广义Falkner--Skan问题的硬约束物理信息神经网络与自适应区域残差平衡
Hard-Constrained Physics--Informed Neural Network with Adaptive Regional Residual Balancing for the Generalized Falkner--Skan Problem
- Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出硬约束物理信息神经网络结合自适应区域残差平衡求解广义Falkner--Skan问题，精确嵌入边界条件，显著降低残差和壁面剪切力误差。
AI中文摘要：
我们提出了一种用于广义Falkner--Skan边值问题的物理信息神经求解器，该求解器将精确的边界容许试验表示与自适应区域残差平衡（ARRB）相结合。三个规定的边界条件被解析地嵌入，消除了边界条件惩罚项，同时允许有限域流函数值和壁面剪切力由控制方程确定。残差被划分为壁面区域、中间区域和尾部区域，并使用区域长度分数从区域均方误差重构物理全局残差。带保护机制的指数移动平均逆梯度系数在Adam优化过程中自适应地平衡区域贡献。使用确定性的800点残差监视器进行检查点选择，随后进行两个确定性的L-BFGS阶段，最小化物理全局残差。该方法在Blasius、有利压力梯度Falkner--Skan和Pohlhausen案例上进行了测试。对于(β₀,β₁)=(0.75,0.50)，预测的壁面剪切力为f''(0)=0.8997161394，而有限域参考值为0.8997168085，绝对误差为6.691×10⁻⁷。残差MSE为6.357×10⁻⁹，而f'和f''的相对L₂误差分别为2.848×10⁻⁶和4.188×10⁻⁵。匹配的单种子消融研究表明，相对于全局残差训练，ARRB将残差MSE降低了52.81%，相对于等区域加权降低了50.24%。壁面剪切力误差分别降低了77.30%和72.37%。这些结果支持ARRB作为一种面向精度的残差调节策略，而多种子实验仍然是量化优化变异性所必需的。
英文摘要：
We present a physics--informed neural solver for the generalized Falkner--Skan boundary-value problem that combines an exact boundary--admissible trial representation with adaptive regional residual balancing (ARRB). The three prescribed boundary conditions are embedded analytically, eliminating boundary-condition penalty terms while allowing the finite--domain streamfunction value and wall shear to be determined by the governing equation. The residual is divided into wall, middle, and tail regions, and a physical global residual is reconstructed from regional mean-squared errors using region-length fractions. Safeguarded exponential--moving--average inverse--gradient coefficients adaptively balance the regional contributions during Adam optimization. A deterministic 800-point residual monitor is used for checkpoint selection, followed by two deterministic L--BFGS stages minimizing the physical global residual. The method is tested on the Blasius, favourable-pressure--gradient Falkner--Skan, and Pohlhausen cases. For \((β_0,β_1)=(0.75,0.50)\), the predicted wall shear is \(f''(0)=0.8997161394\), compared with the finite-domain reference \(0.8997168085\), giving an absolute error of \(6.691\times10^{-7}\). The residual MSE is \(6.357\times10^{-9}\), while the relative \(L_2\) errors in \(f'\) and \(f''\) are \(2.848\times10^{-6}\) and \(4.188\times10^{-5}\). A matched single--seed ablation shows that ARRB reduces residual MSE by 52.81\% relative to global--residual training and by 50.24\% relative to equal-regional weighting. Wall--shear errors are reduced by 77.30\% and 72.37\%, respectively. These results support ARRB as an accuracy-oriented residual-conditioning strategy, while multi--seed experiments remain necessary to quantify optimization variability.
