Krzysztof Graczyk Homepage
Electromagnetic and Weak Structure of the Nucleon Investigated with Bayesian Neural Networks

Nucleon axial form factor from a Bayesian neuralnetwork analysis of neutrinoscattering data,
Luis AlvarezRuso, Krzysztof M. Graczyk, Eduardo SaulSala,
submitted
Abstract:

The Bayesian approach for feedforward neural networks has been applied to the extraction of the nucleon axial form factor from the neutrinodeuteron scattering data measured by the Argonne National Laboratory (ANL) bubble chamber experiment. This framework allows to perform a modelindependent determination of the axial form factor from data.. When the low $0.05 < Q^2 < 0.10$ GeV$^2$ data is included in the analysis, the resulting axial radius disagrees with available determinations. Furthermore, a large sensitivity to the corrections from the deuteron structure is obtained. In turn, when the low$Q^2$ region is not taken into account, with or without deuteron corrections, no significant deviations from the dipole ansatz have been observed. A more accurate determination of the nucleon axial form factor requires new precise measurements of neutrinoinduced quasielastic scattering on hydrogen and deuterium.

Zemach moments of proton from Bayesian inference,
Krzysztof M. Graczyk and C. Juszczak,
Phys. Rev. C91, 045205 (2015)
Abstract:

The first and the third Zemach moments are obtained, $\langle r \rangle_{(2)}= 1.1108\pm 0.0021 $ fm and $\langle r^3\rangle_{(2)}=2.889 \pm 0.008$ fm$^3$,
from the Bayesian analysis of the elastic $ep$ scattering data.
The quantitative discussion of the dependence of the results
on the parametrization choice is presented and the corresponding systematic uncertainties are estimated  about 0.6\% and 1.6\% for the first and the third Zemach moments respectively.

Applications of Neural Networks in Hadron Physics,
Krzysztof M. Graczyk and C. Juszczak, J.Phys. G42 (2015) 3, 034019
invited contribution to special issue of J.Phys. G: Nucl. Phys., "Enhancing the interaction between nuclear experiment and theory through information and statistics"
(ISNET).
Abstract:

The Bayesian approach for the feedforward neural networks is reviewed. Its potential for usage in hadron physics is discussed. As an example of the application the study of the the twophoton exchange effect is presented. We focus on the model comparison, the estimation of the systematic uncertainties due to the choice of the model, and the overfitting. As an illustration the predictions of the cross sections ratio $d \sigma(e^+ p\to e^+ p)/d \sigma(e^ p\to e^ p)$ are given together with the estimate of the uncertainty due to the parametrization choice.

Proton Radius from Bayesian Inference,
Krzysztof M. Graczyk and C. Juszczak, Phys. Rev. C90, 054334 (2014).
Abstract:

The methods of Bayesian statistics are used to extract the value of the proton radius
from the elastic $ep$ scattering data in a model independent way.
To achieve that goal a large number of parametrizations
(equivalent to neural network schemes) are considered and ranked by
their conditional probability $P(\mathrm{parametrization}\,\,\mathrm{data})$ instead of using the minimal error criterion.
As a result the most probable proton radii values ($r_E^p=0.899\pm 0.003$ fm, $r_M^p=0.879\pm 0.007$ fm) are obtained and systematic error due to freedom in the choice of parametrization is estimated.
Correcting the data for the two photon exchange effect leads to smaller difference between the extracted values of $r_E^p$ and $r_M^p$.
The results disagree with recent muonic atom measurements.

Comparison of Neural Network and Hadronic Model Predictions of TwoPhoton Exchange Effect,
Krzysztof M. Graczyk, Phys. Rev. C88, 065205 (2013)
Abstract:

Predictions for the twophoton exchange (TPE) correction to unpolarized $ep$ elastic cross section, obtained within two different approaches, are confronted and discussed in detail. In the first one the TPE correction is extracted from experimental data by applying the Bayesian neural network (BNN) statistical framework. In the other the TPE is given by box diagrams, with the nucleon and the $P_{33}$ resonance as the hadronic intermediate states. Two different form factor parametrizations for both the proton and the $P_{33}$ resonance are taken into consideration. Proton form factors are obtained from the global fit of the full model (with the TPE correction) to the unpolarized cross section data. Predictions of both methods agree well in the intermediate $Q^2$ range, $(1,3)$ GeV$^2$. Above $Q^2=3$ GeV$^2$ the agreement is on $2\sigma$ level. Below $Q^2=1$ GeV$^2$ the consistency between both approaches is broken. The values of the proton radius extracted within both models are given. In both cases predictions for VEPP3 experiment have been obtained and confronted with the preliminary experimental results.

TwoPhoton Exchange Effect Studied with Neural Networks,
Krzysztof M. Graczyk, Phys. Rev. C84, 034314 (2011)
Abstract:

An approach to the extraction of the twophoton exchange (TPE) correction from elastic ep scattering data is presented. The crosssection, polarization transfer (PT), and charge asymmetry data are considered. It is assumed that the TPE correction to the PT data is negligible. The form factors and TPE correcting term are given by one multidimensional function approximated by the feedforward neural network (NN). To find a modelindependent approximation, the Bayesian framework for the NNs is adapted. A large number of different parametrizations is considered. The most optimal model is indicated by the Bayesian algorithm. The obtained fit of the TPE correction behaves linearly in ? but it has a nontrivial Q2 dependence. A strong dependence of the TPE fit on the choice of parametrization is observed.

The analytical form of the fits fit.pdf and the covariance matrix
(order of parameters the same as in fit.pdf )
Analysis done with:
 GraNet  the feedworward neural network C++ library (will be avialable soon).

Neural Network Parameterizations of Electromagnetic Nucleon Form Factors,
Krzysztof M. Graczyk, Piotr Płoński, Robert Sulej, JHEP (2010) 053
Abstract:

The electromagnetic nucleon formfactors data are studied with artificial feed forward neural networks.
As a result the unbiased modelindependent formfactor parametrizations are evaluated together with uncertainties.
The Bayesian approach for the neural networks is adapted for chi2 errorlike function and applied to the data analysis. The sequence of the feed forward neural networks with one hidden layer of units is considered. The given neural network represents a particular formfactor parametrization. The socalled
evidence (the measure of how much the data favor given form factor model) is computed with the Bayesian framework and it is used to determine the best form factor parametrization.