We introduce frequency propagation, a learning algorithm for nonlinear physical networks. In a resistive electrical circuit with variable resistors, an activation current is applied at a set of input nodes at one frequency and an error current is applied at a set of output nodes at another frequency. The voltage response of the circuit to these boundary currents is the superposition of an activation signal and an error signal whose coefficients can be read in different frequencies of the frequency domain. Each conductance is updated proportionally to the product of the two coefficients. The learning rule is local and proved to perform gradient descent on a loss function. We argue that frequency propagation is an instance of a multimechanism learning strategy for physical networks, be it resistive, elastic, or flow networks. Multimechanism learning strategies incorporate at least two physical quantities, potentially governed by independent physical mechanisms, to act as activation and error signals in the training process. Locally available information about these two signals is then used to update the trainable parameters to perform gradient descent. We demonstrate how earlier work implementing learning via chemical signaling in flow networks (Anisetti, Scellier, et al., 2023 ) also falls under the rubric of multimechanism learning.

Recurrent backpropagation and equilibrium propagation are supervised learning algorithms for fixed-point recurrent neural networks, which differ in their second phase. In the first phase, both algorithms converge to a fixed point that corresponds to the configuration where the prediction is made. In the second phase, equilibrium propagation relaxes to another nearby fixed point corresponding to smaller prediction error, whereas recurrent backpropagation uses a side network to compute error derivatives iteratively. In this work, we establish a close connection between these two algorithms. We show that at every moment in the second phase, the temporal derivatives of the neural activities in equilibrium propagation are equal to the error derivatives computed iteratively by recurrent backpropagation in the side network. This work shows that it is not required to have a side network for the computation of error derivatives and supports the hypothesis that in biological neural networks, temporal derivatives of neural activities may code for error signals.