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Learn · Surrogate modelling and uncertainty
A surrogate trained on your simulations will answer any question you put to it, including the ones its data cannot support. TensorLynx surrogates carry a band calibrated on designs they never saw, check every request against the region they were trained on, and withhold the number when rules fixed before the run say the evidence is not there. The solver stays the authority.
Abstaining is the default, not an exceptionPass bars fixed before the first measurementRecommendations rest on solver values
Surrogates stand in for runs you cannot afford: hundreds of design points, a sensitivity sweep, an optimiser’s inner loop. The trouble starts quietly. A model fitted to a limited set of simulations draws a smooth curve through regions it has never seen, and most workflows print a value there with as many digits as anywhere else.
The pattern is familiar: a fit score measured on the training points, a band nobody tested on held-out designs, an optimiser that walks to the edge of the box where the data are thinnest. The miss surfaces late, in a full-model re-run or on the test bench.
The gap is not speed. It is a warrant: a reason, written down before the run, for believing a number, or for refusing to give one.
Industry survey NAFEMS × Teratec consultation, June 2026 [1]; IoT Analytics, 2026 [2].
Each gate is a rule the TensorLynx platform applies the same way on every run. Scroll through what happens between the last simulation and the value you read.
Every value carries a status: OK, MISSING, NOT_RUN, FAILED or UNVERIFIED. Only OK may train a surrogate. A failed evaluation stays on record as FAILED with its outputs MISSING; it is left out of every fit and nothing is imputed in its place. Missing is never zero.
In the design-optimisation loop, ten Gaussian-process variants compete for each output. The winner has the lowest leave-one-out prediction error (PRESS), with every fold refitted, hyperparameters included, so selection rewards prediction rather than memory.
A model’s own spread says where it is unsure relative to itself, not whether it is right. The served band is that spread scaled by a conformal quantile of the errors on held-out designs, per stratum, with the whole pipeline refitted for each group left out.
Three numbers from the leave-one-out table decide: Q² for accuracy, Kendall’s τ for ranking, and two-sided coverage of the ±1.96σ band with its Wilson interval. The verdict is the worst of the three. With fewer than 15 cases coverage reads UNKNOWN, and UNKNOWN is never a pass.
Before answering, the surrogate checks the ranges it learned, the distance to its nearest training design and the width of its posterior. At the edge the answer is BORDERLINE, further out OUT_OF_DOMAIN; when the posterior is too wide to support a number it abstains and names the coordinate that put the request there.
A failed or unjudged fit hides the predicted value everywhere, with the reason. The solver’s value is always shown, and the recommended design is the best one the solver evaluated and found feasible. Only a gradient that passed gradient verification may enter a gradient-enhanced fit.
A verdict cannot be built without a rule, and a rule cannot be built without a reference. These are the ones a surrogate answers to.
Q²LOO = 1 − PRESS / PRESSmean, PRESS = Σ (yi − μ−i)²
coverage = share of i with |yi − μ−i| ≤ 1.96 σ−i, with its Wilson 95 % interval
| Rule | PASS | CAUTION | FAIL |
|---|---|---|---|
| Q²LOO | ≥ 0.90 | ≥ 0.70 | < 0.70 |
| Kendall τ-b | ≥ 0.60 | ≥ 0.40 | < 0.40 |
| Coverage | interval contains 0.95, within 10 pp | contains 0.95, beyond 10 pp | interval excludes 0.95, either side |
Method Coverage is two-sided: an over-wide band fails as surely as a narrow one. Fewer than 15 cases reads UNKNOWN, and a PASS is never shown with an undecided metric. The bars were approved before the first measurement and are never moved after one.
band(x) = q · s(x), q = the ⌈(n + 1)(1 − α)⌉-th smallest of |yi − ŷ−i| / s−i
Each held-out score is the error divided by the model’s own spread, so the band is wide only where the model says it is unsure: split-conformal prediction with normalised scores.
Five labels cover every answer: CFD_VERIFIED (this design was solved), IN_DOMAIN, BORDERLINE (advisory), OUT_OF_DOMAIN (reported, not stood behind) and ABSTAINED (no number, and what to run instead). Abstention is a first-class outcome.
| Status | Means |
|---|---|
| OK | a usable value; the only status that may reach a model |
| NOT_RUN | never asked for |
| FAILED | asked for and failed, for example by diverging |
| UNVERIFIED | present, but nobody checked it |
| MISSING | does not exist for this run |
Verdict order: PASS < CAUTION < UNKNOWN < MISSING < FAIL. UNKNOWN means not measured: never a pass, never a failure. Global sensitivity indices on a surrogate pass the same gate, and a refusal returns no indices at all.
In a recorded surrogate-based optimisation of a NACA 0012 airfoil, drag minimised with lift held at CL ≥ 0.5, the platform fitted Gaussian-process surrogates for drag and lift on 19 leave-one-out cases. By accuracy and ranking they cleared their bars. Their bands did not.
