November 2026 · Politecnico di Milano

Data-driven weather forecasts: assimilation, reanalysis and AI models

Atmosphere and Ocean Dynamics — Lecture 10

Prof. Dr. Pedro da Silva Peixoto

Applied Mathematics
Instituto de Matemática e Estatística
Universidade de São Paulo

Visiting Professor at MOX Lab
Politecnico di Milano

Warm-up

1. Why did spectral models dominate global forecasting for forty years, and why are new models leaving them?
2. What sets the time step of an explicit model? And of a semi-implicit semi-Lagrangian one?
3. Lorenz (1963): why are forecasts run as ensembles?
4. Name two dynamical cores of today and their grids.

The models of today, from memory: answer on paper first, 3 minutes. Today: the box of the forecast chain before the model (the initial state) and the newest box (a learned model).

Answers: (1) spherical harmonics: no pole problem, exact derivatives, an easy semi-implicit solve; but the Legendre transforms cost more than the grid points and need global communication: they do not scale on 10^5–106 cores. (2) The fastest wave (CFL, C <= 1 or so); semi-implicit semi-Lagrangian: accuracy only (the departure points), steps of minutes at 9 km. (3) Chaos: small initial errors grow; two weeks of predictability; an ensemble predicts the uncertainty. (4) e.g. IFS (octahedral reduced Gaussian, spectral), ICON (icosahedral triangles, finite volumes), FV3 (cubed sphere), MPAS/MONAN (Voronoi), LFRic (cubed sphere, mixed finite elements).

Where we are and where we are going

Previous knowledge

• Spectral models: no pole problem, but transforms that do not scale; hence quasi-uniform grids
• Explicit steps: the fastest wave (CFL); semi-implicit semi-Lagrangian: steps of minutes
• Chaos (Lorenz 1963): about two weeks of predictability; ensembles
• Dynamical cores of today: IFS (spectral), ICON, FV3, MPAS/MONAN (finite volumes), LFRic (finite elements)
• Least squares: minimizing a cost function; a neural network as a fitted function

Where we are going

1. The forecast chain; observations; data assimilation
2. Reanalysis: ERA5
3. Data-driven forecasts: what is learned
4. Inside a learned model; an example
5. Do they work? What they cannot do
6. The newest: end to end; hybrids
7. A tiny AI weather model

The initial state comes from an inverse problem; the newest models learn the step from it

(Bauer et al. 2015; Hersbach et al. 2020; Lam et al. 2023)

Two minutes. Left: what we take as known (lecture 9 and the methods of lectures 5–8; the warm-up checked it). Right: the road of the day; each item links to its first slide.

From 1950 to today: the forecast chain

• 1950: one equation, one layer, 736 km, North America
• Today: the whole globe, ~10 km, ~100 levels, ocean, land, ice, an ensemble of forecasts

• Skill: about one day per decade: a 6-day forecast today is as good as a 5-day forecast ten years ago

But: where does the initial state come from?

Bjerknes’ programme, industrialized: observe, assimilate, integrate the laws, many times over

(Bauer et al. 2015; Kalnay 2003)

Ask first: which box of the chain did lectures 2 to 8 build? (The dynamical core.) The skill gain comes from better models, better observations (satellites), better data assimilation, and ensembles.

Observations: the global observing system

• Every day ECMWF receives about 800 million observations; about 60 million pass quality control
• Satellites (about 90 instruments) give most of them: radiances for temperature and humidity, GNSS radio occultation, winds from tracked clouds and, over the sea, scatterometers
• In situ: stations, radiosondes, aircraft, ships, buoys, Argo floats: direct and local; they anchor the analysis and expose satellite biases

But: irregular, mostly indirect, and short of winds. How do they become a state on the grid?

Satellites give the numbers, in situ observations the anchor; neither is a model state

(Modigliani et al. 2026) · Figure: after wmo.int: Global Observing System

Own sketch after WMO’s diagram. Satellites cannot measure temperature directly: they measure radiances (spectroscopy). Ask first: which of these measures the wind? Next: from observations to an initial state.

Data assimilation: an inverse problem

• The observations: irregular in space and time, mostly indirect (radiances, not winds)
• 4D-Var: find the initial state whose model trajectory best fits the observations and the last forecast (the background)
• It needs the model (and its adjoint); centres now mix it with ensembles (hybrid)
• Much of the skill gained in 40 years came from better analyses

But: how is the best fit found, with about \(10^9\) unknowns?

The initial state is the solution of an inverse problem, with the model inside

(Kalnay 2003, Sect. 5.6)

Ask first: why not simply interpolate the observations onto the grid? Next: the cost function and how it is minimized.

4D-Var: a minimization problem

\[ J[x(t_0)] = \underbrace{\tfrac12\,[x(t_0)-x^b]^{\mathsf T}\mathbf B_0^{-1}[x(t_0)-x^b]}_{J_b\text{: misfit to the background}} + \underbrace{\tfrac12\sum_{i=0}^{N}[H(x_i)-y^o_i]^{\mathsf T}\mathbf R_i^{-1}[H(x_i)-y^o_i]}_{J_o\text{: misfit to the observations}}, \qquad x_i = M(t_0, t_i)[x(t_0)] \] \(t_0,\dots,t_N\): observation times in the window; \(x(t_0)\): initial state (the unknown); \(x_i = x(t_i)\); \(x^b\): background (last forecast); \(y^o_i\): all observations at \(t_i\); \(H\): observation operator (state → measured, e.g. radiances); \(M(t_0, t_i)\): the model run from \(t_0\) to \(t_i\); \(\mathbf B_0\), \(\mathbf R_i\): error covariances

• Gradient: the model forward, its adjoint \(L^{\mathsf T}\) backward (\(L\): the linearized model); a quasi-Newton step; repeat
• Strong constraint: the analysis is a model trajectory, balanced; the model is taken as perfect
• Incremental: inner loops at lower resolution

But: a reanalysis needs decades of such analyses. With which model?

4D-Var: the model trajectory that best fits the last forecast and the observations

(Kalnay 2003, Sect. 5.6.3, eqs. 5.6.14–5.6.20, Fig. 5.6.2)

Figure: own drawing after Kalnay’s Fig. 5.6.2 (12-h window, observations every 3 h). Without the B₀ term the problem is underdetermined: far fewer observations than unknowns. The gradient costs a few model runs, whatever the number of unknowns.

A reanalysis: decades of the past with one model

• An analysis is the initial state of one forecast; a reanalysis repeats the analyses of the past, every day of decades, with one frozen model and one assimilation system

• ERA5 (ECMWF, Copernicus): 1940 to today, hourly, \(0.25^\circ\) (about 31 km), 137 levels, about 5 PB; 4D-Var with the IFS of 2016 (cycle 41r2). Few observations before the satellites (1979)

Download it: free (CC BY 4.0) from the Copernicus Climate Data Store, cds.climate.copernicus.eu: ERA5 on single levels · ERA5 on pressure levels; from Python with cdsapi

The best estimate of the past atmosphere: one model and all the observations, together

(Hersbach et al. 2020; Bauer et al. 2015) · Data: (Copernicus Climate Change Service 2018b, 2018a)

Ask first: why not use the operational analyses of the past? (Each was made with the model of its day: improvements would look like climate change.) ERA5 is free and openly licensed (Copernicus Climate Change Service): the public good that made data-driven forecasting possible.

ERA5 as training data

• Circular: made by the model being replaced: “AI beats IFS” partly means “AI matches a 2016 IFS”
• Not homogeneous: few observations before the satellite era; the constellation keeps growing

• Not available in real time: ERA5 uses observations up to about 9 h after each time; an operational analysis gets about 3 h: trained on one, run on the other
• Training data = truth: AI models are scored against ERA5, the IFS against its own analysis

A learned model is only as good as the reanalysis it learns from, and that reanalysis is physics

(Hersbach et al. 2020; Lam et al. 2023)

Pedro’s BRICS talk adds: ERA5 analyses use observations from hours after the analysis time (up to about 9 h); an operational analysis must be ready sooner (about 3 h): trained on one, run on the other.

A new revolution: data-driven forecasts

For a century, since Richardson, every forecast discretized the equations of motion. Since 2022 (first preprints on the timeline), models learned from data rival the best physics-based forecasts; on 25 February 2025 ECMWF made its AI model, AIFS, operational.

But: what exactly do they learn?

In three years, learned models went from a preprint to an operational forecast

(Lam et al. 2023; Bi et al. 2023; Lang et al. 2024)

Ask first: what would you need to train a model that forecasts the weather? Keisler (2022) built the first graph-network forecast; FourCastNet: NVIDIA; Pangu: Huawei; GraphCast and GenCast: Google DeepMind; NeuralGCM: Google with ECMWF and MIT; Aurora: Microsoft. Dates: arXiv preprints; the journal papers followed 6–12 months later.

Seven models in three years

Year	Model	Who	What was new
2022	Keisler GNN	Keisler (arXiv)	a graph neural network on a spherical mesh, encoder–processor–decoder; IFS-like skill at 1°
2022	FourCastNet	NVIDIA, LBNL, Caltech	Fourier neural operator; the first at the IFS’s own 0.25°
2022	Pangu-Weather	Huawei	3D transformer; the first to claim to beat the IFS deterministic forecast
2022	GraphCast	Google DeepMind	a graph network on an icosahedral multi-mesh; better than the IFS on 90% of 1380 targets
2023	GenCast	Google DeepMind	diffusion: samples possible futures; better than the ECMWF ensemble on 97% of 1320 targets
2023	NeuralGCM	Google, ECMWF, MIT	hybrid: a differentiable spectral core and learned physics
2025	AIFS	ECMWF	graph transformer; the first operational data-driven model at a major centre (open)

Each year a new kind of network: graph, Fourier operator, transformer, diffusion, hybrid

(Keisler 2022; Pathak et al. 2022; Bi et al. 2023; Lam et al. 2023; Price et al. 2025; Kochkov et al. 2024; Lang et al. 2024)

Years: first preprints; the journal papers followed 6–12 months later. AIFS operational on 25 February 2025 (deterministic); its ensemble later in 2025. After Pedro’s BRICS 2026 talk.

What is learned, and from what

\[x^{n+1} = \Phi_\theta\!\left(x^{n}, x^{n-1}\right), \qquad \theta = \arg\min_\theta \sum_n \bigl\|\Phi_\theta(x^n, x^{n-1}) - x^{n+1}\bigr\|^2\]

\(x^n\): the whole state at time \(n\) (all variables, all points); \(\Phi_\theta\): the network, with parameters \(\theta\)

• One 6-hour step, fitted to pairs of states; a 10-day forecast composes it 40 times
• Trained on ERA5 (1940–today, 0.25°, 137 levels, hourly), made by the physics model and 4D-Var

But: what does \(\Phi_\theta\) look like inside?

One box of the chain replaced: the model is learned, the initial state still comes from physics

(Hersbach et al. 2020; Lam et al. 2023)

Figure: a schematic, the small maps are synthetic waves, not data. Phi_theta is not derived from the equations: it is fitted. x: the whole state; for GraphCast 227 variables (5 at the surface, 6 on 37 levels) at 10^6 points, 2.4e8 numbers. The training data is made by the thing being replaced: “AI beats IFS” partly means “AI matches the IFS of ERA5”.

Five families of learned models

• What disappears: the grid as an approximation, the CFL time-step limit, every conservation law
• What survives: the state vector, and the initial state from 4D-Var

Different networks, one task: a map from the state now to the state 6 hours later

(Pathak et al. 2022; Lam et al. 2023; Bi et al. 2023; Chen et al. 2023; Lang et al. 2024; Price et al. 2025; Kochkov et al. 2024)

After the BRICS talk (“What exactly is being learned?”). AIFS: a graph neural network encoder and decoder with a sliding-window transformer processor (Lang et al. 2024). The CFL limit disappears as a limit on the step; the domain of dependence does not (next slides and the example).

Inside a learned model: GraphCast

Lam et al. (2023), Fig. 1, CC BY 4.0

• (a–c) State → next state → forecast: 227 variables on the 0.25° grid (5 at the surface, 6 on 37 levels); GraphCast applied again and again
• (d) Encoder: the grid → a mesh of icosahedra, refined 6 times
• (e) Processor: 16 rounds of messages along the edges
• (f) Decoder: back to the grid: the change in 6 hours
• (g) Multi-mesh: the edges of all 7 meshes at once: near and far

36.7 million parameters; trained on ERA5 1979–2017, rollouts of 1 to 12 steps; about 4 weeks on 32 TPUs. A 10-day forecast in under a minute on one TPU (the IFS: about an hour on a supercomputer)

But: CFL forbids explicit steps of 6 hours on a 28 km grid. How can it?

Grids are back: a learned operator on an icosahedral mesh, as in finite volumes

(Lam et al. 2023; Keisler 2022) · Figure: arxiv.org/abs/2212.12794

Message passing = a learned stencil, applied 16 times. The multi-mesh lets information cross thousands of km in one step. Other families: neural operators (FourCastNet), transformers (Pangu), diffusion (GenCast).

One box replaced

• Still needs physics to run: the initial state comes from 4D-Var, and 4D-Var needs a model (and its adjoint)
• Training data needs a physics model: ERA5 is IFS + 4D-Var

• Resolution is not the metric: AIFS at 28 km beats the IFS at 9 km on many scores
• The physics–dynamics split is gone: dynamics and parameterizations are learned together; neither can be inspected or swapped

The learned step replaces the model; observations, assimilation and reanalysis remain

(Lang et al. 2024; Hersbach et al. 2020; Bauer et al. 2015)

Physics: about 1200 steps of 12 minutes for 10 days; AI: 40 steps of 6 hours. After Pedro’s BRICS talk (“Comparing frameworks”).

Example: how can an AI model take a 6-hour step?

• How far does a gravity wave travel in 6 hours at 300 m/s? How many explicit steps would a 28 km grid need for those 6 hours (CFL)?
• The learned model takes one step: how far apart are the two ends of an edge of the icosahedron (\(63.4^\circ\) of arc), the coarsest edges of GraphCast’s mesh?
• And for a step of 24 hours, with the jet stream (100 m/s) and the gravity waves?

(Lam et al. 2023; Courant et al. 1928)

Worked out with the class (an example, not a notebook exercise). Answers: 300 m/s × 6 h = 6480 km; dt <= 28 km / 300 m/s = 93 s, about 230 steps. Edge: 63.4 degrees × 6371 km = 7050 km. The multi-mesh has the 40 962 nodes of M6 and the edges of all levels: a node has the edges of the level where it first appears and finer, so only the 12 icosahedron vertices have 7000 km edges (hubs); a typical node (level 6, ~110 km edges) goes up through coarser hubs in a few hops, crosses, and comes down: a few of the 16 rounds. 24 h: 100 m/s × 24 h = 8640 km (two hops of 7050 km); 300 m/s × 24 h = 25 900 km > 20 000 km (half the circumference): the whole globe, a global receptive field (3 hops reach anywhere; GraphCast’s processor makes 16).

What the example shows

• In 6 hours a gravity wave travels 6480 km; an explicit model on a 28 km grid needs about 230 steps of 93 s (CFL)
• One edge of the icosahedron, \(63.4^\circ\) of arc: 7050 km, longer than the domain of dependence of a 6-hour step. Only 12 hub nodes have such edges; a typical node reaches them in a few hops (edges of 110, 220, 450 … km), well within the 16 rounds of message passing
• A 24-hour step: 8640 km for the jet, 25 900 km for gravity waves, more than half the globe: a global receptive field

A learned step is a stencil: it must reach as far as the fastest signal travels in one step

(Lam et al. 2023; Courant et al. 1928)

Same idea as semi-Lagrangian: put the domain of dependence inside the stencil. Three hops reach anywhere; GraphCast’s processor makes 16.

Do they work? Yes, mostly

GraphCast (blue) and ECMWF’s HRES (black), 500 hPa geopotential, forecasts of 2018: error (RMSE, m²/s²) and anomaly correlation (ACC) against lead time; shading: 95% intervals. Lam et al. (2023), Fig. 2a, c, CC BY 4.0

What works

• GraphCast beat HRES on 90% of 1380 targets; GenCast (diffusion) beat the ECMWF ensemble on 97% of 1320
• AIFS, at 28 km, beats 9 km physics-based models on many scores (2026); about 1000 times less energy per forecast

What does not, yet

• Blurring: trained on the mean-square error, they forecast the average of the possible futures
• Extremes are rare in the data; no conservation laws by construction
• They need 4D-Var and a reanalysis; the climate of the training years

Excellent average forecasts; extremes, conservation and the physics they rest on remain open

(Lam et al. 2023; Price et al. 2025; Rasp et al. 2024; Ben Bouallègue et al. 2024; Moldovan et al. 2026) · AIFS: ecmwf.int · Figure: arxiv.org/abs/2212.12794

Scores: AI models against ERA5, the physics models against their own analyses. “Resolution isn’t the metric” (BRICS talk): the coarser AIFS wins on many scores.

What AI models cannot do, yet

1. Extremes: record-breaking heat, cold and wind are forecast better by physics-based models; extremes are rare in the data, and the mean-square error pulls towards the mean
2. Blurring: the best mean-square forecast is the average of the possible futures; fronts and storm centres are smoothed
3. No conservation laws: nothing enforces mass or energy; long runs drift
4. Circular data: ERA5 is a physics product, not available in real time
5. A changing climate: trained on the past (GraphCast: 1979–2017), used in the climate of today

Blurring in the tiny AI model: a forecast trained on the mean-square error, 5 days ahead, loses the amplitude of the waves

Excellent on average; the rare, the extreme and the conserved still need physics models

(Zhang et al. 2026; Price et al. 2025; Kochkov et al. 2024)

Zhang et al. (2026, Science Advances): on record-breaking extremes, physics-based models (IFS-HRES) outperform the AI models. Diffusion models (GenCast) address blurring: they sample, not average. After Pedro’s BRICS talk (“What AI cannot do yet”).

The newest: learning the analysis too

4D-Var: the analysis is the model trajectory that best fits the background and the observations

• Every model so far starts from a physics-based analysis
• End to end: learn the map from the raw observations to the forecast, with no 4D-Var at run time: Aardvark Weather (Cambridge, Turing Institute, 2025), FuXi-Weather (Fudan, 2025)
• Still trained on reanalyses, and still behind the best operational systems on many scores

But: without a physics model anywhere, who checks the learned atmosphere?

The last box of the chain is being learned: from observations straight to the forecast

(Allen et al. 2025; Sun et al. 2025)

Aardvark: an encoder from observations to a gridded state, a processor, decoders to forecasts; FuXi-Weather: data assimilation and forecast cycled by learned models (Nature Communications 2025).

Where it is going: hybrids, and the world

• Hybrid: NeuralGCM keeps a differentiable spectral core and learns the unresolved physics: the equations come back

• Foundation models: Aurora, one network fine-tuned to weather, air quality, ocean waves
• Ensembles: diffusion models (GenCast, FuXi-ENS) sample possible futures instead of averaging them

The likely future: physics and learning together, not one replacing the other

(Kochkov et al. 2024; Bodnar et al. 2025; Price et al. 2025; Zhong et al. 2025)

Marchuk (AIRI, 2026) has no peer-reviewed paper yet (Pedro’s BRICS talk). Ask first: which box of the chain does each column replace?

Notebook A tiny AI weather model

Notebook 10a (open in Colab · download): train and run an AI weather model on a laptop, in numpy.

