Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning
2026-08-31 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how certain physical learning systems use their natural responses to adjust and learn, instead of relying on separate programmed computations. They focus on a network where the total flow is conserved and learning respects this constraint, which affects the learning directions allowed. Their analysis shows that when feedback is balanced (reciprocal), learning follows a modified gradient path, but adding imbalanced feedback can cause the learning path to rotate without immediately improving error. The authors find that such rotations may or may not help overall, depending on factors like curvature and step choices, highlighting different roles of physical properties in the learning process.
physical learninggradient descentlayered transport networkconservationreciprocitynonreciprocityadjoint matchingclosed-loop feedbacklearning trajectorylocal curvature
Authors
Ruiwu Niu, Xiaowen Bi, Michaël Antonie van Wyk
Abstract
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.