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.

Yes Yes Yes (Lemma 4.1 and 4.2) Yes (Theorem 3.7 and 5.8) [11,22,30,34] No No No Yes [17,40] Yes Yes No No [5,18,26,42] Yes Yes Yes No Table 1: A table comparison between this work and related prior works mentioned above or below. Remark 1.1. (About the terminology "consistency") In this paper, we choose to adopt this term from the learning theory instead of directly using the more common term "convergence" mainly due to the following reasons: 1. The term "consistency" emphasizes more on the convergence of the population loss L, and we hope to distinguish the main result (Theorem 3.7) in this work from the convergence results of the empirical loss L [5, 18, 26, 42]; 2. We hope to distinguish this work from the existing and forthcoming results that focus on the convergence rate, because the term "consistency", in general, emphasizes more on general results about convergence under weak conditions, while the convergence rates are stronger results under stronger conditions.