Design a semi-rigorous adaptive strategy balancing surrogate interpolation error and PDE approximation error

Develop a semi-rigorous adaptive sampling and refinement strategy for constructing the bifurcation-boundary surrogate in the hydrodynamic stability classification framework that explicitly balances the interpolation error of the trained neural-network surrogate with the numerical PDE approximation error arising from the time-dependent finite element discretization of the Navier–Stokes/Boussinesq equations. The strategy should adaptively select new parameter points near the bifurcation boundary to reduce surrogate uncertainty while controlling discretization error in a principled manner.

Background

The paper proposes a two-stage methodology for classifying hydrodynamic stability: (i) generate labeled data by time-dependent finite element simulations of the Navier–Stokes/Boussinesq equations with adaptive time-stepping, and (ii) train a shallow neural network to serve as a surrogate classifier that maps parameter values to bifurcation labels. This approach is applied to several benchmark problems, including a differentially heated cavity exhibiting a Hopf bifurcation where the stability boundary depends on both the Rayleigh and Prandtl numbers.

To improve the fidelity of the surrogate near the bifurcation boundary, the authors note the need for an adaptive strategy that refines sampling where predictions are most uncertain. Crucially, they emphasize that such a strategy should be semi-rigorous by balancing two sources of error: the interpolation (surrogate) error and the PDE discretization error from the underlying simulations. Designing this principled adaptive procedure is identified as future work.