# Minimization

The training (also known as learning or optimisation phase) of neural networks is carried out using the gradient descent method in one of its versions such as back-propagation or stochastic gradient descent. In these methods, the determination of the fit parameters (namely the weights and thresholds of the NN) requires the evaluation of the gradients of , that is,

(1)

Computing these gradients in the NNPDF case involves handling the non-linear relation between the fitted experimental data and the input PDFs, which proceeds through convolutions both with the DGLAP evolution kernels and the hard-scattering partonic cross-sections as encoded into the optimised APFELgrid fast interpolation strategy.

The theory prediction for a collider cross-section in terms of the NN parameters reads

(2)

where indicates a convolution over , and stand for the hard-scattering cross-sections and the DGLAP evolution kernels respectively, and sum over repeated flavour indices is understood.

In the APFELgrid approach, this cross-section can be expressed in a much compact way as

(3)

where now all perturbative information is pre-computed and stored in the interpolation tables, and run over a grid in .