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