Learned WENO: an advection scheme that learns where to dissipate
esmax · ocean · 8 · An advection scheme that keeps WENO's stencils, convexity and masks and learns only how to weigh the candidates
Abstract¶
The advection scheme is where a coarse ocean model loses information: it smooths fronts, dissipates eddies, and decides whether a run stays stable. In this page, we keep everything that makes WENO safe (flux form, fixed candidate stencils, convex weights, land masks) and learn only one thing, which is how to weigh the candidates at each face. We use the same Gulf Stream QG example as the other pages.
Where it sits. One concrete hybrid (type T2 of the taxonomy), worked through from the basis functions to the training and the adjoint. The full page is a self-contained document with operator boxes (inputs, outputs, shapes and units), figures and code sketches, which opens on its own and links back here.
TL;DR: Fixed stencils, learned weights¶
Every advection scheme is a centred part plus a dissipation. The centred part moves information and the dissipation removes it. Fifth-order upwinding is exactly sixth-order centred plus a hyperdiffusion , with the velocity at the face (m s⁻¹) and the grid spacing (m).
A single learned kernel cannot work. Godunov’s theorem says that a linear scheme which never creates oscillations is at most first order, so the weights must depend on the local field.
WENO already does that, by hand. It blends three small stencils with weights set by how smooth each one sees the field. Learned WENO adds a learned correction to those weights, starts from zero (so it starts as classical WENO5), and is switched off where the field is smooth.
Boundaries use the same mechanism. A candidate that reads a land cell gets a weight of zero. So near a coast only the safe candidates remain, and the stencil narrows on its own.
It is a familiar machine-learning object: soft attention, or a mixture of experts, over a fixed bank of local polynomial experts which are scored by their roughness. The parallels section works each comparison through with the same numbers.
Every weight is smooth in the state, so the adjoint is the true linearisation of the forecast, which is important for 4D-Var.
On the page¶
The problem
Notation
Flux form: conservation for free
Every scheme is a centred part plus a dissipation
Why a single learned kernel cannot win
Classical WENO5, from local basis functions
Learned WENO
Parallels
As operators
Code sketch
Where it goes in the QG model
Training
For data assimilation
What to watch
Test ladder
Where it lives