Learning second-order TVD flux limiters using differentiable solvers
This work addresses the challenge of designing effective flux limiters for computational fluid dynamics, offering a flexible optimization method that is incremental but improves accuracy for complex flow problems.
The paper tackled the problem of optimizing second-order total variation diminishing (TVD) flux limiters for hyperbolic conservation laws by using differentiable solvers with neural networks, resulting in learned limiters that surpass classical ones in accuracy across problems with shocks and discontinuities, with strong generalizability demonstrated.
This paper presents a data-driven framework for learning optimal second-order total variation diminishing (TVD) flux limiters via differentiable simulations. In our fully differentiable finite volume solvers, the limiter functions are replaced by neural networks. By representing the limiter as a pointwise convex linear combination of the Minmod and Superbee limiters, we enforce both second-order accuracy and TVD constraints at all stages of training. Our approach leverages gradient-based optimization through automatic differentiation, allowing a direct backpropagation of errors from numerical solutions to the limiter parameters. We demonstrate the effectiveness of this method on various hyperbolic conservation laws, including the linear advection equation, the Burgers' equation, and the one-dimensional Euler equations. Remarkably, a limiter trained solely on linear advection exhibits strong generalizability, surpassing the accuracy of most classical flux limiters across a range of problems with shocks and discontinuities. The learned flux limiters can be readily integrated into existing computational fluid dynamics codes, and the proposed methodology also offers a flexible pathway to systematically develop and optimize flux limiters for complex flow problems.