Inability to Model Turbulence

AI does reach the step that carries the measured cost, which neither the telescope nor the publishing chain produced, so the method can return that answer. But the step it reaches was already getting cheaper without AI: the authors of that case put the monetary cost of the GPU route at 'an order of magnitude less than current central processing unit (CPU)-based supercomputing platforms', and its turnaround at 7 hours against 7.5 days — a wall-clock ratio on different hardware rather than a fall in core-hours, which is why it appears nowhere as a number on this chain. And the expectation that a priced step also being an AI step would settle something turns out to be a statement about what has been priced. One caveat on the prediction itself: I wrote it down before building any step and have not changed it since, but I wrote it after fourteen searches had already turned up the main sources, so it is a weaker test than a blind one.

The expectation was also wrong about capability coverage, in the direction of understating it. Convergent attach one capability to this gap, 'Develop New Modeling Frameworks for Turbulence', and reading its five substantive resources rather than its name puts it on four steps: the closure itself, the data the closure is trained on, the run, and - through the tensor-network calculation of turbulence probability distributions - the error bar, though that last mapping is the arguable one, and a second pass and I disagreed about it. The DARPA APAQuS programme behind it is explicitly about producing turbulence data from tabletop quantum simulators at lower cost than wind tunnels, which is the ground-truth step. What the capability set does not touch is deciding what accuracy the decision needs, the mesh, and getting the answer accepted. That is an observation about where a single capability was aimed, not a gap in their map; with one capability attached, against the three this method wants, the coverage reading here is thin by construction and half of what a chain usually produces is unavailable.

The September 2026 Navier-Stokes result acts on none of the seven steps, as registered. OpenAI's agents established statements C and D of the Clay formulation - a finite-time singularity in three-dimensional incompressible flow with a smooth forcing term, from rest - and the unforced question that physicists care about stays open. As of 18 September 2026 the proof had not been independently verified, the Clay Mathematics Institute's page for the problem is unchanged, and priority is contested. Nothing on this chain waits on that answer: every step here is about the cost of approximating a solution, not about whether a smooth one exists. The result is a strong datapoint about AI doing mathematics and a weak one about modelling turbulence, and those are different claims that the coverage has tended to merge.

Where the evidence is thin, and it is thin in three places a reader should know about. One costed workflow carries steps 2 and 5, at a wind-tunnel Reynolds number an order of magnitude below flight. The 360,000 figure is a choice: the same authors call both the medium and the fine grid 'of reasonable accuracy', and the medium grid is 40,000, which would have made a quieter headline on the same arithmetic. And the nulls on steps 4 and 7 are genuine nulls, so 'the unpriced steps cost more' is the practitioners' claim rather than a measurement - a second pass argued step 4 the other way, that a reference dataset is produced once per canonical flow and reused for decades while the 360,000 core-hours recur for every flow condition, which is why that step is no longer marked as carrying the cost.

Appendix: what I predicted before running this

AI acts on three of the seven steps: mesh and geometry preparation, the choice and form of the closure, and the simulation run itself through learned surrogates. It does not act on generating the high-fidelity ground truth that calibration and validation consume, nor on getting a result accepted for a decision. The cost concentrates on the simulation run once the flow must be resolved rather than modelled, and on obtaining ground truth; I expect at least one step carrying cost to also be a step AI reaches, which neither the telescope nor the publishing chain produced. The binding constraint should turn out to be circular rather than absent: the learned surrogate that cheapens the run is trained on data from the ground-truth step it cannot replace, so the cost moves upstream rather than disappearing. Convergent attach a single capability to this gap, 'Develop New Modeling Frameworks for Turbulence', and I expect it to act on the closure step alone, leaving six of seven steps with nothing attached. The September 2026 Navier-Stokes blow-up result acts on none of the seven steps.

