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Figure b shows a toy model example. The microscopic input <math>X_t(X_t^1,...,X_t^6) \in \{0,1\}^6</math> has six dimensions, with each dimension being binary (0 or 1). <math>X_{t + 1}</math> is the output of <math>X_{t}</math> at the next moment. The macroscopic variable is <math>V_{t}=\oplus_{i = 1}^{5}X_t^i</math>, where <math>\oplus_{i = 1}^{5}X_t^i</math> represents the result of adding the first five dimensions of the microscopic input <math>X_t</math> and taking the modulo 2. There exists a probability <math>\gamma</math> that the macroscopic states at two consecutive moments are equal (<math>p(\oplus_{j = 1..5}X_{t + 1}^j=\oplus_{j = 1..5}X_t^j)= \gamma</math>). The sixth dimension of the microscopic input at two consecutive moments is equal with <math>\gamma_{extra}</math> (<math>p(X_{t + 1}^6=X_t^6)= \gamma_{extra}</math>).

Figure b shows a toy model example. The microscopic input <math>X_t(X_t^1,...,X_t^6) \in \{0,1\}^6</math> has six dimensions, with each dimension being binary (0 or 1). <math>X_{t + 1}</math> is the output of <math>X_{t}</math> at the next moment. The macroscopic variable is <math>V_{t}=\oplus_{i = 1}^{5}X_t^i</math>, where <math>\oplus_{i = 1}^{5}X_t^i</math> represents the result of adding the first five dimensions of the microscopic input <math>X_t</math> and taking the modulo 2. There exists a probability <math>\gamma</math> that the macroscopic states at two consecutive moments are equal (<math>p(\oplus_{j = 1..5}X_{t + 1}^j=\oplus_{j = 1..5}X_t^j)= \gamma</math>). The sixth dimension of the microscopic input at two consecutive moments is equal with <math>\gamma_{extra}</math> (<math>p(X_{t + 1}^6=X_t^6)= \gamma_{extra}</math>).

The results show that in the simple example shown in Figure (b), by maximizing <math>\mathrm{\Psi}</math> through the model constructed in Figure a, the experiment finds that the learned <math>\mathrm{\Psi}</math> is approximately equal to the true groundtruth <math>\mathrm{\Psi}</math>. This outcome verifies the effectiveness of model learning. This system is effective in identifying the occurrence of causal emergence. However, this method also has the problem of being difficult to deal with complex multivariate situations. This is because the number of neural networks on the right side of the figure is proportional to the number of macroscopic and microscopic variable pairs. Therefore, the more the number of microscopic variables (dimensions), the more the number of neural networks will increase proportionally, which will lead to an increase in computational complexity. In addition, this method is only tested on very few cases, so it cannot be scaled up yet. Moreover, since the network calculates an approximate index of causal emergence, it inherits the limitations of the earlier approximate algorithms.

The results show that in the simple example shown in Figure b, by maximizing <math>\mathrm{\Psi}</math> through the model constructed in Figure a, the experiment finds that the learned <math>\mathrm{\Psi}</math> is approximately equal to the true groundtruth <math>\mathrm{\Psi}</math>. This outcome verifies the effectiveness of model learning. This system is effective in identifying the occurrence of causal emergence. However, this method also has the problem of being difficult to deal with complex multivariate situations. This is because the number of neural networks on the right side of the figure is proportional to the number of macroscopic and microscopic variable pairs. Therefore, the more the number of microscopic variables (dimensions), the more the number of neural networks will increase proportionally, which will lead to an increase in computational complexity. In addition, this method is only tested on very few cases, so it cannot be scaled up yet. Moreover, since the network calculates an approximate index of causal emergence, it inherits the limitations of the earlier approximate algorithms.

====Neural information squeezer====

====Neural information squeezer====