Attention Is All You Need (to Avoid Spurious Oscillations)

2026-09-11

AI Summary

本文通过开发一种基于CFL条件的注意力通量选择方法，解决了在大时间步长下保持尖锐激波的问题。

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.

Problem

• attention
• shock
• flux
• transport
• conservation

Innovation

• attention flux
• conservative finite-volume scheme
• shock transport
• large time step
• information stencil