Operator learning regularization for macroscopic permeability prediction in dual-scale flow problem

Challenges lie in the optimization of the process due to the lack of understanding of key characteristic of textile fabrics - permeability. The difficulty is mainly related to the nature of multiple lengths scales being involved in the flow kinematics across this type of porous media. At microscale, the resin flows between individual fibres, which has a typical length of micrometer; whereas, larger pores of the size of millimeters exist between the fiber bundles that are woven together, which leads to a clear fluid region at this commonly-called mesoscale. If one considers the problem with a unified length scale, at millimeter order of magnitude, the microscale flow needs to be described with Darcy's law, and the mesoscale flow with Stokes equation. Then, the problem becomes a two-domain problem, for which an interfacial behavior between the Darcy region and the Stokes region has to be introduced. Previous studies [1] have shown that a term may be necessary to be added to the Darcy's law to incorporate the effect of such an interface. This leads to the well-known Brinkman equation [2]. The Stokes-Brinkman equation can then be formulated to convert the two-domain problem into a single domain problem, simplifying the solution procedure, see e.g.