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D-LEGOS / Adrian Dunkley
Reconstructing
Jamaica’s atmosphere
Our retained weather source represents roughly 56 × 66 km at Jamaica. A farm needs a much closer view. This thesis asks how local models can reconstruct detail between sparse observations.
POWER / MERRA-2 meteorology: 0.5° × 0.625°. Kilometres calculated at 18° N.
Read the research01 / The information gap
The source grid is wider than the landscape we need
Satellites, surface instruments and atmospheric models contribute different kinds of evidence. The POWER meteorological data used here are hourly grid averages, with approximately 56 km north–south and 66 km east–west support near Jamaica.
56 × 66 km source support → 1 km local estimates → sub-kilometre research components.
Where these numbers come from ↗One large-scale value covers many places
This outlined box is chosen to enclose Jamaica. A coarse field cannot directly tell us the separate conditions in a coastal settlement, an upland farm and a mountain valley.
Illustrative parent: 317 × 167 km.
Actual POWER meteorological support: 0.5° × 0.625°.
Finer grids require more work
In a wave-speed-limited explicit model, more cells also need more time steps. This idealised three-dimensional example refines every spatial direction and reduces the time step accordingly.
Published case: about 1 month on 600 CPU cores for 30 simulated years at 12 km over the contiguous US. Wang et al., 2021 ↗
The control counts idealised work. The published runtime is case-specific; D-LEGOS has no demonstrated speed-up here.
02 / Downscaling
Estimate the unresolved structure
Statistical downscaling relates large-scale fields to local evidence. The finer field is an estimate whose assumptions and errors still need testing.
Constructed example; geographic terrain does not make the temperature field an observation.
03 / Local evidence
On the ground, every station constrains one location
Follow the pulses from the thermometer, rain collector and anemometer. Instruments provide time series at sampled sites. The valleys between them remain only partly constrained.
The station placements, readings and transmission pulses are constructed. They illustrate sparse sampling over the real elevation surface.
04 / Introducing D-LEGOS
Give a small piece of atmosphere its own model
Begin with one local volume above the terrain. Its state describes conditions inside; its faces connect it to neighbouring volumes. D-LEGOS proposes joining these local models to reconstruct a larger atmospheric field.
Each component carries a local field, a time interval and shared boundary exchanges.
How a component is defined ↗05 / Physics-informed learning
Use observations and equations together
A physics-informed neural network (PINN) represents a continuous local field. Training penalises disagreement with observations, governing equations and boundary conditions. Sparse measurements anchor the field; equations constrain its behaviour between samples.
The separated field layers show exact, constructed heat-equation solutions. They explain a physical constraint; no atmospheric PINN is trained in this view.
Read the loss and field definitions ↗06 / Below one kilometre
Refine the volume as you move closer
The grid now divides from 1 km to 500 m and 250 m across the same 8 km terrain patch. Air trajectories follow a prescribed transport field; rain and evaporation cross the selected volume’s faces.
Sub-kilometre spacing is a diagram setting. The water balance is a constructed box calculation; the air paths are prescribed.
07 / The space–time component
A climate cube has a history
Follow one volume from tₙ to tₙ₊₁. The linked cubes are a rotating projection of a four-dimensional hypercube: three spatial coordinates plus time. Air and water cross spatial faces; the temporal faces hold the starting and ending states.
The tesseract is an explanatory projection. The atmosphere has three spatial dimensions; the fourth coordinate represents time.
Represent local volumes through time
D-LEGOS proposes local fields over space–time blocks. Each component has a state, an observation window and interfaces to its neighbours. Separating the blocks makes those relationships inspectable.
A shared transfer enters two budgets
Watch heat move from warmer to cooler cells. One interface flux leaves one cell and enters its neighbour. The two signs cancel when the closed system’s energy is added.
An executed twelve-cell thermal example, separate from a full atmospheric model. Particle direction follows the calculated flux; particle speed and size are schematic.
Learn a rule inside the physical update
The archived neural network estimates a positive interface conductance. The shared-flux update does the accounting. In the wider proposal, local neural fields would also be constrained by observations and atmospheric equations.
These weights were trained on constructed heat fields. This animation is not training or validating a moist atmospheric PINN.
Inspect the executed experiment ↗Test the detail against withheld evidence
The intended output is a reconstruction with physical budgets and stated uncertainty. Its value must be measured against independent observations and matched baselines, including the cost of producing it.
Real Blue Mountain terrain.
One coarse POWER value repeated in the smaller display cells.
Inspect the heat example
The research question
What can physics recover when observations are sparse?
My thesis asks when local models coupled through physical exchanges can improve atmospheric reconstruction over Jamaica. The problem combines limited local observations, imperfect large-scale background information and the cost of resolving finer spatial and temporal behaviour.
The proposed D-LEGOS framework connects local neural representations through shared interfaces and explicit budgets. Its usefulness depends on tests against withheld information at matched data and computational budgets.
Read the theoretical formulation ↗The information problem
Terrain detail is not weather detail
The mountain surface in the animation is sampled from the retained Mapzen/Tilezen elevation bundle for an 8 × 8 km Blue Mountain domain. The display refines from a 250 m mesh to the retained 125 m sample spacing as the camera approaches. It shows real terrain structure; it supplies no new atmospheric measurements.
The archived NASA POWER series covers 1–3 July 2024 UTC. Its meteorological estimates derive from MERRA-2 and represent coarse spatial support, approximately 0.5° latitude by 0.625° longitude. Repeating a parent rainfall estimate inside smaller display cells does not turn it into separate local rain measurements. [1]
A weather station supplies a different kind of support: a local instrument, height, exposure, sampling interval and uncertainty. Real station records and their quality controls would be required to calibrate and evaluate a local reconstruction. The station-like markers in this animation illustrate that need; they are not Jamaica’s station inventory.
Geographic surface structure from a retained elevation source.
Archived regional estimates, not new 250 m weather observations.
Inside one D-LEGOS component
From sparse readings to a local field
In the proposed atmospheric method, a neural field maps position and time, (x, y, z, t), to local atmospheric variables. Observations enter through an operator that matches their spatial footprint and sampling time. A station constrains a sampled location; a coarse reanalysis value constrains an aggregate. Treating either as a complete fine-scale field would add unsupported information.
The data term measures disagreement with available evidence. The physics term measures residuals of the chosen equations. Boundary terms impose the specified initial and boundary conditions; interface terms coordinate adjacent components. The λ coefficients set the relative weights. The original PINN formulation combines data and equation residuals in this way. Minimising these penalties does not by itself prove accurate reconstruction or exact conservation.
The animated heat surface uses u(x,z,τ) = exp(−2κπ²τ) sin(πx) sin(πz), with κ = 0.1 and normalised coordinates x,z ∈ [0,1]. It satisfies ∂τu = κ(∂xxu + ∂zzu) and zero boundary values. Its colours and sample markers are constructed reference values, rather than predictions from a trained PINN.
At the closer level, the selected 250 m component is enlarged for inspection. An illustrative water column starts with 2 kg/m² and follows dW/dt = P − E − Q, where P = 0.02, E = 0.006 and Q = 0.004 kg/m²/s. The displayed storage and residual follow this stated box budget over a repeating 30-second interval. Air trajectories are a separate prescribed, divergence-free horizontal field. Neither demonstration solves coupled moist atmospheric dynamics.
A space–time block is Ωᵢ × [tₙ,tₙ₊₁]. The tesseract projection exposes its two temporal faces and spatial connections. The later twelve-cell example executes a learned conductance inside a shared thermal update; it is a completed component test. Applying local neural fields and coupled moisture budgets to an atmospheric reconstruction remains proposed. Read the formulation ↗
The proposed method
Local fields, connected interfaces
A D-LEGOS space–time block is a spatial control volume Ωᵢ followed through a finite interval. Local neural fields would represent resolved variables within these blocks. Observations constrain the state through an observation operator; differential equations and boundary conditions constrain its evolution. The formulation in the linked working paper remains a proposal for full moist reconstruction. [2]
A shared outward transfer enters neighbouring budgets with opposite orientations. That bookkeeping can enforce cancellation of an internal exchange. It cannot establish that the learned transfer matches the atmosphere, that missing processes are represented, or that the reconstructed state is accurate.
U is a cell’s extensive stored amount; Φ is a transfer integrated over one face and the time interval; S is the integrated source. Spatial scales and physical time keep their own units. The tesseract analogy identifies the spatial volume crossed with time; it does not imply a physical rotation between seconds and metres.
Where emulators and PINNs fit +
An emulator approximates a model relationship after training. A physics-informed neural network adds differential-equation, initial-condition or boundary-condition terms to a learning objective. Neither a small network nor a low sampled residual automatically guarantees faster end-to-end inference, exact conservation or accurate climate behaviour. The thesis needs to compare error and computational cost against appropriate numerical and statistical baselines.
The animated 3–8–1 network is the narrower, already executed conductance example. It receives three constructed edge descriptors, uses eight tanh units and maps one output through softplus to obtain a positive G. Node colour shows the activation for the selected interface. It is an interface closure, not a trained atmospheric PINN. The separate laboratory below contains an inspectable two-feature physics-informed objective.
Evidence and next tests
Separate what ran from what is proposed
| Component | Current evidence | What it does not establish |
|---|---|---|
| Geographic context | Retained terrain and historical POWER records. | Fine-resolution atmospheric truth. |
| Shared heat exchange | Executed closed twelve-cell graph with archived learned conductances. | Skill on real meteorology or unseen geography. |
| Statistical example | Constructed regression pairs and exact parent-mean correction. | Accuracy of the inserted local detail. |
| Jamaican rainfall component | Recorded retrospective CHIRPS tests: the hard constraint improved a matched learner; seasonal allocation remained stronger in the listed tests. | Independent fine-scale rainfall truth or a full atmospheric reconstruction. |
| Jamaican temperature component | Recorded ERA5-target tests: a constrained ensemble improved on an autoregressive baseline in 2026, while TimesFM 2.5 remained more accurate. | Uniform superiority of the physics-informed correction. |
| Full moist reconstruction | Proposed state, interfaces, objectives and evaluation programme. | An operational forecast or demonstrated speed-up. |
The executed development experiment uses twelve thermal cells and eleven internal edges. Capacities vary from 1 to 1.4 J/K; conductances are in W/K. The browser evaluates the archived neural weights and advances a closed heat equation with Δt = 0.8 minᵢ Cᵢ / Σⱼ Gᵢⱼ. Its capacities and geometry are constructed for verification and do not approximate kilometre-scale atmospheric heat capacities. [3]
The rainfall and temperature outcomes above are source-reported in Paper 3 and Paper 5, both working manuscripts. This page does not rerun those empirical studies. Validation of the broader thesis should withhold stations, time windows or other independent observations; compare uncoupled, coupled and established reconstruction methods; measure boundary and budget errors; and report training, inference and total computational costs on stated hardware. A sharper-looking field is insufficient evidence.
Open the experiment, weights and run record ↗Animation and calculations
How to read this visualisation
The Earth uses NASA Earth Observatory’s January 2004 Blue Marble composite. Cloud patterns, satellite geometry and orbit are illustrative. The generalised Jamaica outline comes from Natural Earth; the closer mountain mesh comes from the retained elevation bundle. Local boxes and instruments are enlarged for visibility. The heat graph’s 3D arrangement is explanatory: only the eleven displayed graph connections exchange heat.
The grid-cost control assumes a fixed physical domain, uniform refinement r in all three spatial directions, a time step proportional to cell width, and constant work per cell update. Its relative work is r³ × r = r⁴. This is an arithmetic illustration of one explicit time-stepping regime, not a universal climate-model cost law. Refining only horizontal directions would instead give r² × r. [4]
The coloured downscaling example fits the four existing synthetic station regressions, interpolates their offsets and centres the fine field so it recovers its parent mean. It is displayed as a schematic field above the terrain. It has not been calibrated to the Blue Mountain domain. The final archived POWER readout retains its coarse support and never supplies those synthetic offsets as observations.