Lorenz-96: an “atmosphere” of 40 numbers around a latitude circle

1. Lorenz-96: an “atmosphere” of 40 numbers, chaotic
2. A “reanalysis”: 11 years of 6-hourly states
3. Learn the 6-hour step: linear, then a small network with a local stencil
4. Forecast by composing the step ❗ 4.1
5. Why a learned forecast blurs

Predict, run, explain: which forecasts better, the linear model or the small network?

Train and run an AI weather model on a laptop, in numpy

(Lorenz 1996; Lam et al. 2023)

In class, 25 min: sections 1 to 5; the optional exercises at home. About one minute in total. 0.05 time units = “6 hours” (Lorenz 1996).

What the notebook showed

• The local network (1300 parameters) beats the linear model (1640): it learns the dynamics, a stencil like a finite difference
• Its stencil must cover where information comes from in one step: CFL, learned
• Nothing beats the true model from a slightly wrong start for long: chaos

A learned step is a numerical method whose coefficients were fitted instead of derived

(Lorenz 1996)

The linear model decays to the mean: its error stops at the climatological spread. Exercises: stencil width 3 fails, 5 works (❗ 4.1, to hand in); a 1-day step needs a wider stencil (🧩 4.2); ensembles (optional).

Key points

1. The initial state: data assimilation, an inverse problem with the model inside (4D-Var)
2. Reanalysis: decades of analyses with one frozen model; ERA5 is what AI models learn from
3. Learned models: a 6-hour step fitted to ERA5; it must see as far as the fastest signal goes: CFL, learned
4. One box replaced: the observations, the assimilation and the reanalysis remain physics
5. Do they work? Excellent average forecasts; extremes, conservation and a changing climate remain open
6. The newest: learning the analysis too (end to end); hybrids keep the equations

Notebook 10a (open in Colab · download): an AI emulator to train and test.

AI replaces one box of the chain: the observations and the physics around it remain

Reading: (Bauer et al. 2015; Hersbach et al. 2020; Lam et al. 2023; Rasp et al. 2024; Zhang et al. 2026)

The last lecture’s key points. Next: the course in one chain.

The course in one chain

Topic	What you can now do
History	tell the story from Halley to the 1950 forecast, and why rotation matters
Laws	derive the equations, reduce them to shallow water, find their waves and balances
Vertical	treat the vertical: buoyancy, internal waves, convection, mountain waves
Grids	discretize them with finite differences and volumes; stability, CFL, conservation
Spectral, FE	spectral methods and finite elements: the cores of IFS and LFRic
Models	follow the models from 1950 to today: grids, the vertical, the tests
AI	read the chain of a modern forecast and what learned models change in it

Thank you! The problem is old; the tools are yours now

(Bjerknes 1904; Bauer et al. 2015)

The rows are the lectures in order: 1, 2–3, 4, 5–6, 7–8, 9, 10.

Backup slides

Slides kept for reference, not shown in class.

AI forecasts in the BRICS countries

China

• Pangu-Weather (Huawei), FengWu (Shanghai AI Laboratory), FuXi (Fudan): 10–15-day forecasts
• FuXi-ENS: learned ensembles; FuXi-Weather: from observations

Russia

• “Marchuk” (AIRI, 2026): flow matching in a latent space, after LaDCast; 276 million parameters

India

• BharatBench, IndiaWeatherBench: open regional data and baselines; nowcasting at IMD

Brazil, South Africa

• GraphCast evaluated over Brazil (2026); plans with MONAN
• South Africa: the Met Office’s Unified Model and its partnership

Good models need good reanalyses and benchmarks: built by those who own the observations

(Chen et al. 2023; Zhong et al. 2025; Nguyen et al. 2025; Rowell and Kupssinskü 2026) · Marchuk: habr.com: AIRI

From Pedro’s BRICS 2026 talk. Also: Bi et al. (2023, Pangu), Chen et al. (2023, FengWu), Zhuang & Duraisamy (2025, LaDCast), Choudhury et al. (2024, BharatBench). Marchuk has no peer-reviewed paper yet (an AIRI announcement). Monsoon, Amazon, Siberia: regional benchmarks only the countries that own the observations can build well.