PDE-Constrained Learning-Driven Autonomy with Belief, Policy, and Actuation Uncertainty

Autonomous systems in distributed physical environments must transform incomplete and stale observations into actions that remain useful under transition uncertainty and imperfect execution. We introduce PDE-Constrained Learning-Driven Autonomy (LDA), a closed-loop framework that separates observations, beliefs, candidate-conditioned predictive distributions, intended controls, executed actions, and feedback. LDA combines masks and measurement age with modality-specific random Fourier features, reliability-weighted kernel means, and temporal attention; conditions predictions on context, control history, and candidate plans; and evaluates plans with actuation-aware mean–CVaR control and soft safety tightening. Screening experiments on hourly traffic and irregular groundwater records, together with model-conditional traffic-control simulations, show that no representation dominates point prediction, calibration, risk discrimination, safety, and policy quality. At 30% traffic missingness, imputation gives the lowest belief-screening RMSE (0.0474), Mask+Age is closest to nominal 90% coverage (0.909), and reliability attention gives the strongest exceedance AUROC (0.980). Ridge gives the lowest traffic prediction RMSE, whereas last-state pooling gives stronger policy-oriented behavior. In simulation, Mean–CVaR MPC lowers mean and tail loss relative to expected-cost MPC, while executed-action modelling exposes command–execution trade-offs. Persistent sampling, transition, actuation, approximation, and numerical errors lead to an error-dependent neighborhood rather than exact optimality. The evidence supports modular beliefs and explicit actuator models.