No pages match that search.
Improve · Gradient verification
A wrong adjoint gradient does not crash. The optimiser converges, the history plot looks healthy, and the design walks confidently in the wrong direction. The TensorLynx platform tests each gradient route against noise-sized finite differences on two independent directions, fits the Taylor remainder, and withholds a route that fails before any optimiser may use it.
Bars fixed before the first measurementTwo random directions, never oneUnresolved reads Unknown, never Pass
The cost of the old way
On one duct family, the sensitivity of pressure drop to its throat parameter came out three ways. The adjoint read on its own, without the chain through mesh movement, gave −48.42 Pa per unit. A same-mesh central finite difference gave −240. The adjoint carried through the mesh chain gave −242.9, within 1.22 % of the reference. Measured duct shape family, recorded benchmark
Every solve converged; no residual plot separates the first number from the third. An optimiser fed the first sees a fifth of the true sensitivity, while other parameters are wrong in their own ways, so the direction itself turns.
How TensorLynx does it
The adjoint optimisation chapter computes one gradient for every surface point at the price of one extra solve. This chapter decides whether that gradient is allowed to move the design. Five steps, run before the optimiser starts.
Every bar is declared with its rationale and a cited reference before the first gradient is computed, and never moved afterwards. The direction bar admits about 18 degrees, which a line search absorbs, and cautions down to 37, which it does not. Method
An iterative solver’s objective carries noise. The platform estimates that floor and sizes the finite-difference step to it, after Moré and Wild. A parameter whose whole effect sits below the floor is reported unresolved, not judged.
The projected gradient g·d is compared with a finite difference along two random directions. Equal ratios mean a scale error: the sign holds and the step still descends. Ratios that differ or flip sign mean an error as large as the gradient. No scaling repairs that.
If g is the true derivative, the remainder J(x+hd) − J(x) − h g·d shrinks with h2; if it is not, with h, whatever its size. The fitted order is read only over steps where the remainder stands at least three times above the noise floor, because the order of noise is not an order.
A passing check labels the gradient Direction-checked (usable for optimisation), never “verified”. A failing route is withheld and the router falls back to finite differences while 2n solves fit the budget, or refuses and names both costs. A failed check stays on the result, with its run id.
Inside the engine
For the analyst who signs the study: what is measured, and why each bar sits where it does.
A central difference carries truncation error falling with h2 and cancellation error growing as εJ/h, where εJ is the objective’s noise. Their sum is a V on log-log axes; agreement exists only in its trough. An iterative solver’s εJ sits far above machine precision, so a default step can land on either wall.
The complex step, f′ ≈ Im f(x + ih)/h, has no subtraction and so no rising wall. Where a code path supports complex arithmetic it is the sharper reference: the platform’s free-form-deformation geometry Jacobians match it to 4.5e-16 Measured. Flow-solver gradients are checked against noise-sized central differences.
Direction. cos θ = g·gFD / (‖g‖ ‖gFD‖). If the error obeys ‖e‖ ≤ ε‖g‖ then cos θ ≥ √(1 − ε2); a usability threshold of ε = 0.15 follows from that arithmetic.
Magnitude. The regression slope of the reference on the adjoint, the statistic the cosine cannot see.
Correctness. R0(h) = |J(x+hd) − J(x)| = O(h); R1(h) = |J(x+hd) − J(x) − h g·d| = O(h2). The median R1/R0 sizes the systematic directional error the order only detects.
| Check | Pass | Caution | Why there |
|---|---|---|---|
| Resolvable | disagreement above the reference’s noise | — | Otherwise Unknown: wrong adjoint and noisy reference are indistinguishable. |
| Cosine | ≥ 0.95 | ≥ 0.80 | 18° is absorbed by a line search; 37° is not. |
| Regression slope | 1 ± 0.10 | 1 ± 0.25 | Catches a gradient uniformly too large or too small. |
| Taylor order | 2 ± 0.2 | 2 ± 0.5 | Beyond 0.5, second order and first blur. |
| Systematic error R1/R0 | ≤ 5 % | ≤ 15 % | 15 % is where a gradient is merely usable for optimisation. |
| Adjoint convergence | ≥ 4 decades | — | One stalled adjoint equation spoils the sensitivity it feeds. |
| Adjoint vs primal shortfall | ≤ 2 orders | ≤ 3 orders | Primal down six orders, adjoint down two: a confidently wrong gradient. |
| Gradients used as training labels | cosine ≥ 0.99 | ≥ 0.95 | A regression has no line search to absorb a direction error. |
dJ/dp runs from the objective through the flow adjoint and surface sensitivity, through mesh movement, to a CAD or free-form-deformation parameter. The platform names every route and the mesh mover it assumes, and admits it only when the forward run’s mover matches. A mixed chain, or a remeshed forward run, is refused by identity before any number is computed.
Each route is gated on a resolvable laminar twin of its case: same mesh, markers and objective, solved to round-off. Where the reference cannot resolve, the gate reads Unknown. A gradient is tied to its rank count and solver build; change either and the check is due again.
