arXiv:2608.18019 2026 Architecture 2 ideas extracted · analyzed Sep 1, 2026
The paper's transferable asset is a data-driven construction of directed local Markov kernels whose infinitesimal generator contains a controllable drift term without requiring the underlying velocity field or potential to be known. An ordering function converts weak ordinal information into an asymmetric perturbation of an otherwise isotropic Gaussian diffusion kernel, while row normalization produces a stable stochastic operator. This suggests replacing symmetric graph attention or message passing with an order-tilted local kernel, allowing a network to represent directed transport and diffusion from unordered point clouds. A second opportunity is to compose several such kernels with Strang splitting to model anisotropic transport while controlling composition error.
Use a learned scalar ordering function to turn a symmetric local Gaussian graph kernel into a directed, row-stochastic message-passing operator. The asymmetric tilt lets neighboring nodes communicate preferentially along an inferred dynamical direction, while the Gaussian factor retains locality and diffusion-like smoothing.
Approximate anisotropic diffusion in a neural operator by composing several ordered local propagation steps rather than learning one unrestricted dense attention matrix. Each directional step uses its own ordering function and bandwidth, and symmetric composition reduces the leading splitting error.