Recent advances in deep learning have led to efficient solutions of initial and boundary value problems governed by partial differential equations defined in a domain \(\Omega \subset \mathbb {R}^d\) for \(d\ge 2\) . These approaches recast the problem as optimization problems, where the objective is to learn suitable neural network parameters that approximate the solution. In this expository article, we focus on neural network-based solvers namely, Deep Ritz Method, Physics-Informed Neural Network, and Variational Physics-Informed Neural Network for the Poisson problem with Dirichlet boundary condition. We discuss a comprehensive error analysis for each of these methods and conduct numerical experiments to assess their computational efficiency.
