更改

====Neural information squeezer====

====Neural information squeezer====

In recent years, emerging [[artificial intelligence]] technologies have overcome a series of major problems. Machine learning methods incorporate carefully designed [[neural network]] structures and [[automatic differentiation]] technologies. These technologies enable the approximation of any function in a vast function space. Therefore, Zhang et al. proposed a data-driven method using neural networks to identify causal emergence from time series data <ref name="NIS">Zhang J, Liu K. Neural information squeezer for causal emergence[J]. Entropy, 2022, 25(1): 26.</ref><ref name=":6" />. This method can automatically extract effective coarse-graining strategies and macroscopic dynamics, overcoming various deficiencies of the Rosas's method <ref name=":5" />.

In recent years, emerging [[artificial intelligence]] technologies have overcome a series of major problems. Machine learning methods incorporate carefully designed [[neural network]] structures and [[automatic differentiation]] technologies. These technologies enable the approximation of any function in a vast function space. Therefore, Zhang et al. proposed a data-driven method using neural networks to identify causal emergence from time series data <ref name="NIS">Zhang J, Liu K. Neural information squeezer for causal emergence[J]. Entropy, 2022, 25(1): 26.</ref><ref name=":6" />. This method can automatically extract effective coarse-graining strategies and macroscopic dynamics, overcoming various deficiencies of the Rosas's method <ref name=":5" />. This framework is based on Erik Hoel's theory of causal emergence.

In this work, the input consists of time series data <math>(X_1,X_2,...,X_T )</math>, where <math>X_t\equiv (X_t^1,X_t^2,…,X_t^p )</math> and <math>p</math> denotes the dimensionality of the input data. The author assumes that this set of data is generated by a general [[stochastic dynamical system]]:

In this work, the input consists of time series data <math>(X_1,X_2,...,X_T )</math>, where <math>X_t\equiv (X_t^1,X_t^2,…,X_t^p )</math> and <math>p</math> denotes the dimensionality of the input data. The author assumes that this set of data is generated by a general [[stochastic dynamical system]]:

|{{EquationRef|2}}}}

|{{EquationRef|2}}}}

[math]\mathcal{J}[/math] represents the dimension-averaged <math>EI</math> (see the entry [[effective information]]), and <math>\mathrm{\phi}</math> is the coarse-graining strategy function. <math>f_{q}</math> denotes the macroscopic dynamics, and <math>q</math> represents the dimension of the coarsened macroscopic state. [math]\hat{X}_{t + 1}[/math] is the prediction of the microscopic state at time <math>t + 1</math> by the entire framework. This prediction is obtained by applying the inverse coarse-graining function ([math]\phi^{\dagger}[/math]) to the macroscopic state prediction of the macroscopic state <math>\hat{Y}_{t + 1}</math> at time <math>t + 1</math>. Here [math]\hat{Y}_{t + 1}\equiv f_q(Y_t)[/math] represents the prediction of the macroscopic state at time <math>t + 1</math> by the dynamics learner, based on the macroscopic state [math]Y_t[/math] at time <math>t</math>. The macroscopic state [math]Y_t\equiv \phi(X_t)[/math] at time <math>t</math> is obtained by coarse-graining the microscopic state [math]X_t[/math] using the function [math]\phi[/math]. Finally, the difference between [math]\hat{X}_{t + 1}[/math] and the real microscopic state data [math]X_{t + 1}[/math] is compared to obtain the microscopic prediction error.

[math]\mathcal{J}[/math] represents the dimension-averaged <math>EI</math> (see the entry [[effective information]]), and <math>\mathrm{\phi}</math> is the coarse-graining strategy function. <math>f_{q}</math> denotes the macroscopic dynamics, and <math>q</math> represents the dimension of the coarsened macroscopic state. [math]\hat{X}_{t + 1}[/math] is the prediction of the microscopic state at time <math>t + 1</math> by the entire framework. This prediction is obtained by applying the inverse coarse-graining function ([math]\phi^{\dagger}[/math]) on the macroscopic state <math>\hat{Y}_{t + 1}</math> at time <math>t + 1</math>. Here [math]\hat{Y}_{t + 1}\equiv f_q(Y_t)[/math] represents the prediction of the macroscopic state at time <math>t + 1</math> by the dynamics learner, based on the macroscopic state [math]Y_t[/math] at time <math>t</math>. The macroscopic state [math]Y_t\equiv \phi(X_t)[/math] at time <math>t</math> is obtained by coarse-graining the microscopic state [math]X_t[/math] using the function [math]\phi[/math]. Finally, the difference between [math]\hat{X}_{t + 1}[/math] and the real microscopic state data [math]X_{t + 1}[/math] is compared to obtain the microscopic prediction error.

The entire optimization framework is shown below:

The entire optimization framework is shown below: