Differential Equations for Feynman Integrals and Physics Informed Neural Network

This thesis explores a novel research line that combines Quantum Field Theory and Computer Science, by focusing on the use of Deep Learning and Neural Networks techniques for the numerical evaluation of multi-loop Feynman integrals, which are the building blocks for the calculation of Scattering Amplitudes in Perturbation Theory. Feynman integrals are known to obey systems of first-order partial differential equations (PDEs), and owing to this property, as recently observed in the literature, their numerical evaluation can be addressed by means of Physics-Informed Neural Networks (PINNs). Within PINNs, differential equations become an intrinsic theoretical constraint to be implemented into the machine learning process, by adding, during the training phase, a physics-motivated term to the loss function. This approach directly accommodates the PDEs governing Feynman integrals, facilitating the learning algorithm to capture the correct solutions. In this thesis, We begin by demonstrating the application of PINNs to solve classical differential equations, such as the hypergeometric and harmonic oscillator equations, showcasing the potential efficiencies and accuracies of PINNs. Elaborating on the recent ideas proposed in the literature, we apply PINNs to systems of PDEs obeyed by one- and two-loop Feynman integrals, taken from scattering reactions of increasing complexity, whose analytical solutions, around four space-time dimensions, ranges from classical polylogarithms to generalised polylogarithms and to elliptic functions, and propose novel strategies to improve the NN reconstruction. Through this exploration, we highlight the possible powerful synergy between Machine Learning and Feynman caclculus, suggesting possible advancements in the area of Computational Quantum Field Theory, and more generally for problems in Applied Mathematics and Computational Science benefitting from the development of new classes of numerical solvers for PDEs.