Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning

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.

Untrainable elements determine what physical learning remembers

Physical learning rules such as equilibrium propagation (EP), coupled learning (CL), and adjoint coupled learning (AL) train resistive networks through local measurements. The learned function is decided by where on the solution manifold training lands. Two properties could decide it, and they have not been separated: the circuit's invariance under rescaling every conductance, and the rule's conservation of the mass K = (1/2) sum_e kappa_e^2. We separate them. When every element is trainable, all three vector fields are homogeneous in the conductances, so the initialization scale is provably inert. An element the rule does not adjust breaks that homogeneity whatever its constitutive law. Across twenty topologies the learned function moves with the initialization scale by a median of twelve percent with fixed rectifiers and eight with fixed linear resistors, against 3e-8 when every element is trainable; a single fixed rectifier produces the whole effect. The conservation law is not what protects the function: AL, which we prove dissipates the mass at exactly twice its own loss, remembers its initialization as strongly as the rules that conserve it, and the memory survives in runs where K is conserved to 1e-4. Raising the fixed-element count from one to eight multiplies the conservation drift by five thousand and leaves the memory unchanged, while the all-trainable circuit under AL drifts comparably and remembers nothing. What the rule's conservation structure does control is solution quality: at matched training loss AL is worse than EP and CL in four of six small circuits, by a median of three to seven percent, though the ordering is not stable across checkpoints and does not reproduce at fifty nodes. Physical learning therefore carries two independent inductive biases, one belonging to the circuit and one to the rule, and only the first is a memory of how the device was built.

Perturbative Contrastive Physical Learning

Responses to perturbations are key to understanding physical systems. The ability to contrast such responses by comparing how a system reacts under slightly different conditions provides a mechanism for learning. Here, we introduce Perturbative Contrastive Physical Learning (PCPL), a general framework in which learning emerges from measurable contrasts between physical states produced by controlled changes to inputs, boundary conditions, parameters, or interpreter functions. PCPL unifies and extends prior approaches: Equilibrium Propagation is rooted in contrasts between free and nudged equilibria in energy-based systems, while Frequency Propagation corresponds to contrasts extracted from sinusoidally driven, frequency-demodulated responses. We show that contrast-driven updates can reflect either local sensitivities or global inverse-problem structure, yet do not require centralized gradient computation. Instead, effective learning geometry emerges implicitly from the system's own physical response, allowing learning behavior to arise without an external processor or explicit backpropagation. We demonstrate PCPL in two platforms: (i) spring networks that update bond stiffness using measured displacements and forces, and (ii) continuous-variable photonic circuits trained via x quadrature measurements and finite-difference estimates of the Jacobian. Both platforms successfully learn classification tasks. We further show that a continuous-variable photonic circuit can be trained to implement analog multiplication, illustrating a step toward more autonomous physical learning systems.