Energy Function Choice for Training Nonlinear Resistive Neural Networks with Equilibrium Propagation

Equilibrium Propagation (EqProp) is an alternative to backpropagation for minimizing an energy function. Existing EqProp implementations for nonlinear resistive networks typically rely on a pseudo-power formulation, defined as the integral of the local current-voltage characteristic. However, pseudo-power is only applicable to two-terminal components and cannot be extended to dependent or multi-terminal elements, limiting its conceptual consistency across the circuit. This work proposes using the true electrical power, defined as I • V , as the governing energy function for analog resistive networks. Theoretically, the true electrical power satisfies the equilibrium condition ∂P/∂V = 0 and aligns with Kirchhoff's laws across both linear and nonlinear devices. Ngspice and PySpice simulations of the XOR task, averaged over 100 randomly initialized resistance values, show that the true-power formulation achieves stable convergence to approximately -7dB error, compared to -4dB for pseudo-power. When controlled-source or diode contributions were excluded, both formulations failed to learn and diverged toward +10dB. These results demonstrate that EqProp learning in analog hardware requires an energy function that accounts for all circuit components; in addition, the true electrical power provides a physically consistent formulation with lower computational overhead than pseudo-power.