The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations

Machine Learning tools for global PDF fits

The use of machine learning algorithms in theoretical and experimental high-energy physics has experienced an impressive progress in recent years, with applications from trigger selection to jet substructure classification and detector simulation among many others. In this contribution, we review the machine learning tools used in the NNPDF family of global QCD analyses. These include multi-layer feed-forward neural networks for the model-independent parametrisation of parton distributions and fragmentation functions, genetic and covariance matrix adaptation algorithms for training and optimisation, and closure testing for the systematic validation of the fitting methodology.Comment: 12 pages, 9 figures, to appear in the proceedings of the XXIIIth Quark Confinement and the Hadron Spectrum conference, 1-6 August 2018, University of Maynooth, Irelan