This book chapter introduces the integration of Machine Learning (ML) methods with traditional modeling and simulation approaches in engineering and materials science. As computational problems grow in complexity and scale beyond the capacity of traditional methods, engineers increasingly require interdisciplinary expertise spanning partial differential equations, numerical methods, and machine learning algorithms. While ML has achieved remarkable success in fields such as image recognition and medical diagnostics, its systematic application to engineering problems remains underdeveloped despite the availability of vast datasets from simulations and real-time monitoring. The chapter provides an overview of ML fundamentals, categorizing approaches into supervised, unsupervised, and reinforcement learning, with special emphasis on physics-informed neural networks that combine data-driven techniques with governing physical equations. Key advantages of artificial neural networks for solving PDEs are discussed, including automatic differentiation for higher-order derivatives, reduced meshing requirements, and ability to handle high-dimensional problems. Two primary approaches for solving differential equations using ANNs are presented: collocation methods based on strong-form residual minimization and energy minimization techniques using variational formulations. The book focuses on Deep Neural Network solutions for engineering PDEs, particularly in solid mechanics, while also covering ML-based surrogate modeling and optimization methods, with all concepts supported by practical code implementations.