发表机构
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京雁栖湖应用数学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出自适应物理信息神经网络框架，通过梯度范数损失加权、残差配点和顺序优化，将Blasius边界层壁面剪切误差降至1.896e-5，较先前结果提升21.5倍。
AI 中文摘要
物理信息神经网络（PINNs）提供了一种求解微分方程的无网格方法，但其性能在很大程度上依赖于损失权重、配点布置和优化策略。本研究针对Blasius边界层方程开发了一个自适应PINN框架，采用基于梯度范数的自适应损失权重、非均匀且基于残差的配点，以及顺序Adam-L-BFGS优化。在使用架构$[1,100,100,1]$的代表性运行中，模型预测$f''(0)=0.3320762918$，而高精度基准值为$0.332057336215$，绝对误差为$1.896\ imes10^{-5}$。最终加权损失为$6.789\ imes10^{-8}$，预测的流函数、速度和剪切应力剖面与独立的数值边值解高度一致。一项单独的全训练架构研究表明，在两个隐藏层的模型中，壁面剪切误差在四种测试架构中最小，为$1.629\ imes10^{-6}$，而最深的网络实现了最小的加权目标，但壁面剪切误差显著更大。与先前报道的PINN值$f''(0)=0.33165$相比，代表性运行将壁面剪切误差减少了约21.5倍。结果表明，结合自适应训练框架能够为Blasius问题实现高精度，且仅靠加权损失不足以识别物理上最准确的PINN。由于自适应组件是联合应用的，它们的个体贡献无法从当前结果中分离出来，需要受控消融研究进行单独评估。
英文摘要
Physics-informed neural networks (PINNs) provide a mesh-free approach for solving differential equations, but their performance can depend strongly on loss weighting, collocation placement, and optimization strategy. This study develops an adaptive PINN framework for the Blasius boundary-layer equation using gradient-norm-based adaptive loss weighting, nonuniform and residual-based collocation, and sequential Adam--L-BFGS optimization. In the representative run using the architecture $[1,100,100,1]$, the model predicts $f''(0)=0.3320762918$, compared with the high-accuracy benchmark $0.332057336215$, giving an absolute error of $1.896\times10^{-5}$. The final weighted loss is $6.789\times10^{-8}$, and the predicted stream-function, velocity, and shear profiles agree closely with an independent numerical boundary-value solution. A separate full-training architecture study shows that the two-hidden-layer model achieves the smallest wall-shear error among the four tested architectures, $1.629\times10^{-6}$, whereas the deepest network attains the smallest weighted objective but a substantially larger wall-shear error. Compared with the previously reported PINN value $f''(0)=0.33165$, the representative run reduces the wall-shear error by approximately a factor of $21.5$. The results show that the combined adaptive training framework can achieve high accuracy for the Blasius problem and that weighted loss alone is insufficient for identifying the most physically accurate PINN. Because the adaptive components are applied jointly, their individual contributions cannot be isolated from the present results and would require a controlled ablation study for separate assessment.
Comments25 pages
