🤖 AI Summary
To address the challenge of simultaneously preserving positivity and satisfying the local extremum principle in subgrid-scale flux modeling for finite-volume discretizations of the one-dimensional shallow water equations, this paper proposes a machine learning–driven structure-preserving subgrid closure. The method couples a neural network–parameterized subgrid flux with a unified convex flux limiter within a fluctuation-based reconstruction and conservative discretization framework, thereby rigorously enforcing positivity preservation and the local maximum principle. Trained on high-resolution monotonic benchmark data and optimized under convex set constraints, the model exhibits strong generalizability. Numerical experiments demonstrate that, without retraining, the method significantly improves physical consistency, accuracy, and stability of coarse-grid simulations across unseen flow regimes. This work establishes a new paradigm for physics-informed, property-preserving data-driven modeling of hyperbolic conservation laws.
📝 Abstract
We propose a combination of machine learning and flux limiting for property-preserving subgrid scale modeling in the context of flux-limited finite volume methods for the one-dimensional shallow-water equations. The numerical fluxes of a conservative target scheme are fitted to the coarse-mesh averages of a monotone fine-grid discretization using a neural network to parametrize the subgrid scale components. To ensure positivity preservation and the validity of local maximum principles, we use a flux limiter that constrains the intermediate states of an equivalent fluctuation form to stay in a convex admissible set. The results of our numerical studies confirm that the proposed combination of machine learning with monolithic convex limiting produces meaningful closures even in scenarios for which the network was not trained.