arXiv:2608.30778cs.LGnlin.AO2026-08

物理学习中,互反性决定梯度流是否转向,影响学习路径效率。

Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning

  • 通过构建保质层状传输网络,研究物理学习中的响应对称性机制。
  • 非负模式反馈产生互反闭环响应,等效于重加权梯度流,而边界反对称项引入旋转运动。
  • 揭示了守恒、互反与非互反在物理学习中的不同作用,适合研究物理启发学习的学者。

物理学习使可训练材料或网络利用自身物理响应传递误差信号,减少对独立反向计算的需求。我们探究何种因素决定系统遵循传统梯度下降,还是沿本质不同的学习轨迹演化。模型为定向分层传输网络,每个节点均分配固定流量,确保学习过程保持正性和总质量守恒。在此框架下,守恒仅约束允许的学习方向。在所研究的匹配响应类中,伴随匹配使物理输出响应呈现对称形式;非负模式反馈生成互反闭环响应,导致重加权梯度流。引入反对称边界分量则使闭环响应具有旋转特性:学习路径可转弯,同时误差仍在即时减小。转弯并非自动有益,其有限步效应由局部曲率决定,累积效应还依赖步长选择及路径上新状态的访问情况。数值一致性检验重现了精确响应结构,预测了新网络族中局部效应符号,并展示了轨迹漂移可能抵消局部优势。这些结果清晰分离了守恒、互反性与非互反性在物理学习中的作用。

原文摘要 · Abstract (English)

Physical learning lets a trainable material or network use its own physical response to carry error signals, reducing the need for a separately programmed backward computation. We ask what determines whether such a system follows conventional gradient descent or evolves along a genuinely different learning trajectory. Our canonical model is a directed layered transport network in which every node redistributes a fixed amount of flow, so learning preserves positivity and total mass. In this model, conservation constrains only the allowable learning directions. Within the matched response class studied here, adjoint matching gives the physical output response a symmetric form. Non-negative mode-wise feedback then produces a reciprocal closed-loop response and a reweighted gradient flow. Adding an antisymmetric boundary component makes the closed-loop response rotational: the learning path can turn while the error driving that update still decreases at that moment. Turning is not automatically beneficial. Its finite-step effect is set by local curvature, and its accumulated effect also depends on step selection and on the new states visited along the path. Numerical consistency checks reproduce the exact response structure, predict the sign of the local effect across new network families, and show how trajectory drift can negate a local advantage. These results separate the roles of conservation, reciprocity, and nonreciprocity in physical learning.

物理学习梯度流互反性旋转学习

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