发表机构
City University of Hong Kong; Hong Kong Baptist University; The University of Sydney（香港城市大学; 香港浸会大学; 悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究基于边界强制PINNs求解椭圆型狄利克雷边值问题，通过结合近似理论与统计学习论证推导先验误差界，明确对\(\rho\)的依赖，确定边界自适应PINNs子类，数值实验验证，还得出相关新的VC维界与近似界。
AI 中文摘要
受有界域\(\Omega\subseteq\mathbb{R}^d\)上随机过程平均逃逸时间（MET）\(\tau:\Omega\to\mathbb{R}\)数值计算的启发，我们使用边界强制物理信息神经网络（PINNs）研究椭圆型狄利克雷边值问题（BVPs），其中通过将网络输出与预定义的到边界近似\(\rho\)相乘来精确施加狄利克雷条件。结合对整流二次单元（ReQU）和双曲正切（tanh）网络的近似理论和统计学习论证，我们推导了先验误差界，明确了对\(\rho\)的依赖性。特别地，我们表明仅精确的边界强制不足以获得\(H^2(\Omega)\)误差界，并且一个充分且基本必要的条件是\(\rho\)是一种光滑的距离近似且“一阶归一化”，如在arXiv:2104.08426 [math.NA]中构建的那样。我们由此确定这种“边界自适应”PINNs子类为解决狄利克雷BVPs的合适神经网络假设。数值实验支持该理论，表明\(\rho\)的适当选择可提高精度和收敛性，而选择不当的距离函数会严重降低解的质量。我们的证明还为ReQU和tanh网络高阶导数的假设空间产生了新的VC维界，以及高阶Sobolev范数下浅ReQU网络的新近似界，所有这些都具有重要的独立研究价值。
英文摘要
Motivated by the numerical computation of the Mean Escape Time (MET) $τ:Ω\to\mathbb{R}$ of a stochastic process from a bounded domain $Ω\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $ρ$. Combining approximation-theoretic and statistical-learning arguments for Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we derive a priori error bounds that make explicit the dependence on $ρ$. In particular, we show that exact boundary enforcement alone is not enough for $H^2(Ω)$ error bounds, and that a sufficient and essentially necessary condition is for $ρ$ to be a smooth distance approximation $\textit{normalized to first order}$, of the kind constructed in arXiv:2104.08426 [math.NA]. We thereby identify this subclass of $\textit{boundary-adapted}$ PINNs as the appropriate neural network ansatz for solving Dirichlet BVPs. Numerical experiments support the theory, showing that appropriate choices of $ρ$ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.
