Envelope-Max Neural Operator

Integrate the operator at the one-step transition used in a robust dynamics model, HJB solver, or uncertainty-aware rollout predictor. Let the action set be a[1:M], either a fixed grid or outputs of a small action proposal network. For each action compute a function-valued branch v_j = Phi_theta(f, a_j) and a scalar running cost c_j = eta_psi(a_j); form q_j = v_j - h*c_j pointwise and return y = max_j(q_j). Pseudocode: for j in actions: v[j]=Phi_theta(f,a[j]); q[j]=v[j]-h*eta_psi(a[j]); y=max(q,dim=action). During training use tau*logsumexp(q/tau) with tau decreasing from 0.1 to 0.01, then evaluate with the hard maximum. To exploit the compactness argument, estimate action coverage empirically: for each minibatch compute the largest nearest-neighbor distance in action space and increase M until branch outputs for neighboring actions differ by less than epsilon. The paper's delta-epsilon condition is the target criterion, while delta and epsilon are estimated from sampled branches. Enforce monotonicity in f when required using nonnegative branch weights or monotone scalar couplings. First cheap test: train a 1D or 2D robust control value-function model on synthetic dynamics with randomly varying drift, comparing an ordinary neural operator and a max-over-actions model at the same branch budget. Measure Bellman residual, worst-case rollout cost, calibration under unseen disturbances, and sensitivity to action-grid refinement. Success is lower worst-case cost and stable refinement as M increases, rather than oscillatory predictions from an unconstrained model.