The router. Finite differences while 2n solves fit the budget; beyond it an admitted adjoint; otherwise a refusal that names both costs. On one turbulent duct case, a gradient run took 544 s against 40 s for a primal, so the crossover sat near seven variables. Measured
An adjoint gradient answers a local question: which way to move from this design. Variance-based indices answer a global one across the box. The first-order index Si is the share of output variance one input explains alone; the total index STi adds its interactions. Morris screening ranks inputs cheaply by mean absolute elementary effect.
The Ishigami test function shows why both matter: its third input explains nothing alone and a quarter of the variance through interaction. A one-at-a-time study would discard it.
Thousands of evaluations make these indices a surrogate calculation. OLTRION’s discipline: the surrogate passes its fit verdict before any index is computed, and every index is labelled as a statement about the surrogate. That is the hand-off to trust-aware surrogates.
Evidence you can check
Every figure below comes from a recorded verification run, judged against bars declared before it.
| Case | Cosine | Norm ratio |
|---|---|---|
| Pipe bend, surface route, laminar twin | 0.9999999952 | 1.0000734 |
| Duct, CAD-parameter route, laminar twin | 1.0000000000 | 1.0000000 |
| Duct, turbulent, wall functions | 0.99999999 | 1.00020 |
| Duct, thermal adjoint (heat flow), laminar twin | 0.9999999990 | 0.99999 |
| Structural compliance derivative | 1 − 1.2e-12 | — |
A mesh-warping adjoint route missed its gradient-norm band by 0.15 %. It was withheld and the router used finite differences. The cause: a derivative term that vanished exactly at the rest shape. Fixed, the turbulent duct passed all six route bars. No bar was moved. Measured
The throat example above is now a refusal: a mover that does not match the forward run is rejected before any number exists. The refused chain is kept as evidence. Measured
A remeshed bracket with objective noise of 3.9e-5 was refused for finite differencing below its floor rather than reported as agreement. Measured
A passing check reads “Direction-checked”; “Verified” is kept for a candidate confirmed by an independent re-run. A failed check stays on the label as “last check FAIL” with its run id. Method
What it changes for your programme
A proposed shape change arrives with the check behind its gradient, so the review argues about physics, not derivatives.
Step, directions, bars and Taylor fit done the same way on every route, with a verdict and a record, not a script rewritten per project.
Gradient runs go to routes that passed; a withheld route falls back or stops with both costs named before the budget burns.
Intakes and airfoils with hundreds of shape variables: an adjoint is affordable, a checked one is safe.
Bends and cooling passages tuned for pressure loss through mesh movement, the link most easily dropped.
Exchanger and cold-plate passages, where a thermal adjoint is checked before it moves a wall.
Brackets and plates, with compliance shape derivatives checked the same way.
| Typical workflow | With TensorLynx | |
|---|---|---|
| When | Once, by hand, if time allows | Before an optimiser may use the route |
| Step size | A default value | Sized from the measured noise of the objective |
| Directions | One perturbation | Two independent random directions |
| Unresolved check | Read as agreement | Unknown, never Pass |
| Failure | Rescaled or ignored | Route withheld, the failure kept on the label |
| Validity | Assumed to carry over | Tied to rank count, solver build and mesh |
A gradient that passes here drives the design; the result is then confirmed by an independent high-fidelity reassessment, on meshes whose own error is measured in mesh verification. See how the chapters fit in the TensorLynx overview.
FAQ
Convergence says the equations were solved, not that the derivative belongs to what the optimiser evaluates. A dropped mesh-movement term, an adjoint that stalled while the primal converged, or a changed parallel decomposition all leave a converged-looking run with a wrong gradient. The check is where that error becomes visible.
From the measured noise of the objective, not from a default. The platform estimates the noise floor, sizes the step to it, and refuses a parameter whose whole effect over the step is below that floor. A comparison the reference cannot resolve reads Unknown rather than Pass or Fail.
Every check uses at least two independent random directions. If the ratio of projected gradient to finite difference is the same on both, it is a scale error and the direction still descends. If the ratios differ or flip sign, the error is as large as the gradient and no scaling repairs it.
Direction-checked means the gradient passed its check and may drive an optimiser. Verified is reserved for a candidate design that an independent re-run has confirmed against your acceptance limits. The two labels are kept apart in the interface and the report, and a failed check stays on the label with its run id.
Where the code path supports complex arithmetic. The platform's geometry Jacobians for free-form deformation match complex step to 4.5e-16. Flow-solver gradients are checked against noise-sized central finite differences and a Taylor remainder test, on a resolvable twin of the case where needed.
Yes, as a study OLTRION engineers plan with you. Variance-based indices need thousands of evaluations, so they are computed on a surrogate, which must pass its fit verdict first. An index on a poor surrogate describes the surrogate, not your physics, and is labelled that way.
A record of the route, the bars declared before measurement, the steps, cosine, regression slope, fitted Taylor order and verdict, with the inputs, rank count, solver build and mesh hash the gradient depends on. Change any of those and the check is due again.
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.