Wind trails follow a prescribed smooth, divergence-free horizontal field integrated with a midpoint step. They make transport legible; they are separate from the thermal calculation and do not solve atmospheric momentum or rain microphysics. Heat-pulse direction follows the computed interface flux, while pulse size and travel speed are schematic. Replay interpolates stored constructed states; reversing it does not solve an inverse heat equation.
Inspect thermal and cost calculations, archived closure weights and the terrain and POWER source bundle.
Data and sources
- NASA POWER meteorological methodology. Source lineage and 0.5° × 0.625° MERRA-2 spatial support. The retained request explicitly uses UTC.
- Dunkley: space–time control volumes for Caribbean atmospheric reconstruction. Working manuscript; full three-dimensional moist reconstruction proposed.
- DLEGOS mathematical development laboratory. Executed constructed graph, learned weights, numerical assertions and archived run record.
- Clawpack Riemann Book: finite-volume updates and the CFL condition. Source for shared numerical fluxes and the explicit wave-speed timestep constraint used in the cost illustration.
- NASA Earth Observatory: Blue Marble, January 2004. Geographic Earth texture; image credit NASA Earth Observatory.
- Mapzen / Tilezen Terrain Tiles and source attribution. The retained bundle records source tiles, decoding, sampling and retrieval notes.
- Natural Earth public-domain generalised country outlines, distributed through D3 Graph Gallery.
- Wang et al. (2021), Fast and accurate learned multiresolution dynamical downscaling for precipitation. The abstract reports roughly one month on 600 CPU cores for a 30-year, 12 km conventional dynamical simulation over the contiguous US. This external case does not benchmark Jamaica or D-LEGOS.
Research and direction: Adrian Dunkley. AI-assisted educational implementation. NASA provides referenced imagery and data; no NASA affiliation or endorsement is implied. This page is a research explainer, not an operational weather service.
Interactive laboratoryExplore the calculations and the community
DLEGOS exchange, downscaling, emulators and physics-informed learning, with the farms, buildings and residents below.
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Explore the calculations and the community
DLEGOS exchange, downscaling, emulators and physics-informed learning, with the farms, buildings and residents below.
DLEGOS climate laboratory
Change wind and rain, then inspect the cells and the community below.
From orbit to an observation footprint
DLEGOS
Exchange between cells
Inspect
Choose a cell in the model.
Select a farm plot or building for local details.
Select a resident in either view, or use the resident list.
Ground view
At street level
Full laboratory methods and original teaching examples +
Methods
What the model calculates
Definitions, equations and limits for each experiment.
01 DLEGOS: connect local pieces +
Adrian Dunkley’s DLEGOS research explores climate prediction through connected local components. This scene illustrates the idea with neighbouring cells above a stylised island. Each cell stores a scalar state, q. You can think of it as a normalised temperature-like quantity, but it is dimensionless and is not an observed air temperature. The coloured parcels and travelling pulses make local exchange visible. In that framework, a spatial cell followed through a time interval forms a space–time block: the tesseract analogy. The linked research papers describe the wider proposed space–time framework and its separate component experiments. [1]
What is actually calculated?
The browser advances a closed, equal-capacity diffusion grid. A warm cell transfers a quantity to a cooler neighbour. Every internal transfer is subtracted from one cell and added to the other. This paired bookkeeping preserves the grid sum, up to floating-point rounding. The island is a visual backdrop; it does not supply a measured terrain dataset or a calibrated atmospheric boundary condition.
qᵢ(t + Δt) = qᵢ(t) + Δt Σⱼ Fᵢⱼ
Here κ is a non-negative exchange coefficient, Δt is the model timestep, and j runs over the side-sharing neighbours of i. Grid spacing and capacities are set to one. The timestep obeys Δt κ d ≤ 1, where d is the largest neighbour count. That bound keeps this explicit diffusion step a convex combination of existing values. No external source or sink is applied after you reset the initial heat pattern.
Where the analogy stops
Real atmospheric cells exchange momentum, heat and moisture, with pressure forces, radiation, phase changes and open boundaries. The moving wind particles in this scene explain transport visually; they do not solve those equations. Conservation of this toy scalar does not establish energy or water conservation in a full DLEGOS implementation. Validation would require documented inputs, physical units, reference simulations and held-out observations. Use the cell inspector to watch local values approach their neighbours as the displayed total remains stable.
