 This view not only unifies previous methods but also allows us to propose novel methods that, in contrast with traditional mean-field approaches, preserve correlations of the system, both in stationary and transient states. By comparing analytical approximations and exact numerical simulations, we show that the proposed methods provide more accurate estimates of covariance evolution in the system than classical equations, and consequently outperform previous mean field theories for solving the inverse Ising problem. In sum, our framework unifies and extends current mean-field approximations of kinetic Ising model, constituting a powerful tool for studying non-equilibrium dynamics of complex systems .
<clarity>  This view not only unifies previous methods but also allows us to propose novel methods that, in contrast with traditional mean-field approaches, preserve correlations of the system, both in stationary and transient states. By comparing analytical approximations and exact numerical simulations, we show that the proposed methods provide more accurate estimates of covariance evolution in the system than classical equations, and consequently outperform previous mean field theories for solving the inverse Ising problem. In sum, our framework unifies and extends current mean-field approximations of kinetic Ising model, constituting a powerful tool for studying non-equilibrium dynamics of complex systems .
 This view not only unifies previous methods but also allows us to propose novel methods that, in contrast with traditional mean-field approaches, preserve the system's correlations. We show that these new methods can outperform previous ones in predicting and assessing network properties near maximally fluctuating regimes .