Robust Barrier Projection for Learned Dynamics

1. Integration point: use this as an inference-time safety wrapper around a neural policy $u_{\mathrm{nom}}=\pi_\theta(o_t)$, or around the action emitted by a world model or reinforcement-learning agent. Define a differentiable safety function $B(x)$, such as a distance-to-obstacle margin, actuator-temperature margin, maximum latent-state norm margin, or predicted rollout constraint. A KAN, MLP, encoder, or recurrent estimator produces $\hat{x}$ from observations. 2. Pseudocode: compute $\hat{x}=E_\phi(o_t)$; evaluate $B(\hat{x})$ and its Lie derivatives using automatic differentiation through a learned vector field $\hat f_\psi$ and input matrix $\hat g_\psi$; form $\eta(\hat{x})$; solve $u^*=\arg\min_u\|u-u_{\mathrm{nom}}\|_R^2$ subject to $L_{\hat f}^rB+L_{\hat g}L_{\hat f}^{r-1}B\,u-\bar\Delta_r\geq-K_\alpha\eta(\hat{x})$ and actuator bounds. Apply $u^*$ and log the unconstrained and projected actions. 3. Compute versus estimate: $K_\alpha$ is selected offline by checking Hurwitz eigenvalues; Lie derivatives are computed by autodiff; $\bar\Delta_r$ is estimated from held-out transition residuals using a high quantile, then inflated by the observed estimator error and a safety factor. The paper supplies the robust inequality, while the residual bound must be calibrated empirically or with a confidence method. 4. First cheap experiment: train a small neural state-space model and PPO policy on a two-dimensional obstacle-avoidance or cart-pole task with hidden velocity and a hard state constraint. Compare an unconstrained policy, nominal CBF projection, and robust projection. Sweep the safety-bound multiplier $c$ in $\bar\Delta_r=cq_{0.999}(|\widehat{\Delta}_r|)$. The quantitative prediction is a sharp safety transition: for $c\geq1$ and no residual exceeding the bound, constraint violations should be zero up to numerical tolerance; for $c<1$, violations should begin specifically on trajectories where $\Delta_r<-\bar\Delta_r$. Also measure projection magnitude: it should remain near zero far from the boundary and grow approximately linearly with the barrier deficit near activation.