  This work explores the application of deep operator learning principles to a
problem in statistical physics. Specifically, we consider the linear kinetic
equation, consisting of a differential advection operator and an integral
collision operator, which is a powerful yet expensive mathematical model for
interacting particle systems with ample applications, e.g., in radiation
transport. We investigate the capabilities of the Deep Operator network
(DeepONet) approach to modelling the high dimensional collision operator of the
linear kinetic equation. This integral operator has crucial analytical
structures that a surrogate model, e.g., a DeepONet, needs to preserve to
enable meaningful physical simulation. We propose several DeepONet
modifications to encapsulate essential structural properties of this integral
operator in a DeepONet model. To be precise, we adapt the architecture of the
trunk-net so the DeepONet has the same collision invariants as the theoretical
kinetic collision operator, thus preserving conserved quantities, e.g., mass,
of the modeled many-particle system. Further, we propose an entropy-inspired
data-sampling method tailored to train the modified DeepONet surrogates without
requiring an excessive expensive simulation-based data generation.
