Field dynamics inference via spectral density estimation

Frank, Philipp; Steininger, Theo; Enßlin, T. A.

doi:10.1103/physreve.96.052104

Cited by 9 publications

(8 citation statements)

References 24 publications

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“…For further details see refs. [ 13,14 ] . The covariance function associated with T takes the form

T false(k, k^{'} false) = \frac{σ^{2}}{2} \sqrt{\frac{π}{2 γ^{3}}} (1 + \sqrt{γ} (|| k - k^{'})) e^{- \sqrt{γ} |k - k^{'}|}

Therefore the real and imaginary part of the Fourier modes

f^{k}

, which are just the functions

f_{1false r / 1false i}

evaluated at

k \in Z

, are defined to be two independent, infinite dimensional Gaussian random vectors with zero mean and covariance

T_{k k^{'}} = T false(k, k^{'} false), k, k^{'} \in Z

…”

Section: Priormentioning

confidence: 99%

Field Dynamics Inference for Local and Causal Interactions

2021

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Inference of fields defined in space and time from observational data is a core discipline in many scientific areas. This work approaches the problem in a Bayesian framework. The proposed method is based on statistically homogeneous random fields defined in space and time and demonstrates how to reconstruct the field together with its prior correlation structure from data. The prior model of the correlation structure is described in a non‐parametric fashion and solely builds on fundamental physical assumptions such as space‐time homogeneity, locality, and causality. These assumptions are sufficient to successfully infer the field and its prior correlation structure from noisy and incomplete data of a single realization of the process as demonstrated via multiple numerical examples.

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“…For further details see refs. [ 13,14 ] . The covariance function associated with T takes the form

T false(k, k^{'} false) = \frac{σ^{2}}{2} \sqrt{\frac{π}{2 γ^{3}}} (1 + \sqrt{γ} (|| k - k^{'})) e^{- \sqrt{γ} |k - k^{'}|}

Therefore the real and imaginary part of the Fourier modes

f^{k}

, which are just the functions

f_{1false r / 1false i}

evaluated at

k \in Z

, are defined to be two independent, infinite dimensional Gaussian random vectors with zero mean and covariance

T_{k k^{'}} = T false(k, k^{'} false), k, k^{'} \in Z

…”

Section: Priormentioning

confidence: 99%

Field Dynamics Inference for Local and Causal Interactions

2021

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“…Here, the full spatial and temporal Fourier power spectrum P ϕ (k, ω), with ω being the temporal frequency, encodes the full dynamics of linear, autonomous system. An inference of this highly structured spectrum from data can be used to learn the system dynamics from system measurements [26,40]. It just requires a more complex spectral prior than discussed here.…”

Section: Information Field Theorymentioning

confidence: 99%

Information Field Theory and Artificial Intelligence

Enßlin

2021

Preprint

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Information field theory (IFT), the information theory for fields, is a mathematical framework for signal reconstruction and non-parametric inverse problems. Here, fields denote physical quantities that change continuously as a function of space (and time) and information theory refers to Bayesian probabilistic logic equipped with the associated entropic information measures. Reconstructing a signal with IFT is a computational problem similar to training a generative neural network (GNN). In this paper, the inference in IFT is reformulated in terms of GNN training and the cross-fertilization of numerical variational inference methods used in IFT and machine learning are discussed. The discussion suggests that IFT inference can be regarded as a specific form of artificial intelligence. In contrast to classical neural networks, IFT based GNNs can operate without pre-training thanks to incorporating expert knowledge into their architecture.

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“…IFT can then be applied for signal inference in all areas, where limitations on the exactness of the measurement are given. DFI [ 14 , 15 , 16 ] utilizes methods from IFT for the inference of signals in a DS. The reconstruction of the signal is advanced by the knowledge on the signal properties, which are specified by the prior covariance of the signal.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Field Inference and Supersymmetry

Westerkamp

Овчинников

Frank

et al. 2021

Entropy

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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.

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Field dynamics inference via spectral density estimation

Cited by 9 publications

References 24 publications

Field Dynamics Inference for Local and Causal Interactions

Field Dynamics Inference for Local and Causal Interactions

Information Field Theory and Artificial Intelligence

Dynamical Field Inference and Supersymmetry

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