An optimization-based equilibrium measure describing fixed points of non-equilibrium dynamics: application to the edge of chaos

Understanding neural dynamics is a central topic in machine learning, non-linear physics, and neuroscience. However, the dynamics are non-linear, stochastic and particularly non-gradient, i.e., the driving force cannot be written as the gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory. Still, the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed-form solutions in general. From the associated Fokker-Planck equation, the steady-state solution is generally unknown. Here, we treat searching for the fixed points as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, can the distribution of the original fixed points be achieved. The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos. Furthermore, the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics.