Watch a cold pulse move through the reactor.
A cold pulse enters through part of the inlet. Flow stretches it downstream while heating and reaction change the temperature and conversion fields. Follow its motion, then compare two heating schedules.
Prescribed shear flow · deterministic inlet pulse · numerical PDE responses
Both controllers encounter the same realized inlet-temperature disturbance. Heatmaps show numerical PDE states; colors use the same fixed scales in every run. Dashed lines separate the four heater zones.
Paired trials
| Trial / controller | Completed | Heating energy¹ | Min. outlet conversion | Max. temperature | Quality / heat violation |
|---|
¹ Integral of mean heater fraction, in seconds. Crosses or incomplete runs are not counted as energy-saving successes. Violations are measured at the 2 s execution grid.
The weighted control update
Surrogate prediction
PDE response under the same forecast
The weights average the 24 latent control parameters. A sigmoid converts the updated parameters to bounded heater settings, and interpolation supplies the heating sequence. That sequence is simulated again to obtain these fields; candidate field images are never averaged. Only the first 2 s control is executed before replanning.
Learned surrogate
Numerical PDE
Surrogate − PDE
Held-out prediction checks
Test episodes are separate from training and checkpoint selection. Criteria: average temperature RMSE below 10 K and average outlet-conversion MAE below 0.02 over 40 s rollouts. Peak underprediction is reported separately.
Model, uncertainty, and recorded computation
This is an illustrative 8 m × 2 m flow-through reactor, discretized into 64 × 32 cells. Each cell stores temperature and remaining reactant fraction. Flow moves left to right at 0.2 m/s. Conversion is one minus the reactant fraction.
Four heating fractions lie between zero and one. The target is at least 95% outlet conversion while keeping all cells at or below 1070 K. The observed inlet fluctuation has a 20 s correlation time and a 15 K stationary standard deviation.
The controller replans every 2 s over 40 s. Its default update uses 128 candidate sequences, four conditional inlet futures, and two weighted updates. Proposed controls are checked with the PDE under conditional mean inlet conditions. This check does not guarantee feasibility under every random disturbance.
The surrogate projects predicted reactant fractions onto [0, 1]; its temperature predictions are unrestricted. This prevents impossible fractions but does not establish prediction accuracy. The validation report includes the raw fraction-bound excess.
| Trial / controller | Mean decision time | Rejected updates | Mean ESS | One-step T RMSE |
|---|
Sampling sensitivity at the same planning state
Three independent planning seeds per setting. The reported cost re-evaluates the chosen sequence in the PDE against 32 fresh shared inlet futures.
| Candidates | Inlet futures | PDE cost · mean ± SD | Median planning time |
|---|
Numerical metadata and diagnostics · Model and regeneration guide · Static comparison figure (PDF)