On the Role of Split Formulations on Aliasing Errors and Entropy Stability of Discontinuous Galerkin Schemes

What the math gives to ML

The paper gives a constructive analysis of how different two-point splittings of a quadratic nonlinearity cancel or amplify collocation aliasing, and shows that the splitting minimizing instantaneous integration error need not be the one that is stable over long time horizons. This is directly relevant to spectral neural operators, Fourier pseudospectral networks, and polynomial neural layers, where pointwise products followed by truncated transforms create the same unresolved-mode aliasing mechanism. The most transferable asset is the entropy-stable skew-symmetric split, especially the (1/3, 2/3) coefficients, which can replace naive pseudospectral quadratic products while retaining discrete energy control. A small ablation over split coefficients can test whether lower aliasing improves rollout stability without the memory and compute cost of full padding-based dealiasing.