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weather-model

numerical weather forecast

ECMWF IFS, ICON

SystemTechniques · 8Primitives · 28 weather-model Halo exchangedistributed · medium: Each rank stores its block plus a border copy of its neighbours' edge cells, refreshed by small messages every step; communication scales with block surface, not volume.›Graph neural networksmachine-learning · medium: Message-passing neural networks over graphs (GCN, GAT, equivariant) for molecules and relationsData assimilation 4dvarnumerics · medium: Fit model initial state to observations by minimizing a cost over a time window with adjoint gradientsEnsemble kalman filternumerics · medium: Kalman update with covariance estimated from an ensemble; localization and inflation for huge statesFinite volume methodnumerics · medium: Conservative cell-averaged discretization with numerical fluxes on structured and unstructured meshesSpherical harmonic transformnumerics · medium: Fast spherical harmonic transforms via FFT in longitude and Legendre in latitudeSemi lagrangian advectionphysics · medium: Trace characteristics back and interpolate; unconditionally stable large time stepsReverse-mode automatic differentiat…ml-training · low: Record the forward computation on a tape, then sweep it backwards once to get the gradient of a scalar loss for all inputs.›L-BFGSnumerics · low: Quasi-Newton minimizer that approximates the inverse Hessian from the last m gradient differences via the two-loop recursion; O(m·n) memory and work per step.›Fast Fourier transformnumerics · low: Computes the DFT in O(n log n) by recursively splitting into even and odd halves; the basis of fast convolution and spectral methods.›Spherical gridsnumerics · low: Lat-lon, cubed-sphere and icosahedral grids for global models; pole and area issuesBit packing and PFORcompression · low: Store blocks of integers with exactly b bits each and patch rare outliers separately (PFOR); the fastest integer codec family, billions of values per second.›Collective communicationml-training · low: The family of group exchanges (broadcast, reduce, all-gather, reduce-scatter, all-to-all) and the latency-plus-bytes cost model that picks ring, tree or direct algorithms by message size and topology.›Floating-point errornumerics · low: How IEEE 754 rounding error accumulates and how to contain it: compensated (Kahan/Neumaier) and pairwise summation, error-free transformations with FMA, ULP comparisons.›Eager rendezvous protocoldistributed · low: MPI point-to-point: eager sends for small messages, rendezvous handshake for large onesRdmanetworking · low: RDMA verbs over InfiniBand/RoCE: queue pairs, memory registration, one-sided opsZero-copy I/Onetworking · low: Move bytes between files, sockets and NICs by passing page references instead of copying through user space: sendfile, splice, MSG_ZEROCOPY, kTLS.›Attention variantsmachine-learning · low: Multi-head, multi-query, grouped-query, sliding-window and latent attention; same softmax(QKᵀ/√d)V core, different trade-offs between quality and KV-cache size.›Embeddingsmachine-learning · low: A learned table mapping discrete ids to dense vectors: gather forward, sparse scatter-add backward, row-wise optimizers to keep huge tables cheap.›Normalization layersmachine-learning · low: Standardize activations per batch, token or group with a learned scale so deep nets train fast and stably; RMSNorm in LLMs, BatchNorm in convnets.›Sparse matrices (CSR)numerics · low: Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.›Conjugate gradientnumerics · low: Krylov iteration for symmetric positive definite Ax = b using one matrix-vector product and two dot products per step; converges in about √κ iterations.›LU and Cholesky factorizationnumerics · low: Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).›+ 1 more techniques+ 12 more primitives

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