Consistency of Physics-Informed Neural Networks for Second-Order Elliptic Equations

The physics-informed neural networks (PINNs) are widely applied in solving differential equations. However, few studies have discussed their consistency. In this paper, we consider the consistency of PINNs when applied to secondorder elliptic equations with Dirichlet boundary conditions. We first provide the necessary and sufficient condition for the consistency of the physics-informed kernel gradient flow algorithm. And then, as a direct corollary, when the neural network is sufficiently wide, we derive a necessary and sufficient condition for the consistency of PINNs based on the neural tangent kernel theory. Additionally, we provide non-asymptotic loss bounds for physics-informed kernel gradient flow and PINN under suitable stronger assumptions. Finally, these results inspire us to construct a notable pathological example in which the PINN method is inconsistent.