• 1. Fix the case and the accuracy the decision needs — Labelled as institutional work because what is missing is an agreed answer to 'how accurate is accurate enough here', which a model cannot supply. Marked guess because a reader could reasonably argue that an LLM reading the regulation and the precedent set would shorten the negotiation, and because the claim that the target is argued from precedent is the practitioners' account rather than something the cited source states; the source is cited only for the definition it does give.
  Evidence: Bekemeyer, Geisbauer, Schaefer, Macomber, Wittenberg, Coggon, Held, Gojny, Denham, Bui and Yoo, 'Challenge Problem Overview for the Certification by Analysis Uncertainty Quantification Discussion Group', AIAA Aviation 2025, NASA NTRS 20250005704, https://ntrs.nasa.gov/api/citations/20250005704/downloads/AIAA_CbA_CoI_Aviation_Paper-forReview_06092025.pdf
• 2. Build the geometry and the mesh — AI reaches this step because adaptive and learned mesh sizing is an active, working line of work and mesh generation is a search problem with a checkable objective. Maturity is Speculative and that is a statement about the axis, not about the technique: on core-hours this step already costs 33, so no application of AI to it can move the measured cost. The gain would be in engineer time, which is the half of the declared axis this chain never shows, and which the same NASA survey calls the dominant cost in human intervention across the workflow. On a labour-cost chain this step would very likely read Working now and might well be the binding one. Marked guess for that reason.
  Evidence: Goc, Lehmkuhl, Park, Bose and Moin, 'Large eddy simulation of aircraft at affordable cost: a milestone in computational fluid dynamics', Flow 1, E14 (2021), doi:10.1017/flo.2021.17. Wall-modelled LES of the JAXA Standard Model high-lift configuration at Re_c = 2x10^6, charLES solver, Table 1. Mesh generation timing from section 4 of the same paper. The human-cost characterisation is Slotnick et al., 'CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences', NASA/CR-2014-218178 (2014), https://ntrs.nasa.gov/api/citations/20140003093/downloads/20140003093.pdf, section 5.3: mesh generation 'constitutes the dominant cost in terms of human intervention'.
• 3. Choose the closure and the fidelity level — This is the step Convergent's capability is named for, and the one where AI has the longest record: field inversion, symbolic regression and neural closures all act here. Maturity is 2-5 years rather than Working now because the September 2026 review identifies the a priori/a posteriori gap as unresolved and traces it to properties of turbulence rather than to architecture, so applying a learned closure to an unseen flow today does not reliably move the step. Guess, because on the narrow classes where a closure has been tuned against matched data it already works now.
  Evidence: Slotnick et al., 'CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences', NASA/CR-2014-218178 (2014), https://ntrs.nasa.gov/api/citations/20140003093/downloads/20140003093.pdf ('RANS models, which have seen nearly stagnant development for 20 years'). Current state of the data-driven alternatives: Hu and Yang, 'Bridging physics and data: A review of machine learning approaches for turbulence closure modeling', Physics of Fluids 38, 091302 (17 September 2026), doi:10.1063/5.0339930.
• 4. Get the high-fidelity data the closure is calibrated and validated against — No current AI capability reaches this step, because a learned model trained on ground truth is not ground truth. Marked guess, and the near-miss is real: DARPA's APAQuS, one of the resources behind Convergent's capability, proposes tabletop quantum simulators to produce turbulence data at lower cost than wind tunnels, with AI-based discovery of governing laws on top. If that works it acts squarely on this step. This step was marked as carrying the cost and then unmarked, and the argument that unmarked it is the right one: on an axis of cost per prediction, a reference dataset is produced once per canonical flow and reused for decades by every closure and every application, so it amortises, while the 360,000 core-hours at step 5 recur for every flow condition. The field names this step as the constraint on every data-driven route, which is a constraint on a technology path and not a share of the cost, and those are different claims.
  Evidence: Yang and Griffin, 'Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation', Physics of Fluids 33, 015108 (2021), doi:10.1063/5.0036515, read as arXiv:2010.15307v2. Data-scarcity characterisation from Slotnick et al., 'CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences', NASA/CR-2014-218178 (2014), https://ntrs.nasa.gov/api/citations/20140003093/downloads/20140003093.pdf, turbulence modelling impediments: 'Typical impediments include test cost, large number of cases needed, and instrumentation limitations.' The characterisation of high-fidelity data scarcity as an open problem is from the abstract of Hu and Yang, 'Bridging physics and data: A review of machine learning approaches for turbulence closure modeling', Physics of Fluids 38, 091302, published online 17 September 2026, doi:10.1063/5.0339930, which closes by examining 'generalizability across flow configurations, interpretability, numerical stability under solver coupling, and the scarcity of high-fidelity data'. Read from the publisher's abstract page in a browser; the full text is paywalled and there is no preprint, so nothing beyond the abstract is relied on here.
