Learning Flux Limiters with Kolmogorov Arnold Networks.

The Euler equations describing compressible inviscid flows remain a standard benchmark for assessing numerical methods. Flows that are characterized by shocks and discontinuities pose a challenge for application of high-order numerical schemes, as they cause non-physical oscillations. Reconstruction by Monotonic Upstream centered Scheme for Conservation Laws (MUSCL) mitigates this issue by using flux limiters that reduce the order of the approximation near large changes in gradients. Choosing a flux limiter is typically heuristic and problem-dependent. This work applies a Kolmogorov–Arnold Networks (KANs) in attempt to discover the optimal flux limiter function for the Euler equations. KANs are boasted to include more interpretability advantages over traditional Multilayer Perceptron (MLP) architectures which makes them of keen interest to fluid flow problems. This paper evaluates the learned KANs limiters against classical flux limiters and MLP-based learned limiters, highlighting the difficulties Neural Networks have with learning discontinuous functions.