基於強制物理資訊神經網路之熱分析：結合遷移學習與自適應取樣

Abstract

隨著積體電路封裝與功率密度的快速提升，精確且高效的熱傳導數值模擬成為現代半導體設計中的一大挑戰。傳統有限元素法（FEM）雖能提供高精度，但在多維度、高解析度問題下計算成本高昂。物理資訊神經網路（PINN）作為一種新興自動微分模型，能夠直接將偏微分方程融入網路訓練，無需網格化生成即可求解熱傳導問題，然而其訓練過程常因殘差不平衡與採樣效率低下而導致收斂緩慢、精度不足。 本論文提出一種「粗網路到細網路」的混合 PINN 解法： 1. 粗網路預訓練：訓練一個小型(隱藏層較少) PINN，以快速擬合初步解形態。 2. 遷移學習初始化：將粗網路的參數部分轉移至更高容量的細網路(隱藏層較多)中，並添加微量噪聲以提高模型魯棒性。 3. 二階段自適應訓練：在 Phase A 凝聚頂層參數，快速優化結構層；Phase B 全參數解凍結合改進的自適應採樣算法（結合誤差與梯度權重），動態引入新樣本以聚焦高誤差區域。 4. 混合精度與自動釋放：全程採用 CUDA AMP 自動混合精度，並在自適應採樣後主動調用減少 GPU 記憶體占用。 5. 以二維二次分佈熱源問題為例，與解析解比較結果顯示，在相同訓練步數下，細網路透過遷移學習與自適應採樣，可將相對 L₂ 誤差從 1e-2 降至 1e-4。 6. 相較於單一 PINN，總訓練時間縮短約 40%；收斂曲線更平滑穩定，實驗結果支持本方法在高精度熱模擬任務中的可行性與優勢。 本研究所提出的混合 PINN 框架兼具效率與精度，並具備良好的模組化設計，可延伸至其他高階偏微分方程或是其他物理問題求解。

With the rapid increase in power density and packaging complexity of integrated circuits, accurate yet efficient thermal simulation has become a critical challenge in modern semiconductor design. While the Finite Element Method (FEM) achieves high accuracy, its computational cost escalates sharply for multidimensional, high-resolution problems. Physics-Informed Neural Networks (PINN) offer a mesh-free alternative by embedding partial differential equations directly into the network’s loss function via automatic differentiation. However, PINN often suffer from slow convergence and limited accuracy due to unbalanced residual terms and inefficient sampling strategies. This thesis presents a coarse-to-fine hybrid PINN framework to accelerate convergence and enhance solution accuracy: 1. Coarse PINN Pretraining: Two lightweight PINNs are trained separately to approximate the homogeneous and particular solutions, capturing the dominant features of the heat transfer field. 2. Transfer Learning Initialization: Selected layers of the coarse networks are transferred into a higher-capacity fine PINN, with small Gaussian perturbations added for robustness. 3. Two-Phase Adaptive Training: Phase A: Freeze shared layers and optimize only the newly initialized layers to rapidly align solution scales. Phase B: Unfreeze all parameters and employ an improved adaptive sampling algorithm—combining K-Means clustering with a weighted error gradient criterion to dynamically introduce new collocation points in regions of high residual. 4. Mixed-Precision and Memory Management: Leverage CUDA Automatic Mixed Precision throughout training after each adaptive sampling step to reduce GPU memory footprint. 5. For a 2D quadratic heat source problem, comparisons against the analytic solution demonstrate that, at equal training iterations: The fine PINN with transfer learning and adaptive sampling reduces L₂ error from 1e-2 to 1e-4. 6. Overall training time is decreased by ~40%, and GPU memory usage is cut by ~30% compared to a baseline PINN. The convergence curves exhibit markedly smoother and more stable behavior. The hybrid PINN framework proposed in this study combines both efficiency and accuracy, and its modular design allows it to be extended to other higher-order partial differential equations or to solve different physical problems.