• 5. Run the simulation — AI reaches this step: learned surrogates and neural operators are aimed exactly here, and this is the step that carries the measured cost. Maturity is Speculative because the accuracy is not there in three dimensions - a current multi-physics foundation model is restricted to 2D and its authors state that 'no model is capable of high-fidelity predictions, let alone replacing numerical solvers' - and because the published speedups are unreliable: of 76 articles claiming to beat a standard numerical method on a fluid PDE, 79% compared against a weak baseline. Marked guess because for a narrowly scoped, heavily sampled configuration a surrogate does move this step today. It is marked as carrying the cost on the arithmetic: this step is more than 99.9% of the cost this chain can measure in its displayed unit. The cost already coming out of this step is attributable to GPUs and solver work rather than to learning; the same authors put that at about an order of magnitude in money. Evidence for the ML caveats: McGreivy and Hakim, 'Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations', Nature Machine Intelligence 6, 1256-1269 (2024), doi:10.1038/s42256-024-00897-5, arXiv:2407.07218. 'Towards a Physics Foundation Model', arXiv:2509.13805v4 (revised 31 May 2026).
  Evidence: Goc, Lehmkuhl, Park, Bose and Moin, 'Large eddy simulation of aircraft at affordable cost: a milestone in computational fluid dynamics', Flow 1, E14 (2021), doi:10.1017/flo.2021.17. Wall-modelled LES of the JAXA Standard Model high-lift configuration at Re_c = 2x10^6, charLES solver, Table 1.
• 6. Turn the model error into design margin — AI reaches this step because estimating model-form uncertainty from features of the flow field is a working line of work with a decade of results behind it. Maturity is 2-5 years because what is wanted is a calibrated error bar a regulator would accept, and the same 18% outlier rate across six workshops says the field does not have one yet. The capability is attached here on a second pass's argument rather than mine: Gourianov et al.'s tensor-network method computes probability distributions of turbulent flow quantities, and that pass argued that a flow-quantity distribution is what a calibrated error bar is made of. I read that resource as acting on the run instead. It is recorded on both steps because overstating a bare step is the more costly error. Guess, for that reason and because the step could reasonably be read as institutional rather than predictive, if what is missing is an agreed convention for reporting error rather than a better estimate of it.
  Evidence: Derlaga and Morrison, 'Statistical Analysis of CFD Solutions from the 6th AIAA CFD Drag Prediction Workshop', AIAA 2017-0209, NASA NTRS 20170001316, https://ntrs.nasa.gov/api/citations/20170001316/downloads/20170001316.pdf
• 7. Get the prediction accepted in place of a test — No current AI capability reaches this step, because what is missing is an agreed basis for trusting a prediction outside its validation envelope, which is a decision by a regulator. Confident, because the certification literature states the blocker in those terms rather than in terms of model accuracy. It is marked as carrying the cost on the practitioners' account rather than on a number: the test is still run outside the narrow approved envelope, so its cost sits on top of everything above. That is a judgment, and the absence of a figure is recorded rather than filled.
  Evidence: Honest null on the cost. Six searches from different angles - published wind-tunnel rates, occupancy-hour cost at a national transonic facility, tunnel hours in a transport development programme, a named programme's tunnel hours, certification-by-analysis progress, and the replacement question directly - found no figure for what a certification decision costs when the prediction is not accepted. Two near-misses, both rejected. The searches returned a widely repeated trade figure of about $20,000 per wind-tunnel hour, carried by an AIAA social post with no primary source behind it, which the project's source ranking excludes; a reader should know it exists and that it was not used. The better-sourced near-miss is Slotnick et al., 'CFD Vision 2030 Study', NASA/CR-2014-218178, case study 4, which reports that on Ares I 'CFD was often less reliable and more expensive than experimental testing, resulting in limited use of CFD principally in specific regions of the flight envelope where testing was not feasible' - a direction without a quantity. Current status and the antenna precedent: Bekemeyer et al., 'Challenge Problem Overview for the Certification by Analysis Uncertainty Quantification Discussion Group', AIAA Aviation 2025, NASA NTRS 20250005704, section II.

Axes I deliberately left out of this chain

Elapsed time from a new closure being proposed to its routine use in production design, which runs over a different set of steps — formulation, solver implementation, workshop validation and certification acceptance — and is measured in decades rather than per prediction. Also excluded: the mathematical question of existence and smoothness of solutions to the Navier-Stokes equations, which is a different kind of work and does not sit on the path to a prediction at all.