Rolling-Horizon Port-Hamiltonian Optimizer

1. Integration point: modify the optimizer update for neural-network parameters, not the forward pass. Flatten each parameter block into \(\theta\), maintain a same-shaped velocity \(v\), and choose \(B\) as either the identity, a per-layer scalar gate, or a low-rank random projector. Compute the ordinary minibatch gradient \(g\) before applying the update. 2. Pseudocode: g = grad(loss, theta); a = (1-h*Gamma)*v - h*g; solve (h*B.T@B + rho/h*I) u = -B.T@a; v = a + h*B@u; theta = theta + h*v. For diagonal or block-diagonal \(B\), solve elementwise; for a rank-r projector use Woodbury, so the control overhead is \(O(r)\) per block. Clip \(h\Gamma\) to below one and use \(\rho>0\) for numerical regularization. 3. Mathematics versus estimates: the rolling-horizon minimizer and energy identity are analytic; \(g\) is the minibatch gradient, while \(\Gamma\) can be fixed initially and later estimated from the ratio of successive velocity norms. The loss linearization is the only approximation, so optionally recompute the candidate loss for two controls and backtrack if actual loss rises. 4. First experiment: train a 3-layer MLP and a small ResNet on MNIST or CIFAR-10 using identical batch sizes and FLOP budgets, comparing SGD with momentum, AdamW, and this optimizer for matched parameter-update counts. Test identity \(B\), layerwise diagonal \(B\), and rank-8 \(B\). Measure training loss at equal FLOPs, gradient/velocity norm spikes, final validation accuracy, and loss increases after updates. Success means faster loss descent or fewer instability spikes than momentum at equal compute; failure is no improvement after tuning \(h,\rho,\Gamma\) on a held-out seed set.