Confidence-Tightened Neural Model Predictive Control

Integrate this into a model-predictive-control loop using a neural dynamics model, or into an offline RL rollout filter. Train an ensemble of small MLPs or a probabilistic recurrent model, but use held-out one-step residuals to estimate $\sigma$ rather than treating ensemble spread as a theorem. At each environment step, compute features $\phi_t$ from the model's final hidden layer, update $V_t$ and the residual covariance, and calculate $r_t(\delta)$. Roll out the nominal action sequence for horizon $H$; propagate uncertainty with the measured or conservatively estimated Jacobian norm $L_f$; then replace every constraint $c\le0$ by $c+L_cR_h\le0$. If no feasible sequence exists, shorten the horizon or use a backup controller. For a differentiable training version, add $\kappa\max(0,c+L_cR_h)^2$ to the rollout loss, with $\kappa$ increased only when empirical coverage falls below $1-\delta$. The paper supplies the uniform-confidence and tightening logic; neural residuals, feature choices, and Jacobian estimates are empirical. First test on CartPole or a two-dimensional point-mass system with noisy dynamics, comparing nominal neural MPC, ensemble MPC, and tightened MPC using identical models and action optimizers. The predicted signature is monotone constraint tightening proportional to $r_t\sum_{j=0}^{H-1}L_f^j$, coverage approximately at least $1-\delta$ on held-out correlated trajectories, and a sharp feasibility collapse as $L_f$ approaches one for long horizons. Report whether empirical violations remain below $\delta$ and whether measured uncertainty growth follows the geometric law within 20%.