This paper develops an Optimality-Informed Neural Network (OINN) approach for the energy-optimal, free-final-time powered descent of a lunar lander from any initial position, velocity, and mass within a bounded operating envelope to a fixed landing site with zero terminal velocity. Building on a recent framework that jointly embeds Pontryagin's minimum principle and the Hamilton-Jacobi-Bellman equation for general nonlinear optimal control, the proposed OINN approach specializes that idea to a lunar landing problem with free time of flight and fixed terminal state. Every boundary and transversality condition is hard-encoded into the network architecture by construction, the closed-form Pontryagin-optimal thrust magnitude and direction law is substituted directly rather than learned, and the remaining state, costate, and an auxiliary value-function output are trained against a physics-residual loss formed entirely from the necessary conditions of optimality, with no precomputed optimal trajectories required. A preliminary theoretical analysis is explored, establishing a stochastic-optimization stationarity guarantee for the offline training procedure, an explicit bound translating the achieved training residual into bounds on touchdown position, touchdown velocity, and flight-time error, and a fixed, input-independent onboard computational and memory cost suitable for real-time deployment. Numerical simulations evaluate the trained policy, with no retraining, against an independently solved indirect-method boundary-value problem at six representative initial states spanning the operating envelope and against eighty additional Monte Carlo simulation runs, demonstrating close agreement with the indirect-method solution and consistently small dynamics and transversality residuals throughout the envelope.

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