Consistent PINNs for Higher‐Order Elliptic PDEs

ABSTRACT In this paper, we develop a consistent physics‐informed neural networks (CPINNs) framework for higher‐order elliptic PDEs, with emphasis on the biharmonic equation under nonhomogeneous Dirichlet boundary conditions. Unlike standard PINNs, whose loss functions are often inconsistent with the natural stability norms associated with higher‐order differential operators and may therefore result in unstable or nonconvergent approximations, the proposed CPINNs formulation is grounded in a norm‐equivalent discrete representation of the underlying variational problem. This structure‐preserving design guarantees convergence in the natural stability norm . Specifically, (i) we introduce a consistent loss functional , (ii) consistency proofs and norm equivalence between the continuous and discrete formulations, (iii) a priori error analysis, and (iv) convergence rates under Besov regularity assumptions using optimal recovery theory. Numerical experiments validate the theoretical predictions and demonstrate the superior performance of the CPINNs formulation over traditional loss designs.