用更精确的边界约束提升物理信息神经网络的求解精度与稳定性
Trace Regularity PINNs: Enforcing $\mathrm{H}^{\frac{1}{2}}(\partial Ω)$ for Boundary Data
- 在边界上使用 $H^{1/2}$ 空间范数，更准确地刻画边界数据
- 数值实验显示，对高频边界条件问题，可比传统PINN提升1~3个有效数字
- 计算高效且收敛更稳定，适合高精度科学计算场景
我们提出一种增强型物理信息神经网络（TRPINN），在边界损失中引入与 $H^1(Ω)$ 对应的正确迹空间 $H^{1/2}(oundary Ω)$ 的 Sobolev-Slobodeckij 范数。通过仅计算半范数的理论必要部分降低计算成本，并避免离散化中的分母求值以提升收敛稳定性。引入精确的 $H^{1/2}(oundary Ω)$ 范数后，证明逼近解在 $H^1(Ω)$ 意义下收敛至真实解；通过神经切线核（NTK）分析表明，TRPINN 可比标准 PINN 收敛更快。在具有高度振荡狄利克雷边界条件的拉普拉斯方程数值实验中，出现标准 PINN 失败而 TRPINN 成功的情况，性能提升达一至三个数量级。
原文摘要 · Abstract (English)
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.
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