Gaussian-Process Stability-Frontier Expansion

Use this as a stability-validation and active-data-collection layer around a recurrent neural network, state-space model, neural ODE discretization, or learned controller. First train the model normally and construct a small positive Lyapunov network V_phi with V_phi(x_star)=0 and V_phi(x)>0. Enforce an initial decrease margin on a small ball by minimizing relu(alpha-(V_phi(x)-V_phi(f_theta(x)))) over sampled states. Initialize a query set with states inside that ball and states on the first outer shell. For every queried state, run the model for T steps, calculate y(x)=min_t[V_phi(x_t)-V_phi(x_{t+1})], and optionally record a binary success label based on terminal error and maximum excursion. Fit an RBF or Matern Gaussian process to these observations. Generate candidate states by sampling random directions and radii, then retain candidates whose Lyapunov value lies in the current shell. Compute the posterior mean and variance and select the candidate maximizing sigma(x)[tau-|mu(x)|]_+. Run its rollout, append the result, and refit the GP. Expand c_k only when q shell samples have failure rate below delta. Compare against uniform and random shell sampling using the same rollout budget. The first experiment should use a tanh RNN on a two-dimensional synthetic stable-system prediction task, T=50, 100 total rollouts, and 10,000 held-out initial states. The predicted signature is that GP queries cluster near mu=0, locate the transition at least three times more efficiently than uniform sampling, and produce a monotonically expanding envelope until the observed shell failure rate reaches delta.