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Published equation contexts

hv(k+1)=ϕ(hv(k),⨁u∈N(v)ψ(hv(k),hu(k),euv))h_v^{(k+1)} = \phi\left(h_v^{(k)}, \bigoplus_{u \in \mathcal{N}(v)} \psi\left(h_v^{(k)}, h_u^{(k)}, e_{uv}\right)\right)

Why this formula appears here

GraphCast , described by Remi Lam and colleagues at Google DeepMind in a 2023 paper published in Science, replaces the equation-integration step with a trained graph neural network operating on an encoder–processor–decoder structure. Weather variables at roughly one million points on a 0.25-degree global grid are first encoded onto a spherical mesh; a processor network then updates each mesh node’s representation by passing messages between it and its neighbours, repeated across the mesh’s structure so that information can propagate across the globe within a single forward pass; a decoder then reads the updated mesh back out to the original grid to produce the forecast. In general form, a…

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hv(k+1)h_v^{(k+1)}

Symbol h_v^(k+1)

h_v^(k+1) is part of the quantity the equation computes from the expression on the right.

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ϕ\phi

Symbol phi

learned functions — the same general operation GNoME’s crystal-graph network applies to atoms and bonds, applied here to points on a sphere and the physical processes connecting them.

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Published contexts (1)

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hv(k+1)=ϕ(hv(k),⨁u∈N(v)ψ(hv(k),hu(k),euv)),h_v^{(k+1)} = \phi\left(h_v^{(k)}, \bigoplus_{u \in \mathcal{N}(v)} \psi\left(h_v^{(k)}, h_u^{(k)}, e_{uv}\right)\right),

Equation 9 · AI for Science

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

GraphCast , described by Remi Lam and colleagues at Google DeepMind in a 2023 paper published in Science, replaces the equation-integration step with a trained graph neural network operating on an encoder–processor–decoder structure. Weather variables at roughly one million points on a 0.25-degree global grid are first encoded onto a spherical mesh; a processor network then updates each mesh node’s representation by passing messages between it and its neighbours, repeated across the mesh’s structure so that information can propagate across the globe within a single forward pass; a decoder then reads the updated mesh back out to the original grid to produce the forecast. In general form, a…

Meanings in this article

  • ϕ\phi: learned functions — the same general operation GNoME’s crystal-graph network applies to atoms and bonds, applied here to points on a sphere and the physical processes connecting them.
  • vv: the message-passing update for a node.
  • ψ\psi: a learned function.
  • euve_{uv}: the edge connecting it to neighbour u.
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