Reciprocity can halve what a mechanical network can learn
Abstract
A passive, reciprocal, linear network driven by forces or currents and read at points it drives cannot be trained below an error floor computed from the target beforehand. The usual parameter count for such networks does not charge for Maxwell-Betti reciprocity. We prove that when such an elastic, resistor or flow network has a symmetric response operator and p driven degrees of freedom are also read, its reachable response blocks lie in a subspace of codimension p(p-1)/2, whatever its size and topology, and the floor is the distance to it. On targets with a reachable symmetric part, a second-order optimiser ends within 0.1% of the floor in 142 of 144 simulated runs, and a contrastive rule with a bond-local update direction, within its step budget, in 22 of 24. Reciprocity charges a list of single drive-read tasks only for the pairs it instruments in both directions, and a rank test on the passive network says in advance whether, and on which bonds, odd couplings locally restore the lost directions, at a cost in non-reciprocity that the target bounds from below. Under an imposed-displacement drive, the one most physical-learning hardware uses, reciprocity survives as a weaker law, an exact balance between forward and reverse transmissions once each is weighted by the driving-point compliance at its input, and an inequality on their product, under which no symmetric positive-definite chain meets both targets of a published robotic metamaterial. All nineteen published layouts we tabulated pay nothing; a force-driven layout that asks one pair to respond differently in its two directions will.
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BibTeX
@misc{vu2026reciprocity,
title = {{Reciprocity can halve what a mechanical network can learn}},
author = {Thai-Son Vu and Hoang-Giang Nguyen and Quoc-Bao Nguyen and Sengaloun Keoalounxay and Bao-Viet Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04169},
eprint = {2609.04169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04169},
}
RIS
TY - UNPB
TI - Reciprocity can halve what a mechanical network can learn
AU - Thai-Son Vu
AU - Hoang-Giang Nguyen
AU - Quoc-Bao Nguyen
AU - Sengaloun Keoalounxay
AU - Bao-Viet Tran
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.04169
UR - https://arxiv.org/abs/2609.04169
AB - A passive, reciprocal, linear network driven by forces or currents and read at points it drives cannot be trained below an error floor computed from the target beforehand. The usual parameter count for such networks does not charge for Maxwell-Betti reciprocity. We prove that when such an elastic, resistor or flow network has a symmetric response operator and p driven degrees of freedom are also read, its reachable response blocks lie in a subspace of codimension p(p-1)/2, whatever its size and topology, and the floor is the distance to it. On targets with a reachable symmetric part, a second-order optimiser ends within 0.1% of the floor in 142 of 144 simulated runs, and a contrastive rule with a bond-local update direction, within its step budget, in 22 of 24. Reciprocity charges a list of single drive-read tasks only for the pairs it instruments in both directions, and a rank test on the passive network says in advance whether, and on which bonds, odd couplings locally restore the lost directions, at a cost in non-reciprocity that the target bounds from below. Under an imposed-displacement drive, the one most physical-learning hardware uses, reciprocity survives as a weaker law, an exact balance between forward and reverse transmissions once each is weighted by the driving-point compliance at its input, and an inequality on their product, under which no symmetric positive-definite chain meets both targets of a published robotic metamaterial. All nineteen published layouts we tabulated pay nothing; a force-driven layout that asks one pair to respond differently in its two directions will.
ER -