Analysis of Dynamical Field Inference in a Supersymmetric Theory

Abstract

The inference of dynamical fields is of paramount importance in science, technology, and economics. Dynamical field inference can be based on information field theory and used to infer the evolution of fields in dynamical systems from finite data. Here, the partition function, as the central mathematical object of our investigation, invokes a Dirac delta function as well as a field-dependent functional determinant, which impede the inference. To tackle this problem, Fadeev–Popov ghosts and a Lagrange multiplier are introduced to represent the partition function by an integral over those fields. According to the supersymmetric theory of stochastics, the action associated with the partition function has a supersymmetry for those ghost and signal fields. In this context, the spontaneous breakdown of supersymmetry leads to chaotic behavior of the system. To demonstrate the impact of chaos, characterized by positive Lyapunov exponents, on the predictability of a system’s evolution, we show for the case of idealized linear dynamics that the dynamical growth rates of the fermionic ghost fields impact the uncertainty of the field inference. Finally, by establishing perturbative solutions to the inference problem associated with an idealized nonlinear system, using a Feynman diagrammatic expansion, we expose that the fermionic contributions, implementing the functional determinant, are key to obtain the correct posterior of the system.