Dynamical Field Inference and Supersymmetry

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.

Information Field Theory and Artificial Intelligence

“…is defined as well as the prior and permits us to answer questions about the field, like its most probable configuration ϕ MAP = argmax ϕ P (ϕ|d, I) (MAP = maximum a posteriori), its posterior mean m = ϕ (ϕ|d,I) , or its posterior uncertainty dispersion D = (ϕ − m)(ϕ − m) † (ϕ|d,I) . IFT exploits the formalism of quantum and statistical field theory to calculate such posterior expectation values [1,28,[36][37][38]. These formal calculations, however, should not be the focus here.…”

“…Output of a generative IFT model for a 2D tomography problem in simulation (top row) and reconstruction (bottom rows) mode. The model is depicted in Figure 1 and described by Equations (32)-(38) with the modification that in Equation (36) the exp-function is replaced by a sigmoid function to obtain more cloud-like structures. Run in simulation mode, the model first generates a non-parametric power spectrum (top right panel) from which a Gaussian realization of a statistical isotropic and homogeneous field is drawn (top left, after procession by the sigmoid function).…”

Information Field Theory and Artificial Intelligence

“…is defined as well as the prior and permits to answer questions about the field, like its most probable configuration ϕ MAP = argmax ϕ P(ϕ|d, I) (MAP = maximum a posteriori), its posterior mean m = ϕ (ϕ|d,I) , or its posterior uncertainty dispersion D = (ϕ − m)(ϕ − m) † (ϕ|d,I) . IFT exploits the formalism of quantum and statistical field theory to calculate such posterior expectation values [1,28,[36][37][38]. These formal calculations, however, should not be the focus here.…”