Mitigating numerical stiffness in least-squares formulations of elliptic PDEs for physics-informed neural networks

Abstract Residual-based physics-informed neural networks (PINNs) combine interior PDE and boundary residuals using mean squared errors. For elliptic boundary value problems, these formulations can show numerical stiffness causing slow training. We study a residual formulation that combines an H 0 −1 interior residual with an H 1/2 boundary residual for nonzero Dirichlet boundary data. For finite-dimensional test spaces in H 1 (Ω), we introduce two finite-dimensional approximations, the subspace-induced negative-order Sobolev seminorm (SINOS) and its dual counterpart (D-SINOS), and quantify their stiffness through a stiffness ratio defined with respect to the H 1 -norm. We establish continuity and coercivity estimates for both formulations and show that, for polynomial spaces of degree n, the stiffness ratios of SINOS and D-SINOS grow at most as O(n 2 ) and O(1), respectively. Numerical experiments recover rates of n 2.00 and n 0.05 , compared with n 6.95 and n 5.68 for MSE and L 2 -based residual losses. PINN experiments for the Poisson equation, a stationary convection-diffusion-reaction problem, and the stationary incompressible Navier-Stokes equations show that SINOS achieves consistently strong convergence and accuracy, while D-SINOS improves over MSE but does not attain the performance of SINOS. These results demonstrate that negative-order Sobolev residuals substantially reduce the stiffness of least-squares PINN formulations. At the same time, the superior PINN performance of SINOS despite the smaller stiffness ratio of D-SINOS demonstrates that stiffness mitigation alone does not determine optimization and approximation quality.