Analysis and Approximation of Flux Functions in Gravity-Driven Particle-Laden Thin Films

We study the approximability of the 2 × 2 system of hyperbolic conservation laws governing the dynamics of viscous slurries on an incline, for which negatively-buoyant particles migrate in a thin film of fluid due to a combination of gravity-induced settling and shear-induced migration. As the numerical construction of the flux functions is costly, we approximate them via a sparse representation of data points in the state space and perform a spline polynomial regression. We then investigate how the choice of the loss function could either preserve or dismantle relevant physics associated with the system of conservation laws, which includes strict hyperbolicity, phase transition between the ridged and the settled regime, and the shock and rarefaction dynamics of solutions to the associated Riemann problem. In particular, we prove an approximation theorem using the implicit function theorem on Banach spaces signaling the preservation of shock and rarefaction solutions for a sufficiently good approximation. The above analysis serves as a primer to physics-informed machine learning aimed at picking the right loss function to preserve relevant physical structures, especially so with shock and rarefaction dynamics associated with systems of hyperbolic conservation laws.