Trace Regularity PINNs: Enforcing $\mathrm{H}^{\frac{1}{2}}(\partial Ω)$ for Boundary Data
Doyoon Kim, Junbin Song
TL;DR
This work introduces TRPINN, a physics-informed neural network that enforces boundary data in the Sobolev–Slobodeckij space $\mathrm{H}^{1/2}(\partial\Omega)$ for divergence-form elliptic PDEs. By discretizing only the essential portion of the $\mathrm{H}^{1/2}$ semi-norm and avoiding problematic denominators, TRPINN achieves boundary regularity consistent with theory while maintaining comparable computational cost to standard PINNs. Theoretical results show that zero loss implies convergence in $\mathrm{H}^1(\Omega)$, and NTK analysis indicates faster convergence rates relative to conventional PINNs, especially for high-frequency boundary data. Numerical experiments on Laplace problems with oscillatory and sharply peaked boundary data demonstrate substantial accuracy gains and robustness to weight choices, validating the practical impact of enforcing $\mathrm{H}^{1/2}(\partial\Omega)$ boundary regularity in PINN frameworks.
Abstract
We propose an enhanced physics-informed neural network (PINN), the Trace Regularity Physics-Informed Neural Network (TRPINN), which enforces the boundary loss in the Sobolev-Slobodeckij norm $H^{1/2}(\partial Ω)$, the correct trace space associated with $H^1(Ω)$. We reduce computational cost by computing only the theoretically essential portion of the semi-norm and enhance convergence stability by avoiding denominator evaluations in the discretization. By incorporating the exact $H^{1/2}(\partial Ω)$ norm, we show that the approximation converges to the true solution in the $H^{1}(Ω)$ sense, and, through Neural Tangent Kernel (NTK) analysis, we demonstrate that TRPINN can converge faster than standard PINNs. Numerical experiments on the Laplace equation with highly oscillatory Dirichlet boundary conditions exhibit cases where TRPINN succeeds even when standard PINNs fail, and show performance improvements of one to three decimal digits.