  Traditional numerical schemes for simulating fluid flow and transport in
porous media can be computationally expensive. Advances in machine learning for
scientific computing have the potential to help speed up the simulation time in
many scientific and engineering fields. DeepONet has recently emerged as a
powerful tool for accelerating the solution of partial differential equations
(PDEs) by learning operators (mapping between function spaces) of PDEs. In this
work, we learn the mapping between the space of flux functions of the
Buckley-Leverett PDE and the space of solutions (saturations). We use
Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any
paired input-output observations, except for a set of given initial or boundary
conditions; ergo, eliminating the expensive data generation process. By
leveraging the underlying physical laws via soft penalty constraints during
model training, in a manner similar to Physics-Informed Neural Networks
(PINNs), and a unique deep neural network architecture, the proposed
PI-DeepONet model can predict the solution accurately given any type of flux
function (concave, convex, or non-convex) while achieving up to four orders of
magnitude improvements in speed over traditional numerical solvers. Moreover,
the trained PI-DeepONet model demonstrates excellent generalization qualities,
rendering it a promising tool for accelerating the solution of transport
problems in porous media.
