Value-Gradient Trajectory Collocation

1. Exact integration point: use this sampler between optimizer updates for any PINN, neural ODE residual model, or value/world-model network. Maintain a replay buffer containing a fixed fraction of uniform domain points and a steerable buffer containing states from current forward rollouts. At a sampled (x,t), evaluate V_theta and its input gradient with autodiff. For box-bounded controls and disturbances, first implement the cheap affine case: calculate g_u=(grad_x V_theta)^T f_u and g_d=(grad_x V_theta)^T f_d, then choose each component of u* and d* at the corresponding bound according to the sign of its coefficient; for nonlinear dynamics use 3-5 projected-gradient steps or a small random candidate set. 2. Pseudocode: initialize uniform points; repeat: compute p=grad_x(V_theta(x,t)); solve u_star=argmin_u p^T f(x,u,d) and d_star=argmax_d p^T f(x,u,d); set x_next=x+Delta_t f(x,u_star,d_star)+sqrt(2 sigma^2 Delta_t) epsilon, where epsilon is standard Gaussian noise; append x_next and intermediate states to the trajectory buffer; draw a batch with alpha trajectory points and 1-alpha uniform points; minimize mean(R_theta^2)+lambda_B mean(boundary_residual^2); periodically refresh the buffer because the policy induced by V_theta changes. Clip states to the PDE domain and reject or reflect invalid points. 3. Computed from the paper's mechanism: the feedback direction from the current value gradient, forward trajectory steering, and injected exploration. Estimated empirically: the occupation density and suitable alpha, Delta_t, and sigma. Use detached gradients for sampling so the sampler does not add second-order terms to the training loss. 4. First cheap experiment: solve a 2-D Burgers or linear-advection PINN and a 2-D double-integrator reachability HJ equation. Compare uniform Sobol sampling, residual-based adaptive sampling, and this method at equal network evaluations. Use a 4-layer tanh MLP, 10,000 collocation points, alpha=0.7, and noise levels sigma in {0, 0.01, 0.05}. The quantitative prediction is that the steered sampler lowers mean squared residual in the top 10 percent largest-error states by at least 25 percent at equal sample count and produces a sharper concentration of points within a prescribed distance of the learned zero level set. Sweep the control gain or steering strength: excessive steering should cause mode collapse and worse global error, while uniform mixing should prevent it. Measure the collapse transition by the effective sample size of occupied spatial bins; the useful regime should have high boundary coverage and lower residual, whereas sigma=0 should show visibly lower coverage and greater run-to-run variance.