Fourier Neural Differential Equations for learning Quantum Field Theories

By multiplying the representations of the interaction vertices, propagators, and particle lines known as Feynman rules [1], particle scattering amplitudes are derived from the interaction Hamiltonian. This is an example of a phenomenological connection between what is theoretically derived and what is empirically observed [2]. Neural Networks have been used to learn physical problems, including Hamiltonians [3], higher-order behaviour [4], and Fourier representations [5], where physical constraints are applied to network architecture, improving convergence and explainability. Neural Differential Equations (NDEs) [6] take the continuous time limit of a Residual Neural Network (RNN) to learn differential equations. Integrating the learnt function through time outputs the network's hidden state to a continuous depth. NDEs have so far been applied to various quantum systems [7] [8] [9] [10], but not yet to scattering processes in Quantum Field Theory (QFT). In this paper, we look for how NDEs can be used to bridge the phenomenological connection between experiment and theory by training these models on particle scattering data to learn scalar quantum field theories. The objectives are twofold: apply Neural Ordinary Differential Equations (NODE) to learn particle scattering.