02 Downscaling: reveal local structure +
A large climate grid box can cover coast, lowlands and mountains together. Downscaling estimates finer spatial structure conditional on large-scale information. Dynamical downscaling uses a regional physical model; statistical downscaling learns or specifies relationships between large-scale predictors and local conditions. A finer grid can represent more detail, but smaller pixels alone do not demonstrate improved accuracy. [2]
Change the resolution to split each coarse cell into smaller cells. The local-detail slider adds a constructed spatial pattern. Each parent cell’s children are then corrected so their mean equals that parent’s value. This is an equal-area aggregate constraint. It illustrates consistency across scales, without claiming that the inserted detail has been observed or learned.
meanₚ(q̂) = Qₚ
Qₚ is the coarse parent value, rᵢ is a synthetic fine-scale pattern and a controls its strength. Because the centred detail sums to zero inside each parent, aggregation recovers the original coarse field. The diagnostic reports the maximum parent-mean mismatch. For unequal cell areas you would need area weights. For rainfall, an additive correction can produce negative values, so a physically appropriate positive constraint would also be necessary.
What would establish useful downscaling?
Compare against interpolation and established downscaling methods, with independent station or satellite observations. Separate spatial and temporal test periods from training, and examine extremes and coastal gradients as well as mean error. Several distinct fine-scale fields can share the same coarse average. An ensemble can represent that ambiguity; a single attractive reconstruction cannot remove it.
03 Climate emulators: learn an update rule +
An emulator is a surrogate for a more expensive model. It learns a relationship between inputs and model outputs, then approximates that relationship for new inputs. Climate emulation may target statistics, spatial fields or repeated state transitions. Hybrid systems can combine learned terms with physical numerical solvers. NeuralGCM is a published example of combining differentiable atmospheric dynamics and machine learning, with stated limits on extrapolation to substantially different climates. [3]
Press “Fit surrogate” to estimate the diffusion coefficient from constructed one-step training pairs. The browser performs a least-squares fit, using a training coefficient of 0.16. The illustrated surrogate has one fitted parameter. It is a deliberately small regression example, rather than a trained neural climate emulator. The separate reference and surrogate states then roll forward on a different initial spatial pattern.
κ̂ = Σ(XY) / Σ(X²), X = Lq Δt, Y = Δq
L is the grid Laplacian: it sums neighbour differences. The RMSE diagnostic compares the two evolving fields cell by cell. Before fitting, the surrogate deliberately uses the wrong coefficient. After fitting, the reference and surrogate share the same operator family, so close agreement is expected in this constructed problem. That is a check of the implementation, not evidence of skill on climate observations.
Why long rollouts matter
Small update errors can accumulate when an emulator repeatedly consumes its own outputs. A useful climate evaluation checks seasonal distributions, extremes, physical budgets, stability and sensitivity to changed forcing. Runtime must be measured on stated hardware against a comparable baseline. This page makes no speed-up claim and does not equate the speed of an animation with scientific computational efficiency.
04 PINNs: let equations shape learning +
A physics-informed neural network uses a differentiable model to approximate a field and penalises violations of specified governing equations during training. Data can constrain measured values; initial and boundary conditions constrain the problem; a differential-equation residual checks whether the learned field behaves consistently between those points. The original PINN work develops this approach for forward and inverse differential-equation problems. [4]
This scene uses a two-feature differentiable trial model so you can inspect the optimisation directly. One feature satisfies a one-dimensional heat equation. The other adds an equation-violating component. The train button updates their two output weights with gradient descent. The displayed loss terms are calculated from synthetic data and analytical residuals. This restricted feature model demonstrates the objective; it does not train a general deep PINN or solve the atmosphere.
k = κπ²; r = ∂ₜu − κ∂ₓₓu
Loss = mean((u − y)²) + λ mean(r²)
The domain is x ∈ [0,1], t ∈ [0,1], κ = 0.1, with zero boundary values. The data come from the first feature with amplitude one, plus a constructed b = 0.3 disturbance. That disturbance creates tension between fitting the supplied targets and reducing the equation residual. Increasing λ changes the compromise. The scalar residual is not a probability of physical correctness.
