Attention Is All You Need (to Avoid Spurious Oscillations)

Abstract:Can attention move a shock across several cells in one update without breaking it? We develop a conservative, fixed grid finite-volume scheme in which a CFL-conditioned attention flux selects upstream information according to the transport required by the current time step. One-dimensional inviscid Burgers transport is used as the central mechanism test: the same learned flux remains reliable in the conventional small-step regime and, with a time step four times larger, preserves sharp shocks while using one stage per update. A standard fifth-order WENO scheme with third-order strong-stability-preserving Runge-Kutta time integration (WENO-5+SSP-RK3) is included alongside controlled Forward Euler comparisons to separate flux selection from time integration. The learned attention shifts upstream with the local transport reach and becomes more selective near shocks; inference-time interventions and retrained ablations show that transport-scale information and state-dependent selection contribute directly to performance. Directional two-dimensional scalar Burgers transport and the one-dimensional shallow-water system then test whether the conservation-scale-selection principle transfers beyond the original scalar setting. The results support attention as a learnable information stencil for conservative large-step shock transport, while identifying finite candidate reach and problem-dependent robustness as the present limits.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Fluid Dynamics (physics.flu-dyn)

arXiv:2609.13531 [cs.LG]

(or arXiv:2609.13531v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.13531 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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