudging for both discrete and continuous time. EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. Assuming the existence of a solution, we identify a coercivity condition, expressed as a rank condition on a matrix built from the network's incidence structure, under which the training loss decays exponentially and the parameters converge to the solution manifold. We show that coercivity can fail by exhibiting a kite circuit in which a symmetry causes the coercivity constant to degenerate on the solution manifold, but prove using Sard's theorem that such degeneracies are non-generic: coercivity holds at every point of the solution manifold for almost every choice of desired output.

翻译：物理学习方法利用系统物理特性处理全局信息传递，仅通过局部更新规则训练物理网络执行计算任务。我们在小扰动极限下（离散与连续时间），首次提供了三种此类方法的局部收敛性分析——平衡传播、耦合学习以及我们提出的新方法伴随耦合学习——针对线性电路。EP与AL在自然损失函数上进行梯度下降，而CL遵循带有三阶修正项的修正动力学。在假设解存在的条件下，我们识别出矫顽性条件（表现为网络关联结构所构建矩阵的秩条件），确保训练损失指数衰减且参数收敛至解流形。通过构造对称性导致矫顽性常数在解流形上退化的风筝电路，我们证明矫顽性可能失效；但运用萨德尔定理证明此类退化是非典型的：对于几乎所有期望输出选择，矫顽性在解流形每一点均成立。