One synthetic receiver-function run on our cluster, a single forward model turning an assumed crustal structure into the waveforms a station would record, used to cost about $12 in cluster hours. A trained neural-operator surrogate answers the same question for roughly a cent.
That ratio is the headline, and it is genuine. What follows is the part that decides whether it helps you, which is not the ratio.
The cost, not the physics, set the ceiling
Our forward modelling is accurate and well understood. RAYSUM computes the receiver-function response of a stack of anisotropic, dipping layers; a finite-difference solver steps through the full elastic wave equation when we need effects RAYSUM cannot capture. Both are trustworthy. Both are expensive per run.
No serious crustal-imaging decision hinges on a single run. Inverting station data for Moho and lithosphere-asthenosphere depth, quantifying how anisotropy trades off against depth, running ensembles across plausible velocity models: all of it needs hundreds or thousands of forward simulations. At cluster-hour prices a proper uncertainty study across a network ran into five or six figures, so it did not get commissioned. The science was not the constraint. The invoice was.
A neural operator learns the forward map directly, predicting what the solver would have produced without stepping through the wave physics. Train it once against a batch of high-fidelity runs spanning the structures you care about, then evaluate it for a fraction of a cent. It is not a lookup table; it generalises to models it has not seen, which is what makes dense sweeps and Monte-Carlo ensembles possible at all. We have written elsewhere about where such a surrogate is faithful and where it is not, on late multiples and thin layers, and that piece is the one to read before trusting it. This one is about what it costs.
The number everyone quotes, and the one that decides
$50,000 buys 4,166 solver runs or 2,600,000 surrogate evaluations, a multiple of 624.1x. The per-run ratio is about 1000 to 1, and the reason this is not that is the $24,000 training investment, which comes out of the same budget.
Across the whole board, 25% of study-and-budget combinations are reachable only with a surrogate, and 10% sit in the dead zone where only the solver can run anything.
| study | $10,000 | $25,000 | $50,000 | $150,000 | $500,000 |
|---|---|---|---|---|---|
| a single best-fit crustal model (1 runs) | solver only | either | either | either | either |
| a sensitivity sweep on one station (200 runs) | solver only | either | either | either | either |
| full ensemble uncertainty on Moho and LAB (10,000 runs) | neither | surrogate only | surrogate only | either | either |
| the same ensemble across a station network (250,000 runs) | neither | neither | surrogate only | surrogate only | surrogate only |
The usual way to justify a surrogate is the break-even: the ensemble size at which its total cost first undercuts the solver's. On our figures that lands at about 2,000 runs. It is a tidy number and it decides almost nothing, because it never asks whether the study you want is reachable at all.
Set the exhibit to the smallest budget and the problem is visible immediately. At $10,000 the solver runs 833 scenarios. The surrogate runs zero, because the training batch has to come out of the same budget and it costs more than the budget holds. There is no partial credit here: buying a surrogate at this size is strictly worse than not buying one, and the break-even figure gives no hint of it. That red column is the dead zone, and a team that budgets from the per-run ratio walks straight into it.
Now step the budget up. At $25,000 the surrogate affords 100,000 evaluations against the solver's 2,083, and full ensemble uncertainty on Moho and lithosphere depth moves from impossible to routine. At $50,000 the same comparison is 2.6 million against 4,166, and a network-wide ensemble appears.
Watch the multiple as you do it: 48x at $25,000, 624x at $50,000, 1008x at $150,000. The per-run ratio is about 1000 to 1 throughout. The effective ratio is not, because until the budget dwarfs the training investment the training investment is most of the budget. The 1000x is an asymptote, and quoting it at $25,000 overstates the advantage by a factor of twenty.
Scope, not savings, and the band that proves it
The amber band across the middle of the block is the real argument. Those are studies that exist only because the surrogate does: at these budgets the solver cannot reach them at any allocation the study would plausibly get.
This is why "it made our simulations cheaper" undersells it and also misdescribes it. The same study does not get cheaper in any way that matters, because at most budgets the same study was already affordable. What changes is which class of question is on the table. Instead of handing a client a single best-fit crustal model, you can hand them a distribution: a depth with an honest error bar and an explicit statement of which features the data constrain and which are assumptions wearing a confident face. For anyone weighing a CCS or geothermal site on crustal structure, the move from a point estimate to a bounded distribution is the entire value.
The surrogate does not replace the solver. It is trained by it and periodically checked against it, and those checks are a running cost rather than a one-off. What it changes is what you can afford to do between the checks.
How to decide, in three questions
Ask how many forward runs the question needs, because that is what sets the class of study, and the run counts differ by orders of magnitude rather than by percentages.
Ask what the training batch costs, in solver runs, and treat that as a floor on the budget rather than a line item. Below it the surrogate is not a cheaper option, it is not an option.
Then ask what the effective multiple is at your budget, not at infinity. It is the only one of the three numbers that describes the decision in front of you, and it is usually a lot smaller than the one in the headline.
Key takeaways
- A high-fidelity forward run costs about $12 in cluster hours and a surrogate evaluation about a cent, but the ratio is a marginal cost, not a budget outcome.
- The break-even at roughly 2,000 runs is the figure usually quoted and it decides nothing, because it never asks whether the study you want is reachable.
- Below the training investment the surrogate buys zero evaluations: at $10,000 the solver runs 833 scenarios and the surrogate runs none, so buying one is strictly worse.
- The effective multiple is 48x at $25,000, 624x at $50,000 and 1008x at $150,000. The headline 1000x is a large-budget asymptote.
- The payoff is a change of category rather than a discount: full ensemble uncertainty and network-wide ensembles are reachable only with a surrogate at these budgets.
- The surrogate is trained and periodically validated by the solver it stands in for, so those checks are a running cost and not a one-off.
References
- Where the surrogate is faithful and where it is not: Neural operator surrogates for receiver-function forward modelling.
- The same trade in a different equation: A neural operator for the mantle-flow Stokes boundary-value problem.
- Z. Li, N. Kovachki, K. Azizzadenesheli, et al. Fourier Neural Operator for Parametric Partial Differential Equations. ICLR 2021. arXiv:2010.08895
- A. Frederiksen, M. Bostock. Modelling teleseismic waves in dipping anisotropic structures. Geophysical Journal International, 141, 401 to 412, 2000. doi:10.1046/j.1365-246x.2000.00090.x
