A Thermodynamically Consistent Cahn-Hilliard-Navier-Stokes Model for Tumor Growth

What the math gives to ML

The paper's transferable asset is the Multiple Scalar Auxiliary Variable (MSAV) treatment of nonlinear energy terms: a difficult nonconvex contribution is represented through scalar variables, allowing each time step to reduce to linear solves while retaining an unconditional discrete energy law. This suggests an optimizer for neural-network training that separates a quadratic or locally linearized parameter energy from the nonlinear loss and updates an auxiliary scalar alongside the parameters. The most practical test is an SAV/MSAV optimizer on small MLPs and transformers, checking whether it prevents loss explosions at learning rates where Adam diverges.