Kinetic Ising models are powerful tools for studying the non-equilibrium dynamics of complex  discrete systems and  analyzing their experimental recordings. However, the behaviour of the model is in general not tractable for large networks; therefore, mean field theories are frequently used to approximate its statistical properties. Many variants of the classical naive and TAP (i.e., second-order) mean field approximations have been proposed, each of which makes unique assumptions about time evolution of the system's correlation structure. This disparity of methods makes it difficult to systematically advance the mean field approach over previous contributions. Here, we propose a unified framework for mean field theories of the dynamics of asymmetric kinetic Ising systems based on information geometry. The framework is built on Plefka expansions of the model around a simplified model obtained by an orthogonal projection to a sub-manifold of tractable probability distributions. This approach not only unifies previous methods but  allows us to define 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 for the evolution of equal-time and delayed covariance structures 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.
Kinetic Ising models are powerful tools for studying the non-equilibrium dynamics of complex , discrete systems and for analyzing their experimental recordings. However, the behaviour of the model is in general not tractable for large networks; therefore, mean field theories are frequently used to approximate its statistical properties. Many variants of the classical naive and TAP (i.e., second-order) mean field approximations have been proposed, each of which makes unique assumptions about the temporal evolution of the system's correlation structure. This disparity of methods makes it difficult to systematically advance mean field approaches beyond previous contributions. Here, we propose a unified framework for mean field theories of the dynamics of asymmetric kinetic Ising systems based on information geometry. The framework is built on Plefka expansions of the model around a simplified model obtained by an orthogonal projection to a sub-manifold of tractable probability distributions. 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.
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