Uno studio del metodo delle equazioni differenziali per gli integrali a loop di Feynman in relazione al Scientific Machine Learning

Scattering amplitudes are physical observables of particular interest, being a link between theoretical particle physics predictions and collider experimental data. They are the probability amplitudes of scattering processes in particle physics and are computed in perturbative series in the coupling constants. Feynman diagrams give us prescriptions to construct expressions for scattering amplitudes at each order in the coupling. These contain integrals, known as Feynman loop integrals, whose evaluation is a major challenge in scattering amplitude computations. They are integrals of rational functions over momentum space, and are often analytically expressed using special functions such as logarithms, polylogarithms, and their generalizations. A state-of-the-art method for computing Feynman loop integrals analytically involves differential equations which they satisfy. In principle it is understood how to obtain such equations, but in many cases solving them analytically or obtaining numerical values from them is a serious challenge. The goal of this work is to introduce the most important elementary concepts related to Feynman integral calculations (with particular attention to the method of differential equations) and to assess to what extent Scientific Machine Learning techniques, namely Physics Informed Neural Networks, can contribute to that end.