偏微分方程在各種不同的領域皆有許多應用，因此如何解微分方程成為一
大重要領域。近數十年來，仰仗於電腦科技的快速發展，從前只存在於理論中
的數值方法因此而發揚光大，深度學習就是其中一項。在本文中我們將同時
使用深度學習與數值分析領域的工具來解經典的微分方程，柏松方程(Poisson
equation)。在微分方程領域，解柏松方程經常會使用到的技巧便是使用格林函
數(Green’s function) ，一旦掌握了與方程對應的格林函數，便可輕易地解決同
種類的問題，於是問題便轉移到了如何找到對應的格林函數。
在本文中，我們將使用人工神經網路來學習並近似目標函數，其中主要的
核心概念是來自由Raissi 所提出的物理信息神經網絡(physics-informed neural
network)，並透過近似狄拉克δ函數或利用其本身的性質來繞過原本單純使
用物理信息神經網絡無法通過的障礙，並展示其方法的可行性。

Partial differential equations have many applications in various fields, so how
to solve differential equations has become an important field. In recent decades,
relying on the rapid development of computer technology, some numerical methods
have been developed, and deep learning is one of them. In this article we will use
tools from the fields of deep learning and numerical analysis to solve the classic
differential equation, Poisson equation. In the field of differential equations, a
technique often used to solve Poisson’s equation is to use Green’s function. Once
you know the Green’s function corresponding to the equation, you can easily solve
the same type of problems, so the problem shifts to how to find the corresponding
Green’s function.
In this article, we’ll use artificial neural networks to learn and approximate
the target function, the main core concepts are derived from the physics-informed
neural networks (PINNs) by Raissi et al. And by approximating the Dirac
delta function or using its own properties, we can bypass obstacles that were
originally insurmountable by simply using physical information neural networks,
and demonstrate the feasibility of our method.

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