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ABSTRACT: Partial differential equations (PDEs) are fundamental in modeling complicated systems across various scientific and engineering disciplines. This presentation will introduce deep learning methods aimed at improving the approximation of PDE solutions. I will discuss two core deep learning strategies: 1) Physics-Informed Neural Networks (PINNs), which integrate physical laws into the learning algorithm, and 2) Deep Operator Networks (DeepONet), which learn mappings from PDE parameters to their solutions. The talk will present the Augmented Lagrangian Physics-Informed Neural Networks (AL-PINNs), which adaptively refine the learning process to focus on more challenging regions of the domain. Furthermore, I will feature a graph neural network-based model grounded in DeepONet for simulating time-dependent PDEs on arbitrary grids. Experimental results indicate that the proposed model enhances the prediction of system dynamics beyond the time of training with improved accuracy.