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The article conducts experiments on datasets including [[SIR dynamics]], the [[Boids model]], [[Game of Life]], and fMRI signals from the [[brain nervous system]] of human subjects. This section focuses on the bird flock and brain signal experiments for detailed analysis.

The article conducts experiments on datasets including [[SIR dynamics]], the [[Boids model]], [[Game of Life]], and fMRI signals from the [[brain nervous system]] of human subjects. This section focuses on the bird flock and brain signal experiments for detailed analysis.

The following figure shows the experimental results of NIS+ learning the flocking behavior of the Boids model. (a) and (e) give the actual and predicted trajectories of bird flocks under different conditions. Specifically, the author divides the bird flock into two groups and compares the multi-step prediction results under different noise levels (<math>\alpha</math> is 0.001 and 0.4 respectively). The prediction is good when the noise is relatively small, and the prediction curve will diverge when the noise is relatively large. (b) shows that the mean absolute error (MAE) of multi-step prediction gradually increases as the radius r increases. (c) shows the change of [[causal emergence measure]] <math>\Delta J</math> and prediction error (MAE) under different dimensions (q) with the change of training epoch. The author finds that causal emergence is most significant when the macroscopic state dimension q = 8. (d) is the attribution analysis of macroscopic variables to microscopic variables, and the obtained significance map intuitively describes the learned coarse-graining function. Here, each macroscopic dimension can correspond to the spatial coordinate (microscopic dimension) of each bird. The darker the color, the higher the correlation. Here, the microscopic coordinates corresponding to the maximum correlation of each macroscopic state dimension are highlighted with orange dots. These attribution significance values are obtained by using the [[Integrated Gradient]] (referred to as IG) method. The horizontal axis represents the x and y coordinates of 16 birds in the microscopic state, and the vertical axis represents 8 macroscopic dimensions. The light blue dotted line distinguishes the coordinates of different individual Boids, and the blue solid line separates the two bird flocks. (f) and (g) represent the changing trends of causal emergence measure ΔJ and normalized error MAE under different noise levels. (f) represents the influence of changes in external noise (that is, adding observation noise to microscopic data) on causal emergence. (g) represents the influence of internal noise (represented by α, added by modifying the dynamics of the Boids model) on causal emergence. In (f) and (g), the horizontal line represents the threshold that violates the error constraint in Equation {{EquationNote|1}}. When the normalized MAE is greater than the threshold of 0.3, the constraint is violated and the result is unreliable.

The following figure shows the experimental results of NIS+ learning the flocking behavior of the Boids model. (a) and (e) give the actual and predicted trajectories of bird flocks under different conditions. Specifically, the author divides the bird flock into two groups and compares the multi-step prediction results under different noise levels (<math>\alpha</math> is 0.001 and 0.4 respectively). The prediction is good when the noise is relatively small, and the prediction curve will diverge when the noise is relatively large. (b) shows that the mean absolute error (MAE) of multi-step prediction gradually increases as the radius r increases. (c) shows the change of the degree of [[causal emergence]] <math>\Delta J</math> and prediction error (MAE) under different dimensions (q) with the change of training epoch. The author finds that causal emergence is most significant when the macroscopic state dimension q = 8. (d) is the attribution analysis of macroscopic variables to microscopic variables, and the obtained significance map intuitively describes the learned coarse-graining function. Here, each macroscopic dimension can correspond to the spatial coordinate (microscopic dimension) of each bird. The darker the color, the higher the correlation. Here, the microscopic coordinates corresponding to the maximum correlation of each macroscopic state dimension are highlighted with orange dots. These attribution significance values are obtained by using the [[Integrated Gradient]] (referred to as IG) method. The horizontal axis represents the x and y coordinates of 16 birds in the microscopic state, and the vertical axis represents 8 macroscopic dimensions. The light blue dotted line distinguishes the coordinates of different individual Boids, and the blue solid line separates the two bird flocks. (f) and (g) represent the changing trends of causal emergence measure ΔJ and normalized error MAE under different noise levels. (f) represents the influence of changes in external noise (that is, adding observation noise to microscopic data) on causal emergence. (g) represents the influence of internal noise (represented by α, added by modifying the dynamics of the Boids model) on causal emergence. In (f) and (g), the horizontal line represents the threshold that violates the error constraint in Equation {{EquationNote|1}}. When the normalized MAE is greater than the threshold of 0.3, the constraint is violated and the result is unreliable.

This set of experiments shows that [[NIS+]] can learn macroscopic states and coarse-graining strategies by maximizing EI. This maximization improves the model's generalization ability to situations outside the training data range. The learned macroscopic state effectively identifies the average [[group behavior]] and can be attributed to individual positions using the gradient integration method. The degree of causal emergence increases with external noise but decreases with increase of internal noise. This result indicates that the model reduces external noise through coarse-graining but cannot mitigate internal noise.

This set of experiments shows that [[NIS+]] can learn macroscopic states and coarse-graining strategies by maximizing EI. This maximization improves the model's generalization ability to situations outside the training data range. The learned macroscopic state effectively identifies the average [[group behavior]] and can be attributed to individual positions using the gradient integration method. The degree of causal emergence increases with external noise but decreases with increase of internal noise. This result indicates that the model reduces external noise through coarse-graining but cannot mitigate internal noise.