Drag 0.211 [0.085, 0.433] and lift 0.579 [0.363, 0.769] against a 0.95 target: both bands too narrow. Q² and Kendall τ passed; the worst of three decides. Measured
Run verdict FAIL. The predicted drag at the final pick, and the lift constraint’s prediction, read “Hidden · fit FAIL”; the solver’s value at that design stayed on screen. Measured
Recommended: evaluation 24, solver CD 0.019390 at CL 0.50038, every constraint met. The surrogate’s own pick, evaluation 26, measured CD 0.020652. The rule postdates the run, so this is derived from its record. Measured
A separate validation review recomputed every metric from the recorded leave-one-out table; all agreed to the printed digits. Of 26 evaluations, 6 failed and were left out, not imputed. Measured
Why this matters. The accuracy score said yes, the calibration said no, and nobody had to notice by hand. The run used a 4,032-cell process mesh: it demonstrates the workflow’s behaviour, not airfoil physics. Its bars were approved the day before it ran. Measured
You know which fast answers to use, which to treat as advice, and which cost a solver run.
Every surrogate result arrives with its verdict, its band and the rule that judged it. Sign-off rests on solver values.
No hand-built parity plots to defend a model: leave-one-out tables, coverage and the reason for every withheld value are recorded with the run.
Fewer late surprises. Abstentions show up as routing to the solver, so the cost of trust is planned, not discovered.
Drag under a lift constraint, where the optimiser pushes to the edge of the box and a miscalibrated band would quietly pick the wrong design.
Pressure-loss campaigns over a parametric family, each prediction labelled for its domain before it reaches a trade-off chart.
Thermal resistance against pressure drop for cold plates and exchangers, with untrained operating points flagged before use.
| Question | Typical workflow | With TensorLynx |
|---|---|---|
| Which model? | picked by habit, judged on training points | ten variants compete on leave-one-out error |
| How sure? | the model’s own spread, unchecked | conformal band, coverage tested two-sided |
| Outside the data? | a number like any other | borderline, out-of-domain or abstained, by label |
| Poor fit? | noticed later, if at all | prediction hidden, reason stated |
| Failed runs? | dropped silently or filled in | FAILED on record, never imputed |
| Final design? | the surrogate’s predicted best | best solver-evaluated feasible design |
Where the story goes next: whole fields rather than single numbers in full-field reduced-order prediction; constraint-aware search driven by these surrogates in MDAO; and the independent re-run that turns a candidate into a verified design in high-fidelity reassessment. The TensorLynx overview shows the whole loop.
Method grounding: Gaussian-process surrogates in engineering design, expected-improvement search, sample size for computer experiments.
A score beside a number still invites its use. In the TensorLynx design-optimisation loop, a surrogate whose fit verdict is FAIL, UNKNOWN or MISSING does not show its predicted value at all: the label reads “Hidden · fit FAIL” or names why the fit was not judged. The solver’s value at the chosen design is always shown.
The request goes back to the solver. The recommended design is the best one the solver evaluated and found feasible. If none is feasible, nothing is recommended and the reason says so. A surrogate’s own proposal is labelled confirmed only after a solver run lands within its ±1.96σ band and within 2 %.
The fit verdict needs at least 15 leave-one-out cases before coverage can be judged; below that, coverage reads UNKNOWN and the prediction is not shown. Global analyses on the surrogate, such as sensitivity indices, need at least ten training points per design variable and a leave-one-out Q² of 0.90, or they are refused with the reason and a fix.
The model’s own spread gives only the shape. The served band scales it by a conformal quantile of the errors on held-out designs, per stratum. Coverage is then checked two-sided: a band too wide fails, as does one too narrow.
Not always, and the label says which case applies. Near the edge of the training range a prediction is BORDERLINE and advisory. Further out it is OUT_OF_DOMAIN: reported, but not stood behind. When the posterior spread exceeds a quarter of the predicted value, or a model without a posterior is asked outside its range, the surrogate abstains and names the coordinate responsible.
Gaussian-process models are the core: Matérn and other kernels, reduced-dimension variants, gradient-enhanced and multi-output models including co-kriging, with ridge and power-law baselines alongside. In the optimisation loop ten Gaussian-process variants compete per output and the lowest leave-one-out error wins.
No. A surrogate decides where the next simulation is worth running. Recommendations rest on solver values, a design is called verified only after an independent re-run, and simulation supports engineering decisions without replacing the tests and sign-off your programme requires.
Every capability hands its result to the next on the same model, the same units and the same record: geometry feeds physics, physics feeds sensitivities and learning, optimisation proposes, verification confirms, and the history keeps every step reproducible.
TensorLynx is built and run by the engineers who use it on client work. You can start small, with one geometry and one question, and grow into a workflow your own team owns.
Send a geometry, the objective and the constraints. We tell you which TensorLynx route answers it, what it needs from you, and what you will get back.
Our engineers run the loop on your problem: physics, sensitivities, optimisation and full-model verification, handed over with every setting on record.
A TensorLynx workflow, surrogate and verification gates set up around your product family, so the next design cycle starts where the last one ended.
Your team on TensorLynx, trained on your own cases, with OLTRION engineers behind it for the questions that need a specialist.
Simulation and AI predictions support engineering decisions; they do not replace the testing, approval and sign-off a product requires. Figures marked measured come from recorded TensorLynx runs on the stated case; figures marked synthetic or schematic illustrate a method.
