When physics encounters artificial intelligence ...

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This article will introduce physical information neural networks from the perspective of physics to artificial intelligence.

Let's take the first step from here:

We know how the world in physics runs. By using scientific methods, we can propose assumptions, we explain how a specific phenomenon happens, and then design a controllable experiment to confirm or falsify the hypothesis through experimental data.

It can be said that the process of physics and natural evolution is closely related. I remembered that a professor in college started his own report with the following sentence:

Herax said, "Everything is moving." I was convinced of this. But how to exercise?

And this is a question that physics is trying to answer. How does "everything" "move"?

People use a specific equation -well -known differential equation to describe the way of all things. Let's first try to understand what the differential equation is.

1. Physics and differential equations

The "slightly divided" describes one category of things related to the subtraction.

1.1 guide number

The guide number of a function has special significance in physics.

For example, speed represents the number of spaces for space for time. Let's consider the following experiments: an object moves along a one -dimensional line.

In other words, this small blue ball is moving along the X -axis. Usually, we set the initial point to 0. When this small ball is moved, its position will change over time.

So this movement can be described as clearer:

A. From 0 to 5 times, position changes from 0 to 9: Small ball move forward.

B. Moment 5 to 15, the position has not changed: the small ball is still.

C. The time is 15 to 17, and the position is moved from 9 to 3: small balls.

D. The position 17 to 47 is from 3 to 6: The small ball has moved forward again.

When does this small ball change your position is clear and quantitative? Even the location of the location is clear. In fact, this is the information of speed. We use the following expression to describe:

It looks a bit complicated. But in fact, the speed corresponds to the two very similar moments (the limit of the limit of 0 in the expression) is divided by the difference (h) of the two moments, so we use the differential division equation to describe it. In other words, the speed is the instantaneous change in the position after the time interval is returned.

If there is a fierce increase at a specific moment, it means that the number of guides is very large; when the position does not change, the guide number is zero; the position is reduced, and the guide number is negative.

In the above examples, the changes in each position are linear, which means that the speed of the time segment 0-5, 5-15, 15-17, 17-47 is unchanged.

Take T = 0 to T = 5 as an example. During the period of T, the instantaneous changes in the position are 9/5, and the above functions can also be understood in other times.

1.2 solution

The above example is very simple. Let's consider the following example:

Such a trajectory is difficult to model, and there is no analysis method to calculate the number of guidance: you can only calculate in a numerical method -at all time points (1) give the guide. By the way, I think you are completely ready to face the cruel reality:

All the differential equations in the real world are solved with numerical software.

But numerical solutions may need to perform tens of millions of iterations. Not only that, but also a high-level method to solve the differential equation. Like the famous RUNGE-KUTTA method, it is often integrated in very complicated software (such as POGO for the finite element method). These software The

High calculation cost

High economic cost

Need professional knowledge

High time cost (take the finite element method as an example, it often takes several minutes to a few hours)

1.3 Disopular issue

The above problems are not even worst. The sadness is that Ill-POSED will occur. Let me give an example below.

For example, the problem of solving X and Y:

Is it quite simple?

• x+y = 4

• So x+2y = x+y+y = 8

• 4+y = 8

• Get y = 4

• x = 0

The solution of the equation group is (x, y) = (0,4). At this time, this problem is appropriate. Because the solution to meet this condition is the only one, that is, the solution obtained above.

But the following questions are different:

The two equations are actually the same! Although (x, y) = (0,4) can also be used as a solution, (1,3) is also possible. This equation group actually has infinite multiple solutions. This problem is a problem of discomfort. For such a certain problem, there are more than one solution (in fact, there are endless multiple solutions). If it is solved the following problem:

This kind of so -called anti -discrimination problem can be used to derive the speed chart from the displacement diagram, and the above -mentioned anti -discrimination problem can be used in an erosion defect in the material in the ultrasonic detection material. Even the most perfect experimental settings (unlimited multiple sensors), this problem is unsuitable. In other words, information cannot derive a unique and reliable speed distribution graph ( ).

2. Artificial intelligence and neural network

Now talk about artificial intelligence (AI).

This sentence can simply introduce artificial intelligence:

The artificial intelligence algorithm can perform specific tasks without a specific program.

