发表机构
University of Europe for Applied Sciences; Technische Universität Chemnitz; quasi digital（欧洲应用科学大学; 开姆尼茨工业大学; quasi digital）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究对比固定权重、梯度归一化和基于规则的自适应PINN训练方法，发现自适应控制虽降低PDE残差但削弱边界条件约束，提出应使用组件级指标而非加权总损失评估PINN训练。
AI 中文摘要
物理信息神经网络（PINN）训练最小化偏微分方程（PDE）、边界条件和初始条件损失的加权组合。由于自适应方法在训练过程中修改这些权重，其加权总损失并不总是直接可比较的。我们在匹配设置下，在热方程基准和一个简化的直拉法（Czochralski）取向热流体问题上，比较了固定权重PINN、梯度归一化PINN（GNPINN）和基于规则的自主控制器（AgenticPINN）。在晶体生长MLP实验中，自适应控制将PDE残差从$10^{-5}$量级降低到$10^{-6}$，而边界条件损失从$10^{-5}$量级增加到$10^{-2}$量级。在热方程基准上，GNPINN实现了最低的相对$L_2$场误差（0.054），而AgenticPINN获得了最小的PDE残差但相对$L_2$误差为1.368。此外，使用修正后的直拉法CFD参数扫描的逐案例留出测试评估了高斯过程代理模型。温度扫描的温度场误差约为6%，而晶体旋转扫描的轴向速度误差约为42%。这些发现表明，自适应控制可以改善方程满足度，同时削弱其他物理约束。因此，PINN训练应使用单独的PDE、边界条件和解误差指标进行评估，而不是仅使用加权总损失。
英文摘要
Physics-informed neural network (PINN) training minimizes a weighted combination of partial differential equation (PDE), boundary-condition, and initial-condition losses. Because adaptive methods modify these weights during training, their weighted total losses are not always directly comparable. We compare fixed-weight PINN, gradient-normalized PINN (GNPINN), and a rule-based adaptive controller (AgenticPINN) under matched settings on a heat-equation benchmark and a simplified Czochralski-oriented thermal-fluid problem. In the crystal-growth MLP experiment, adaptive control reduced the PDE residual from the order of $10^{-5}$ to $10^{-6}$, while the boundary-condition loss increased from the order of $10^{-5}$ to $10^{-2}$. On the heat-equation benchmark, GNPINN achieved the lowest relative $L_2$ field error (0.054), whereas AgenticPINN obtained the smallest PDE residual but a relative $L_2$ error of 1.368. Gaussian-process surrogates were additionally evaluated using case-wise holdout tests on corrected Czochralski CFD parameter sweeps. The temperature-field error for the temperature sweep was approximately 6%, whereas the axial-velocity error for the crystal-rotation sweep was approximately 42%. These findings show that adaptive control can improve equation satisfaction while weakening other physical constraints. PINN training should therefore be evaluated using separate PDE, boundary-condition, and solution-error metrics rather than weighted total loss alone.
Comments7 pages, 4 figures
