A Unified Framework for the Error Analysis of Physics-Informed Neural Networks

We prove a priori and a posteriori error estimates, also known as the generalization error in the machine learning community, for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. Our results give insight into the potential of neural networks for high dimensional PDEs and into the benefit of encoding constraints directly in the ansatz class. The provided estimates are -- apart from the Poisson equation -- the first results of best-approximation and a posteriori error-control type. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.