Trace Regularity PINNs: Enforcing H^frac12(partial Omega) for Boundary Data

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 H1/2(partial Omega), the correct trace space associated with H1(Omega). 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 H1/2(partial Omega) norm, we show that the approximation converges to the true solution in the H1(Omega) 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.