So what we did do in the last chapter was to think about the causal, retro causal explanation

of dynamical systems.

Because the data that you have are always the same and for such applications you do

not get more data.

Even if you, after some years, would have longer times here, then it's really questionable

if for markets such old time series represent anything about what's going on in the dynamics

nowadays.

So you have to stay with a very limited amount of data.

And the only thing which you can add to the model building then is your thinking.

And in the past chapters you have seen several of these points where it is the improved thinking

which was an improvement of the final forecast.

And now is this ending?

For sure it's not ending, but let's do another step.

The causal-retro-causal neural network story was such an interesting thing.

Let's try to do it in a different way.

When you look at types of mathematics where you have causal and retro-causal descriptions,

one obvious thing in our lecture here is backpropagation itself.

So in backpropagation you have a sequence of matrices which go in the one direction.

Or even if it's only the same matrix here, you would say it's a sequence of matrices

going in one direction.

And then you have the opposite movement here, backward.

And in this backward flow you always have to transpose of this matrix here.

Here you transpose matrix.

And this is not only in the backpropagation calculus.

See, if you know something about control theory, control theory is a mathematical theory which

very fast was developing in the 1960s because it's a very important part of the possibility

to do rocket development and flight to the moon and satellite control and all of this.

What's the key idea in it?

You have the dynamical system which in this case is given by the physicists.

So there you do not have the problem to find the dynamical system.

So this is given by the physical laws known since many years.

But then you have a reward function which means you want to position a satellite at

this and this place in space or you want to fly to the moon or whatever it is.

And the combination of both things includes that you find optimal variables here which

you have under control such that you really come, that you really end up at there where

you want to be or with whatever reward you want to have.

Maybe you want to fly to the moon with minimum energy or whatever.

And if you do so then in the one direction you have a causal information flow which is

the physical law.

But in addition to this you have a so-called adjoint equation system or a retro-causal

information system here.

It's going from the future back to the interdirection of the past.

It starts at the end of the reward function here and then it goes step by step backwards.

And the superposition of both sides here is coming up in form of such equations here.

We will speak about such equations later in this lecture again when I speak a little bit

about control theory and reinforcement learning.

But in the moment I simply want to tell you a superposition of the retro flow and of the

causal flow here.

This is giving you a possibility to compute optimal behavior, the optimal control variables

here.