∂ₜT = κ₀ ∂z[g(θ) ∂zT]
The paper’s assumed heat equation uses observed moisture θ in a prescribed multiplier g. Given moisture, it is linear in temperature.
Assumed equation in P6Choose how to read
The full view retains the technical content, figures and sources. The plain-language explanation introduces the ideas without changing the findings.
Recovering a hidden soil-temperature record
How to use this companion
Use the controls to inspect the relationship described by the paper. The interactive display is explanatory; it does not rerun the archived training experiment or supply independent evidence of forecast skill. Read the linked full manuscript for the executed comparison.
What the study asks
Can a small neural replacement for a known heat-transfer calculation improve estimates of soil temperature at a hidden depth?
How the comparison works
The experiment hides the temperature at 20 cm and uses other measurements across the same time window to reconstruct it. It compares the exact heat-flux calculation, a learned replacement, a guarded replacement and a conventional numerical reconstruction.
What the result means
The neural replacement does not improve reconstruction. The safeguard keeps its local calculation error within the stated budget but has slightly higher average reconstruction error. A rule can pass one mathematical check without making the final estimated temperature better.
Where the evidence stops
The safeguard can consult the exact calculation, which would limit its practical advantage. Only three stations and ten uneven seasonal blocks qualify. This is reconstruction using a full observation window, not a demonstrated real-time forecast or Caribbean field validation.
Use the full view for the complete technical discussion, figures, interactive elements and sources.
P6 · INTERACTIVE 3D COMPANION
Test soil heat flux and temperature error
Watch heat move through a synthetic soil column. Compare the hidden temperature with estimates made using a perturbed flux and a local safeguard.
The question
Real geographic contextCalculated synthetic fields
ReferenceEstimate
Loading the 3D scene…
The WebGL 3D view is unavailable
A 2D fallback may appear in its place. You can also follow the study with these steps and the full paper.
- The question Does a more tightly checked heat-transfer calculation recover a hidden temperature more accurately? Follow the observed depths and the hidden target in the constructed soil column.
- The method A reduced heat model compares a known flux with a perturbed replacement. A gate rejects local errors above its assumed budget. Change flux perturbation and gate budget while watching the transfer and the temperature profile.
- Check the result Compare the local flux check with the final target-temperature error. A local bound and a good reconstruction are different outcomes. The reference and difference views let you inspect both.
Assumed diffusivity multiplier for the synthetic column.
Assumed replacement error, not the error of a paper checkpoint.
Local operator-error budget in the synthetic column’s time units; not a measured operating threshold.
Drag to rotate. Focus the scene to zoom with the wheel; arrow keys rotate and + / − zoom. Click a cell to inspect its transfers.
The question
Does a more tightly checked heat-transfer calculation recover a hidden temperature more accurately? Follow the observed depths and the hidden target in the constructed soil column.
Synthetic soil profile; the displayed temperatures and errors are not results from the paper’s USCRN stations.
Data, views and methods
Calculated values
What this browser model represents
The two columns show a constructed 0–50 cm soil profile and the selected comparison field. The outlined depth is the hidden 20 cm target. Gold markers beside the estimate show the calculated model flux; they are illustrative transfer markers, not individual molecules.
The browser perturbs an exact heat flux and checks the local operator error. Its constructed reference also contains a source omitted from the comparison models, so a small operator error need not give a small temperature error. No paper neural-flux checkpoint is trained or loaded.
The paper’s task is retrospective reconstruction with full-window auxiliary observations. This viewer does not demonstrate operational forecasts or a sensor-failure deployment.
Method in the paper
local operator error ≠ recovered-field error
The gate checks a known local map and its derivative. That does not bound model mismatch or the final temperature reconstruction.
Teacher and gate in P6Geography and figure design
Terrain is a sampled Mapzen/Tilezen elevation mesh, with a 62.5 m display spacing over an 8 km square. Source tiles have approximately 36 m pixel spacing at this latitude; display spacing does not establish survey accuracy. Colours show the calculated field, while terrain heights provide location context. WebGL displays a 129 × 129 terrain grid; the CPU compatibility view resamples the same elevations to 33 × 33 for responsiveness. Neither display resolution is an accuracy claim. The location selector does not change the numerical model. P6 soil columns and P7 touching heat cells use diagram dimensions. Changing the cube count changes the synthetic lattice and replays the elapsed scenario; it does not refine an atmospheric forecast. P8 markers use requested pilot coordinates, with illustrative model values. P1 column heights repeat rainfall, or absolute error in Difference view, at diagram scale; they are not millimetres on the terrain. P8 column heights repeat the probability or absolute error; their signs come from the colour scale and inspector. P6 shows zero in the reference column when displaying error. Coastline extrusion, sensor shapes, heat-volume sizes and particle speeds are explanatory symbols; they are not surveyed buildings, physical parcel sizes or measured motion.
Sources: Mapzen/Tilezen Terrain Tiles; USGS and NOAA terrain attribution; Natural Earth, public domain. Regional outlines are at 1:10 million, displayed with an equirectangular projection at 17°N. Terrain uses a local equirectangular display. Elevation exaggeration is shown above the scene.
Sequential field colours follow viridis; signed errors use a centred blue–orange scale. The colour limits stay fixed as the example runs; values outside them are counted in the legend. The plan view removes perspective. Figure design follows Crameri et al. (2020); pause and reduced motion follow WCAG motion guidance. The browser export retains a source caption. No atmospheric model skill follows from this animation.
Asset versions and checksums · Coordinate definitions · Numerical model source
Source sections
Neural flux stencils and oracle-assisted safeguards for soil-temperature reconstruction: a controlled negative result
Terms explained
Select a term to read its meaning in this study.
Reconstruction
Estimating missing states using information from an observation window. When later observations help estimate an earlier state, the task differs from forecasting the future.
Flux
The rate at which a quantity crosses a surface. A transfer over a time interval is the accumulated amount that crosses it.
Oracle
An exact or specially informed calculation available inside an experiment. A method that needs this information may not be usable when the correct answer is unknown.
PINN
A physics-informed neural network: a learned function trained partly by penalising disagreement with specified equations. A small penalty does not prove that every prediction is physically correct.
RMSE
Root-mean-square error. It measures the size of prediction errors and gives larger errors extra weight. Lower is better when the data and units are the same.