Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow

What the math gives to ML

The paper augments a double saddle-point system with a constraint penalty that vanishes on exactly feasible solutions, while making the primal Schur block positive definite and robust to mesh and viscosity parameters. This structure transfers to equality-constrained neural-network optimization, including physics-informed training, continual-learning constraints, and adapter tuning with linearized side conditions. The most promising adaptation is a matrix-free augmented KKT optimizer that separately preconditions parameter and constraint variables instead of applying one generic optimizer to all variables. The field-of-values conditions also provide a practical diagnostic for whether approximate block solves preserve stable Krylov convergence.