Sakana trains 1,000-layer networks with no backpropagation.

Jeffrey Seely and Julian Gould published PC-ALM on September 14, adding dual neurons as Lagrange multipliers to each layer so its local recurrence acts as a PI feedback controller (Sakana AI). That converts backprop's forward-then-backward phase lock into layer-local dynamics that still distribute credit through 1,000-layer residual MLPs, nearly matching backprop where standard predictive coding decays away. In the linear limit the dual neurons converge to exact backprop credit signals using only local computation. The practical case is neuromorphic hardware, where simulating dynamical systems is cheaper than a GPU.