Physics-Informed Neural Networks: Theory and Applications

Methods that seek to employ machine learning algorithms for solving engineering problems have gained increased interest. Physics-informed neural networks (PINNs) are among the earliest approaches, which attempt to employ the universal approximation property of artificial neural networks to represent the solution field. In this framework, solving the original differential equation can be seen as an optimization problem, where we seek to minimize the residual or some energy functional. We present the main concepts and implementation steps for PINNs, including an overview of the basics for defining and training an artificial neural network model. These methods are applied in several numerical examples of forward and inverse problems, including the Poisson equation, Helmholtz equation, linear elasticity, and hyperelasticity.