This chapter explores machine learning-based solutions for partial differential equations (PDEs), focusing on physics-informed neural networks (PINNs) and deep energy methods (DEM) within the framework of scientific machine learning. Unlike traditional computational methods that require large datasets, these approaches leverage known physical models and constitutive relationships to solve PDEs in small-data regimes. The chapter presents the mathematical foundations of feed-forward neural networks and their application to PDE solving through two primary strategies: the collocation-based PINN approach, which minimizes PDE residuals at discrete points, and the variational DEM approach, which minimizes total system energy at quadrature points. The numerical examples include elliptic (Poisson equation, linear elasticity, hyperelasticity), parabolic (heat equation, Navier-Stokes), hyperbolic (wave equation), and coupled problems (phase-field fracture, Allen-Cahn equation). Comparative analysis with traditional methods—finite element method (FEM), meshfree methods, and isogeometric analysis (IGA)—reveals that while conventional approaches solve linear systems with convergence guarantees, machine learning methods tackle nonconvex optimization problems. Despite lacking convergence proofs, ML-based approaches offer distinct advantages: natural handling of high-dimensional problems, built-in frameworks for uncertainty quantification and inverse analysis, mesh-free formulation, and superior performance on ill-posed problems where traditional methods struggle. The chapter provides detailed implementations, benchmark comparisons, and practical guidance for applying these emerging techniques to engineering problems.