The beauty of mathematics 14: The importance of Mathematical Models

The beauty of mathematics 14: The importance of Mathematical Models

Poster: Wu Jun, Google researcher

[Note: readers who have been paying attention to the beauty of mathematics may have discovered that we are always looking for accurate mathematical models for any problem. To illustrate the importance of the model, I used a whole class in Google China in October. The following content is the abstract of my lecture.]

In all astronomy homes, including gobaini, Galileo, and Newton, I admire tolemi, the author of the earth's mind. Although astronomy originated in ancient Egypt, and in ancient Babylon, people observed the trajectory of the five planets (gold, wood, water, fire, earth, and the moving of the planet in the recent day is faster than that in the distant day. (The trajectory of Venus seen on Earth. readers who have read The Da Vinci Code know that Venus draws a star every four years in the sky .)

However, what really created astronomy and computed the trajectory of many celestial bodies was tolemi, two thousand years ago in the ancient Rome era. Although today we may laugh at the simple mistakes made by torremi, those who really understand torremi's contribution will respect him. Tolemi invented the spherical coordinates and defined the meridian line including the equator and zero longitude line. He proposed the yellow track and invented the radian system.

Of course, his biggest and most controversial invention is the land of mind. Although we know that the Earth is moving around the sun, at that time, from people's observations, it is easy to come to the conclusion that the Earth is the center of the universe. From the perspective of the Earth, the moving trajectory of the planet is irregular. the greatness of tolemi is that it accurately calculates the trajectory of all planetary movements using 40 small circles and large circles. (Tolemi inherits some ideas of bidacath, and he also thinks that circles are the most perfect ry .) The high accuracy of the toremi model surprised all scientists in the future. Even today, with the help of computers, it is difficult for us to solve forty equations of circles that are nested together. Every time I think of it, I sincerely admire torlemi. In the past one thousand five hundred years, people have decided on agricultural projects based on their calculations. However, after one thousand five hundred years, tolemi's cumulative error in the sun's motion is still a week away.

The earth heart said that Zhang Heng's great earthquake in China is actually the earth heart.

Correct the Earth's mind and say that the mistake is not to rely on the model of tolemi's 40 circles and then put a few more circles on it, but to further explore the truth. Gobaini found that if we use the sun as the center to describe the operation of a star, we only need 8-10 circles to calculate the trajectory of a planet. Unfortunately, the correct assumption of cobaini has not produced a better result than tolemi. The model error of cobaini is much greater than that of tollmidi. This is one of the reasons why the Church and those who thought that the doctrine of cobaini was an evil saying. Therefore, if we were to accept it with confidence, we had to describe the planetary movements more accurately.

To fulfill this mission is kepura. In all the top astronomy homes, Kepler has poor qualifications and has made countless low-level mistakes in his life. However, he has two things that no one else has. He inherits a lot from Digu, the most accurate observation data at that time, and his luck. Kepura was lucky to find that the planet's orbit around the sun was actually an elliptical shape, so there was no need to enclose a large circle with multiple small circles, however, you only need to use an ellipse to clearly describe the motion of the star. It's just that the knowledge and level of keper is not enough to explain why the planet's orbit is elliptical. Finally, Newton, a great scientist, explained the problem with gravitation.

The story seems to end here. However, many years later, there was a small wave of waves. Astronomers found that the actual trajectory of Uranus was not quite consistent with that calculated using an elliptical model. Of course, the lazy method is to use a small circle to set up a large circle, but some serious scientists are trying to find the real reason. Adams in Britain and Verrier in France have independently discovered Neptune that attracts Uranus from its orbit.

Before the lecture, I and Google China engineers summarized the following conclusions:
1. A correct mathematical model should be simple in form. (Tolemi's model is obviously too complicated .)
2. A correct model may not be as accurate as an elaborate incorrect model at the beginning. However, if we identify the general direction as correct, we should stick to it. (The beginning is not accurate .)
3. A large amount of accurate data is very important for R & D.
4. the correct model may also be affected by noise, but it may not be accurate. In this case, we should not use a co-occurrence correction method to make up for it, but to find the source of noise, this may lead to major discoveries.

In the research and development of network search, the frequency index (TF/IDF) and page rank (page rank) of the single text frequency/inverse text frequency index mentioned above) they are equivalent to the "elliptical model" in network search. They are easy to understand.