本文探討了物理信息神經網絡 (PINNs) 在求解熱方程中的應用。傳統神經網絡在高效處理剛性問題方面遇到挑戰。為了解決這個問題，本文提出了幾種方法，包括用已知解增強神經網絡，使用離散和連續時間模型的混合方法，以及優化具有固定空間維度的神經網絡。

這些方法的有效性通過兩個例子進行了展示：一個涉及具有可變擴散係數的熱核，另一個則具有空間變係數。結果展示了所提出的方法在準確逼近熱方程解方面的能力，即使在存在剛性項和複雜初始條件的情況下也是如此。

我們討論了潛在的未來研究方向並承認了本研究的局限性。儘管我們的研究集中在熱方程上，但所提出的方法可以擴展到求解在各種領域中遇到的廣泛剛性偏微分方程 (PDEs)。

關鍵詞：PINNs、熱方程、剛性問題、混合建模、偏微分方程、計算物理學、科學計算

This paper explores the application of Physics-Informed Neural Networks (PINNs) \cite{raissi2019physics} in solving the heat equation. Traditional Neural Networks encounter challenges in efficiently handling stiff problems. To address this, the paper proposes several methods, including augmenting Neural Networks with known solutions, using discrete and continuous-time models in a hybrid approach, and optimizing Neural Networks with fixed spatial dimensions.

The effectiveness of these methods is demonstrated through two examples: one involving a heat kernel with a variable diffusion coefficient and another with spatially varying coefficients. The results showcase the capability of the proposed approaches in accurately approximating solutions to the heat equation, even in the presence of stiff terms and complex initial conditions.

We discuss potential future research directions and acknowledge the limitations of our study. While our focus is on the heat equation, the proposed methods can be extended to solve a wide range of stiff partial differential equations (PDEs) encountered in various domains.

Keywords: PINNs, heat equation, stiff problems, hybrid modeling, PDEs, computational physics, scientific computing

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