Kernel-Zubov Trust Region for Neural Rollouts

Apply this as a wrapper around a learned simulator, neural ODE integrator, model-based RL rollout, or latent diffusion sampler. Collect a replay buffer of states and short transitions, preferably including both successful and failed trajectories. Fit a kernel estimator \(u_N(x)=b+\sum_j\alpha_j k(x,x_j)\) with a linear-radial product kernel such as \(k(x,y)=\langle x,y\rangle\psi(\|x-y\|^2)\), plus an explicit constant column. For each transition \((x_i,x'_i)\), solve regularized least squares for \(u_N(x_i)-\gamma_i u_N(x'_i)=0\), where \(\gamma_i=\exp[-\Delta t q(x_i)]\). At inference, predict \(x'=F_\theta(x)\), evaluate \(\Delta v=v_N(x')-v_N(x)\), and accept the transition only if \(\Delta v\le\varepsilon\); otherwise shorten the step by interpolation or reduce action magnitude. The paper supplies the kernel invariant construction and sample-dependent error mechanism. Estimate the threshold \(c\), tolerance \(\varepsilon\), and exponent \(\beta\) on validation trajectories. First test a learned two-dimensional Duffing-oscillator or pendulum model with 100-step rollouts, comparing unconstrained and Zubov-gated prediction. The prediction is a sharp increase in basin-exit probability outside \(\Omega_c\), while in-DOA accepted steps satisfy the inequality in at least 95 percent of cases. Increasing samples from \(N\) to \(4N\) should reduce held-out invariance residual approximately by \(4^{-\beta}\); fit the log-residual versus log-sample-size slope and require it to be negative.