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In this paper, we present a discontinuity and cusp capturing physics-informedneural network (PINN) to solve Stokes equations with a piecewise-constantviscosity and singular force along an interface. We first reformulate thegoverning equations in each fluid domain separately and replace the singularforce effect with the traction balance equation between solutions in two sidesalong the interface. Since the pressure is discontinuous and the velocity hasdiscontinuous derivatives across the interface, we hereby use a networkconsisting of two fully-connected sub-networks that approximate the pressureand velocity, respectively. The two sub-networks share the same primarycoordinate input arguments but with different augmented feature inputs. Thesetwo augmented inputs provide the interface information, so we assume that alevel set function is given and its zero level set indicates the position ofthe interface. The pressure sub-network uses an indicator function as anaugmented input to capture the function discontinuity, while the velocitysub-network uses a cusp-enforced level set function to capture the derivativediscontinuities via the traction balance equation. We perform a series ofnumerical experiments to solve two- and three-dimensional Stokes interfaceproblems and perform an accuracy comparison with the augmented immersedinterface methods in literature. Our results indicate that even a shallownetwork with a moderate number of neurons and sufficient training data pointscan achieve prediction accuracy comparable to that of immersed interfacemethods.

A discontinuity and cusp capturing PINN for Stokes interface problems with discontinuous viscosity and singular forces