Autonomous cars can also brake when anyone walks before the car without mathematics and clear training, because it has passed the "training" of millions of people.

To be precise, all artificial intelligence algorithms depend on the Loss Function.

This specific function is used to represent the difference between the target value (target, the values ​​of the output) and the output value of the algorithm, and need to be optimized to achieve the minimum value.

If a characteristic house is given, he hopes to predict its cost (a well -known real estate data set problem). This is a regression problem from input space to continuous space.

Source author

If the predicted price given is 130K and the actual value is 160K, then the average absolute value error MAE is defined as:

30K in this problem.

This formula indicates that our model is named , process the input data < /g> < /g> < /g> < /g> < /g> < /g> /G>, and output a predicted value < /g>. In the above examples, < /g>, and < /> g data-mml-node = "mi" transform = "translate (3194.6, 0)"/> .

If there are many houses, < /g> < /g> With < /g> < /g> < g data-mml-node = "mi"/> This series of prediction costs, then the global loss function will probably grow: the loss function depends on a series of parameter W to create a model. The lower the loss function, the better the model is established.

As a result, the loss function will continue to be optimized, and the lower the lower to a certain sense, the better. The parameters will be continuously iterated to minimize the loss function (the minimum local value).

3.AI+Physics = Physical Information Neural Network

If you see here:

You deserve for a while

You may have doubts: What is the relationship between artificial intelligence (neural network) and differential equations (physical)? Before answering this question, we need to understand another concept -regularization.

3.1 regularization

In the previous part, we learned that all machine learning algorithms were ultimately an optimization problem. This means to find the best parameter group of the minimized loss function .

However, there will be a problem. Maybe this solution is the best in the training concentration, but it is not suitable for test sets (overfit). At this time, the best value is only the best area.

Explain some further:

Assuming that the initial model is < /g> , < g data-mml-node = "mi"/> two parameters are generated, you look for solution in the following space:

Source author

The parameters obtained < g data-mml-node = "msub"> < /g> < /g> , The loss value of the loss function in the training centralized application is close to 0. If this parameter combination is used to test set (a series of new data), the loss function becomes very large. It means that the definition of the loss function is not accurate. In fact, in this problem, the real optimal solution is as follows:

Source author

So how to avoid the error point of the red mark and get the best point (green mark)?

Source author

If the algorithm is only "looking" in the green circle, it will not fall into the wrong red trap (the optimal area).

This is the regularization -modifying the loss function makes its space solve restrictions, which is more likely to solve global optimal rather than the optimal.

3.2 Physical information neural network = regularization!

Remember the anti -discrimination problem expressed in the above equation (2)? Some scholars are trying to solve.

Initially, in some specific positions, they are known to offset (U), and they hope to get V in the unknown area of ​​the algorithm. In other words, the new T, Y, and X given a new displacement can be obtained, so as to obtain a new speed distribution.

There are many controversy about the solution of this equation, because this problem is not appropriate, that is, even if people find a solution, it cannot be determined whether it is unique. Not only that, there are many physics restrictions that cannot be solved.

They hope to generate displacement by restricting displacement to meet the wave function equation (2), combined with the loss function:

< /g> < /g> satisfaction:

< /g> < /g> : Among them, indicates the square difference between the prediction displacement and the target displacement, The value should be as close as 0.

What does this mean? In short, it represents a regularization process:

They find the optimal solution by restricting the loss function range (excluding solutions that do not meet differential equations).

How does neural network get "physical information"? Just formulate it through differential equations.

4 Conclusion

At the end of the article, I want to say:

The neural network of physical information is just a neural network that regularizes the loss function through differential equations.

After a few minutes of reading, this article can be summarized into several aspects:

What is the loss function (part 2)

· What is regularization (part 3.1)

What is the differential equation (part 1)

· What are the physical information (part 3.2)

I hope you have a certain understanding of the physical information neural network ~

Author: Piero Paialunga

Bleak

Translation: zhenni

Grade: CC

Original link:

https://towardSdatascience.com/physics-nd-rTiffial-ntelligence-inTropiness

FU

blessing

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profit

shi

Time

jian

between

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Translation content only represents the author's point of view

Does not represent the position of the Institute of Physics of the Chinese Academy of Sciences

Edit: zhenni

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