An Introduction to PINNs and Their Convergence

Physics-Informed Neural Networks (PINNs) are a framework for approximating partial differential equation (PDE) solutions by embedding governing physical laws directly into the neural network’s loss function via automatic differentiation. Drawing from a seminar reviewing the 2023 paper by Shin, Zhang, and Karniadakis , we introduce a rigorous mathematical framework to analyze the error estimates and convergence properties of PINNs applied to linear PDEs. We explore continuous residual minimization to establish both a posteriori and a priori error bounds as network capacity increases. Furthermore, we discuss practical discrete residual minimization scenarios, utilizing Bernstein-type discrete norm relations and Rademacher complexity to bridge the gap between empirical and continuous losses. We close by demonstrating that under these conditions, the approximated neural network solution rigorously converges to the true PDE solution.

See Error estimates of residual minimization using neural networks for linear PDEs for details.