  Modern analysis on parton distribution functions (PDFs) requires calculations
of the log-likelihood functions from thousands of experimental data points, and
scans of multi-dimensional parameter space with tens of degrees of freedom. In
conventional analysis the Hessian approximation has been widely used for the
estimation of the PDF uncertainties.The Lagrange Multiplier (LM) scan while
being a more faithful method is less used due to computational limitations, and
is the main focus of this study. We propose to use Neural Networks (NNs) and
machine learning techniques to model the profile of the log-likelihood
functions or cross sections for multi-dimensional parameter space in order to
overcome those limitations which work beyond the quadratic approximations and
meanwhile ensures efficient scans of the full parameter space. We demonstrate
the efficiency of the new approach in the framework of the CT18 global analysis
of PDFs by constructing NNs for various target functions, and performing LM
scans on PDFs and cross sections at hadron colliders. We further study the
impact of the NOMAD dimuon data on constraining PDFs with the new approach, and
find enhanced strange-quark distributions and reduced PDF uncertainties.
Moreover, we show how the approach can be used to constrain new physics beyond
the Standard Model (BSM) by a joint fit of both PDFs and Wilson coefficients of
operators in the SM effective field theory.
