  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 $\phi^4$ 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.