A penalty is not a guarantee
A low residual at sampled points does not prove global accuracy, unique solutions or exact conservation. The choice of equation, scales, sample points and optimisation all matter. Real PINNs can be difficult to train, especially across competing loss terms. Compare against numerical solvers and data-only baselines, measure boundary errors separately, and test sensitivity to noise and missing physics. The animated network is a conceptual diagram of information flow, while the two-feature calculation is the executable example.
Weather, soil and residents
The five environmental presets control a separate constructed scene. Trade-wind particles move along prescribed paths; the storm preset uses an analytical circular path. Clouds, tree movement, umbrellas and walking routes are schematic animations. They are not outputs from the DLEGOS, emulator or PINN calculations.
The soil example does execute a dimensionless water budget: S′ = S + P − E − D − R. Rain input P is 0.12 × rain × Δt; potential evaporation is (0.008 + 0.025 × heat) × Δt; drainage is 0.018 × S × Δt. Each removal is limited to available water. Overflow R removes storage above one. The displayed residual checks this arithmetic balance. These coefficients are constructed, not fitted to a catchment.
The coastal-water preset raises a visual water surface. It does not solve hydrodynamics or calculate flood depths, hazards, damage or exposure. Residents have independent scripted routes and articulated walking motion. Rain adds umbrellas; no human-behaviour prediction is claimed.
How the pieces fit together
A possible research pipeline begins with a large-scale climate field, downscales it to local cells, advances those cells with a learned emulator, and checks physical consistency through equations and budgets. DLEGOS supplies a proposed way to organise local coupled components. Each step needs its own benchmark; success in one does not establish success in the others.
“World model” here means an internal representation of state and a rule for evolving that state. Weather forecasting asks about the time sequence of conditions. Climate simulation asks about distributions and behaviour over longer periods under specified forcing. Reconstruction estimates states consistent with past evidence. These are different scientific tasks, even when their visual interfaces look similar.
Laboratory levels and scales
The laboratory’s six levels are camera views of one constructed scene, separate from the geographic descent above. A nominal horizontal conversion of 100 metres per scene unit provides the scale bar and cell-spacing labels. The scale bar is projected at the centre of the current camera view and changes with zoom. It is an illustrative diagram scale, not a surveyed map scale. Buildings, residents, the satellite and vertical separations are enlarged or displaced for readability.
The displayed climate patch spans 780 nominal metres. Its four-by-four coarse parent grid therefore has a nominal spacing of 195 metres. With four subdivisions along each parent side, the sixteen-by-sixteen fine grid has 48.75-metre spacing. These are calculations from this page’s diagram settings, not claims about a real model’s resolution or a satellite’s resolving power. Farm and building close-ups describe the same schematic scene.
Sources and research trail
- Dunkley: proposed space–time climate cells and executable mathematical component studies. The teaching world on this page is a separate synthetic demonstration.
- CORDEX: What is regional downscaling? Institutional explanation of dynamical and empirical statistical downscaling.
- Kochkov et al. (2024), Neural general circulation models for weather and climate, Nature. External research; its results are not reproduced here.
- Raissi, Perdikaris & Karniadakis: Physics Informed Deep Learning; 2019 journal paper.
- Copernicus Climate Change Service: statistical and dynamical downscaling (2021). Source for the predictor-to-local-data relationship illustrated in the descent.
- Natural Earth public-domain, generalised country outlines, distributed through the D3 Graph Gallery world GeoJSON. The original laboratory’s landscape, cloud paths, grid placement and station-like objects are constructed.
Created for Climate Physics AI by Adrian Dunkley. AI-assisted educational implementation. The original laboratory uses constructed scientific data; the new thesis narrative above separately identifies geographic terrain and archived POWER inputs. The laboratory landscape is synthetic; no operational forecast, measured terrain or institutional endorsement is implied. Numerical methods and controls are available in the browser model source.