Recently, physics-informed neural networks (PINNs) have gained attention in the scientific community for their potential to solve partial differential equations (PDEs). However, they face challenges related to resource efficiency and slow convergence. Adaptive sampling methods, which prioritize collocation points with high residuals, improve both efficiency and accuracy. However, these methods often neglect points with medium or low residuals, which can affect stability as the complexity of the model increases. In this paper, we investigate this limitation and show that high residual-based approaches require stricter learning rate bounds to ensure stability. To address this, we propose a Langevin dynamics-based Adaptive Sampling (LAS) framework that is robust to various learning rates and model complexities. Our experiments demonstrate that the proposed method outperforms existing approaches in terms of relative $L^{2}$ error, and stability across a range of environments, including high-dimensional PDEs where Monte Carlo integration-based methods typically suffer from instability.

For machine type communication with random access (RA) protocol, finding optimal policy using deep reinforcement learning (DRL) is being actively investigated for various quality of service requirements. In particular, it was shown that throughput-based reward function in DRL can derive policies that outperform conventional exponential backoff (EB)-based algorithms in terms of throughput and fairness. However, age of information (AoI), which is a measure of the freshness of data, was not addressed in the process of training the DRL agent and only used as a measure of fairness. It has been theoretically proven that, even in the simplest queuing system, maximizing throughput does not guarantee minimizing AoI. In this paper, we proposed a novel DRL scheme to directly optimize AoI in a slotted ALOHA RA channel. By taking into account urgency and packet age as extra local information for a reward, AoI could be improved while preserving the throughput. Numerical simulations showed that the proposed DRL approach could achieve an improvement of 22.04% in AoI performance, with a marginal throughput loss of around 3.95%, compared to the existing throughput-based DRL method.