Trace Regularity PINNs: Enforcing $\mathrm{H}^{\frac{1}{2}}(\partial Ω)$ for Boundary Data
This work addresses a specific bottleneck in PINNs for solving PDEs with oscillatory boundary conditions, offering an incremental improvement over standard methods.
The authors tackled the problem of enforcing correct boundary conditions in physics-informed neural networks (PINNs) by proposing TRPINN, which uses the Sobolev-Slobodeckij norm H^{1/2} for boundary loss, leading to improved convergence and performance gains of one to three decimal digits in numerical experiments.
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.