This minisymposium explores the evolving landscape of neural network-based methods for solving partial differential equations (PDEs), focusing on the interplay between data-driven techniques and classical numerical analysis. While approaches such as physics-informed neural networks (PINNs) [1] and neural operators [2] have gained attention for combining machine learning with physics-based modeling, practical deployments have revealed limitations—including slow convergence, sensitivity to problem settings, and limited adaptability across PDE classes. To address these challenges, recent research has focused on integrating rigorous mathematical insights into neural network solvers. Contributions in this session will showcase a range of methods, including—but not limited to—approaches that incorporate adaptive sampling, structure-preserving architectures, and training strategies to enhance stability and scalability. Attention will also be given to algorithmic innovations that reduce computational cost, enable real-time inference, and improve performance in low-data regimes—for example, in tasks such as sensor placement and inverse problem solving. Combining theoretical developments and application-driven case studies, this session brings together researchers from applied mathematics and computational science and engineering to explore how blending numerical PDE methods with neural architectures can push the frontiers of scientific computing.