# Module 1

## introduction

Now, engineering and science applications may, as you know, have very different demands. Accuracy and reliability may be paramount(=very important). Large data sets might or might not be available. And we might need to use prior physical knowledge with new data-driven insights. We need to really combine physical modeling with what data can tell us.

The computational paradigm change over centuries:

• The experiment based model such as Newton’s Laws of Motion, we just observe carefully and distill the pricinple from thousands of experiments.
• The theory based model. It describe the physical world with a great deal of predictive power, accuary, and generalizability.
• The computational modeling and simulation. Based on the ideas that we use the computer to help us broaden the theory based model.
• The data-driven model. Using the machine learning method which gives us powerful ways of rebuilding models together with making predictions.

# Module 2：常微分方程

they are equations that describe how things vary in time.

they are equation that describe how certain things vary in space.

they can be understood as a very deep limit of a recurrent neural network.

……

## 计算机求解常微分方程

• $f(a)$
• 矩形
• 近似面积三角形

## 代码

### FE vs RK4

RK4 也并非一直都比 FE 更加的精确，看下面的这个例子：

(ps: 实际运算的时候 matlab 和 python 结果有些不同，matlab 更加精确一些，图像看起来也更好看。但图像变换的趋势和结论是一样的)

Ctwo

2021-01-28

2021-02-23