| Abstract: |
| We study the formulation of Physics-Informed Neural Networks (PINNs) for elliptic boundary value problems from the perspective of Sobolev space theory. Standard PINN approaches enforce boundary conditions using discrete $L_2(\partial\Omega)$ penalties, which are not consistent with the trace space of $H^1(\Omega)$. To address this mismatch, we introduce Trace Regularity Physics-Informed Neural Networks (TRPINNs), in which boundary data are enforced in the Sobolev-Slobodeckij space $H^{1/2}(\partial\Omega)$. We develop a computationally efficient approximation of the corresponding semi-norm that retains its essential structure while avoiding numerical instabilities. We show that this formulation yields convergence in the $H^1(\Omega)$ norm and provide numerical evidence demonstrating improved performance, particularly for problems with highly oscillatory boundary data. |
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