Fourier Neural Differential Equations for learning Quantum Field Theories
This work addresses the challenge of connecting experimental data to theoretical models in quantum field theory, representing an incremental advancement by adapting existing neural methods to this domain.
The paper tackles the problem of learning quantum field theories by using neural differential equations to model particle scattering matrices, achieving better generalizability with a new Fourier Neural Differential Equation architecture and enabling extraction of interaction Hamiltonians from network parameters.
A Quantum Field Theory is defined by its interaction Hamiltonian, and linked to experimental data by the scattering matrix. The scattering matrix is calculated as a perturbative series, and represented succinctly as a first order differential equation in time. Neural Differential Equations (NDEs) learn the time derivative of a residual network's hidden state, and have proven efficacy in learning differential equations with physical constraints. Hence using an NDE to learn particle scattering matrices presents a possible experiment-theory phenomenological connection. In this paper, NDE models are used to learn $φ^4$ theory, Scalar-Yukawa theory and Scalar Quantum Electrodynamics. A new NDE architecture is also introduced, the Fourier Neural Differential Equation (FNDE), which combines NDE integration and Fourier network convolution. The FNDE model demonstrates better generalisability than the non-integrated equivalent FNO model. It is also shown that by training on scattering data, the interaction Hamiltonian of a theory can be extracted from network parameters.