Abstract
Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically-driven, dynamically-evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby makes DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.
Highlights
Stochastic differential equations (SDEs) appear in many disciplines like astrophysics [1], biology [2], chemistry [3], and economics [4,5]
The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations
The probability P (φ|η) contains already the two elements that will lead to SUSY in Dynamical field inference (DFI), the delta function, which will be represented with bosonic Lagrange fields and the functional determinant, for which fermionic fields are introduced
Summary
Stochastic differential equations (SDEs) appear in many disciplines like astrophysics [1], biology [2], chemistry [3], and economics [4,5]. In addition to the uncertainty introduced by the stochastic process driving the evolution of the system, any observation of it is noise afflicted and incomplete. This complicates the inference of the system’s state further. We consider arbitrary SDEs and introduce dynamical field inference (DFI) as a Bayesian framework to estimate the state and evolution of a field following a SDE from finite, incomplete, and noise-afflicted data. The dynamical growth rates of the fermionic ghost fields, which are the Lyapuov coefficients measuring the strength of chaos, impact the uncertainty of any field inference. With having connected the DFI and STS formalisms, and shown their mutual relevances, we conclude in Section 6 and give an outlook on future research directions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
