# Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks

## TL;DR

This paper compares Bayesian methods and neural networks for reconstructing parton distribution functions from Ioffe time data in lattice QCD, emphasizing the importance of regularization in solving ill-posed inverse problems.

## Contribution

It evaluates the effectiveness of Bayesian and neural network approaches for reconstructing PDFs from Ioffe time data, highlighting their flexibility and regularization needs.

## Key findings

- Both Bayesian and neural network methods effectively regularize the inverse problem.
- Regularization is crucial for accurate reconstruction of PDFs.
- Neural networks offer a flexible approach for this inverse problem.

## Abstract

The computation of the parton distribution functions (PDF) or distribution amplitudes (DA) of hadrons from first principles lattice QCD constitutes a central open problem. In this study, we present and evaluate the efficiency of a selection of methods for inverse problems to reconstruct the full $x$-dependence of PDFs. Our starting point are the so called Ioffe time PDFs, which are accessible from Euclidean time calculations in conjunction with a matching procedure. Using realistic mock data tests, we find that the ill-posed incomplete Fourier transform underlying the reconstruction requires careful regularization, for which both the Bayesian approach as well as neural networks are efficient and flexible choices.