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In this paper, an unmanned aerial vehicle (UAV)-assisted wireless network is considered in which a battery-constrained UAV is assumed to move towards energy-constrained ground nodes to receive status updates about their observed processes. The UAV's flight trajectory and scheduling of status updates are jointly optimized with the objective of minimizing the normalized weighted sum of Age of Information (NWAoI) values for different physical processes at the UAV. The problem is first formulated as a mixed-integer program. Then, for a given scheduling policy, a convex optimization-based solution is proposed to derive the UAV's optimal flight trajectory and time instants on updates. However, finding the optimal scheduling policy is challenging due to the combinatorial nature of the formulated problem. Therefore, to complement the proposed convex optimization-based solution, a finite-horizon Markov decision process (MDP) is used to find the optimal scheduling policy. Since the state space of the MDP is extremely large, a novel neural combinatorial-based deep reinforcement learning (NCRL) algorithm using deep Q-network (DQN) is proposed to obtain the optimal policy. However, for large-scale scenarios with numerous nodes, the DQN architecture cannot efficiently learn the optimal scheduling policy anymore. Motivated by this, a long short-term memory (LSTM)-based autoencoder is proposed to map the state space to a fixed-size vector representation in such large-scale scenarios. A lower bound on the minimum NWAoI is analytically derived which provides system design guidelines on the appropriate choice of importance weights for different nodes. The numerical results also demonstrate that the proposed NCRL approach can significantly improve the achievable NWAoI per process compared to the baseline policies, such as weight-based and discretized state DQN policies.