A Control-Theoretic Approximation to Predictive Coding Dynamics

Backpropagation remains the dominant method for training deep neural networks, but its reliance on non-local computations has motivated the search for biologically plausible alternatives. Two prominent frameworks, predictive coding (PC) and deep feedback control (DFC), offer distinct approaches to local learning, yet their relationship remains unclear. Here, we show that DFC arises as a local approximation to predictive coding dynamics near fixed points, with exact equivalence in shallow linear networks and controlled deviations arising with depth, nonlinearity, and distance from equilibrium. Building on this connection, we introduce Approx PC, a practical algorithm that replaces layer-wise error propagation in PC with a centralized feedback controller operating over a sequence of intermediate subgoals. Empirically, Approx PC recovers solutions consistent with predictive coding, while converging faster and achieving improved accuracy in our experiments. Our results unify energy-based and control-theoretic perspectives on learning and suggest a computationally efficient approach to training predictive coding networks through control-inspired dynamics.