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Traditional numerical schemes for simulating fluid flow and transport inporous media can be computationally expensive. Advances in machine learning forscientific computing have the potential to help speed up the simulation time inmany scientific and engineering fields. DeepONet has recently emerged as apowerful tool for accelerating the solution of partial differential equations(PDEs) by learning operators (mapping between function spaces) of PDEs. In thiswork, we learn the mapping between the space of flux functions of theBuckley-Leverett PDE and the space of solutions (saturations). We usePhysics-Informed DeepONets (PI-DeepONets) to achieve this mapping without anypaired input-output observations, except for a set of given initial or boundaryconditions; ergo, eliminating the expensive data generation process. Byleveraging the underlying physical laws via soft penalty constraints duringmodel training, in a manner similar to Physics-Informed Neural Networks(PINNs), and a unique deep neural network architecture, the proposedPI-DeepONet model can predict the solution accurately given any type of fluxfunction (concave, convex, or non-convex) while achieving up to four orders ofmagnitude 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 transportproblems in porous media.

Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator