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【Hackathon 5th No.56】A Physics-Informed Neural Network to solve 2D steady-state heat equation #692

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1. Fix $T$ display
Gxinhu authored Oct 18, 2023
commit eac31763221d9386c91e6ab36a4c15e6c789f632
8 changes: 4 additions & 4 deletions hackathon/hackathon_5th/20231009_science_56.md
Original file line number Diff line number Diff line change
@@ -15,10 +15,10 @@
热传导是自然界中的常见现象,广泛应用于工程、科学和技术领域。了解和模拟传热过程对于设计和优化热传导设备、材料和系统至关重要。2D 定常传热方程描述了稳态热传导过程,其数学表达形式为:

$$
\frac{\partial^2T}{\partial x^2} + \frac{\partial^2T}{\partial y^2}=0
\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}=0
$$

其中,$T$ 表示温度分布,传统的求解方法涉及使用数值方法如有限元法或有限差分法,这些方法通常需要离散化领域并求解大规模矩阵系统。近年来,基于物理信息的神经网络(Physics-informed neural networks, PINN)逐渐成为求解偏微分方程的新方法。PINN 结合了神经网络的灵活性和对物理约束的建模能力,能够直接在连续领域中解决偏微分方程问题。
其中, $T$ 表示温度分布,传统的求解方法涉及使用数值方法如有限元法或有限差分法,这些方法通常需要离散化领域并求解大规模矩阵系统。近年来,基于物理信息的神经网络(Physics-informed neural networks, PINN)逐渐成为求解偏微分方程的新方法。PINN 结合了神经网络的灵活性和对物理约束的建模能力,能够直接在连续领域中解决偏微分方程问题。

## 1.2 功能目标

@@ -43,13 +43,13 @@ $$
控制方程为:

$$
\frac{\partial^2T}{\partial x^2} + \frac{\partial^2T}{\partial y^2}=0
\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}=0
$$

在以下区域内:

$$
D = \\{(x, y)|-1\leq{x}\leq{+1},-1\leq{y}\leq{+1}\\}
D = \\{(x, y)|-1 \leq x \leq +1,-1 \leq y \leq +1 \\}
$$

具有以下边界条件: