Module 12 of 14 · one week
Six places where neural operators are deployed or nearly deployed, each with its headline number and the caveat that number needs. Then the semiconductor anchor, where the public literature is empty. Then the company's claims, one at a time, with an evidence grade on each and a statement of what would raise it.
Two of this course's founding decisions meet here. One keeps Accelerated Understanding as a case study with an explicit evidence grade on every claim. The other anchors the course in semiconductors, where the public neural operator literature from the Anandkumar group is empty (research_notes/lineage.md:328-329).
Both need the same skill, and it is a skill rather than a disposition: reading a speed-up or accuracy claim and asking what was measured, against what, on which hardware, and by whom. That skill has a technique, and it is the next section.
A speed-up is a ratio of two times, and both the numerator and the denominator are choices. Six of those choices have to match before the ratio means anything: the hardware on each side, the spatial resolution, the reference solver and its tolerance, the precision, the batch size, and above all the quantity being computed. A model that predicts one scalar per geometry and a solver that produces a full field are not doing the same job, and their ratio is not a speed-up.
Then there is amortization, which is where most published speed-ups quietly become weaker or stronger than they look. A surrogate costs offline time \(T_{\text{off}}\) to generate data and train, and thereafter costs \(T_{\text{mod}}\) per evaluation against the solver's \(T_{\text{ref}}\). It pays for itself only after
\[n^* = \frac{T_{\text{off}}}{T_{\text{ref}} - T_{\text{mod}}}\]
evaluations. Below \(n^*\), running the solver is faster in total, whatever the per-call ratio says.
The FNO paper's own Bayesian inverse problem is the model of how to report this, because it gives every term. One FNO evaluation takes 0.005 s against the pseudo-spectral solver's 2.2 s on a 256² grid, both on GPU. The MCMC chain needs 30,000 evaluations, so the FNO route takes 2.5 minutes and the solver route over 18 hours. The paper also counts the roughly 12 hours of data generation and training against the FNO instead of hiding it (research_notes/pdf-fno-jmlr.md:159-172). Put those into the formula:
\[n^* = \frac{12 \times 3600\ \text{s}}{2.2 - 0.005\ \text{s}} = \frac{43{,}200}{2.195} \approx 19{,}700 \ \text{evaluations}.\]
So the break-even is about 19,700 calls, and the experiment used 30,000. The FNO wins, by a factor of about 1.5 on total time rather than the 440 the per-call ratio suggests. Both numbers are true and they answer different questions. If you needed 5,000 evaluations rather than 30,000, the honest recommendation would be to run the solver.
The two benchmark papers set the standard for saying this out loud. MeshGraphNets: "we find it hard to provide a 'true' hardware-agnostic performance comparison". PDEBench: "Whether or not this constitutes a fair comparison is surely debatable" (research_notes/pdf-mgn-pdebench.md:331).
| Application | Paper | Method | Headline number | Caveat | Grade |
|---|---|---|---|---|---|
| Global weather | FourCastNet, arXiv 2202.11214; FourCastNet 3, arXiv 2507.12144 | Adaptive FNO layers, then spherical harmonics (SFNO, 2306.03838), then a geometric probabilistic ensemble | Week-long forecast at 0.25° in under 2 s. FourCastNet 3: 60-day forecast at 0.25°, six-hourly, in under 4 minutes on one GPU, 8 to 60× faster than the diffusion methods it rivals | Trained on ERA5 reanalysis, which is itself the output of physics-based assimilation. v1 is deterministic; probabilistic only from v3. Initial conditions still come from classical assimilation, and Aurora has the same dependence | B lineage.md:250-271 |
| Tokamak plasma | Gopakumar et al., arXiv 2302.06542; Nuclear Fusion 64, 056025 (2024), arXiv 2311.05967 | FNO on JOREK MHD simulations and MAST camera data | Six orders of magnitude faster than the MHD solver at about \(10^{-5}\) normalized MSE; forecasts filament formation at half the shot duration | Forecasting, not control. No closed-loop control paper was found, so the podcast's "control" framing is a plan, not a result (earlier episode 01:05:30) | B lineage.md:284-304 |
| CO₂ storage | Wen et al., Energy & Environmental Science 16, 1732 (2023), arXiv 2210.17051 | Nested FNOs at several refinement levels | Nearly 700,000× speed-up in flow prediction | The abstract says "existing numerical methods"; the reference solver and its hardware are in the body and were not read here. With no \(T_{\text{off}}\) reported, the break-even count above cannot be computed | B lineage.md:306-315 |
| Catheter design | Zhou et al., arXiv 2304.14554 | FNO surrogate inside a geometry search, then E. coli experiments and 3D-printed prototypes | One to two orders of magnitude less bacterial contamination upstream than a smooth catheter | The journal venue is unverified from the arXiv page. This is also the only row where the deliverable is a physical object and not a number, which is a stronger form of evidence. It appears in the company episode rather than the earlier one (company-episode.md:97-98, :156) | B lineage.md:273-282 |
| Seismic inversion | Yang et al., The Seismic Record 1, 126 (2021), arXiv 2108.05421 | Neural operator trained across random velocity models, then autodiff through it for full waveform inversion | Nearly an order of magnitude faster than conventional full waveform inversion | An inverse problem, which is the use pattern the other five lack and the one where the break-even arithmetic favours a surrogate most, since an inversion makes thousands of forward calls. Grid-free evaluation above training resolution is claimed in the abstract | B lineage.md:317-326 |
| Automotive aerodynamics | GINO, arXiv 2309.00583; UPT, arXiv 2402.12365; Transolver++, arXiv 2502.02414 | Signed distance function plus GNO in, FNO in a latent grid, GNO out; or attention over learned slices | GINO: 26,000× over optimized GPU CFD for drag at Reynolds numbers up to five million, and a one-quarter reduction in error rate against other deep learning methods on unseen geometry. Transolver++: over 20 percent gain on million-point car and aircraft meshes | The speed-up is for the drag coefficient, one scalar per geometry, not the full field, so it fails the "same quantity" test above. UPT separately reports GINO at 900 s per epoch against its own 4 s (foundation.md:66) | B lineage.md:60-69, :137-146 |
The caveat column has a pattern in it. Every one of these results is real, every headline number is larger than the honest number, and the gap is never fraud: it is the difference between a per-call ratio and a total-cost ratio, or between a scalar and a field, or between a forecast and a control loop. That gap is what the ledger below is for.
A search for neural operator work on semiconductor or chip physics from the Anandkumar group returned nothing. Two papers from other groups apply physics-informed operators to EUV lithography mask simulation: "Physics-Informed Neural Operator for Warm-Starting Background-Decomposed and Preconditioned PSFD: Enabling Scalable 3-D EUV Mask Simulation", and "Physics-informed neural networks and neural operators for a study of EUV electromagnetic wave diffraction from a lithography mask". Their arXiv identifiers were never verified, and must be before either is cited (research_notes/lineage.md:328-329). That is an open item, flagged here rather than papered over.
Meanwhile the company names chip design as its first market: physics modelling of chip performance under heat and materials constraints (company-episode.md:126-128; seed episode 20:27 to 22:58). Anandkumar names inverse lithography mask design and quantum-dot gate design as her inverse-design examples in the earlier episode (01:16:00 to 01:17:15, :81). So the market with the loudest claim has the thinnest public record, which is an awkward but useful fact: it means anything you build here is not competing with a published baseline, and also that you have no published baseline to check yourself against.
What this course does about it. It generates its own semiconductor data rather than waiting. Lab 02 in Module 6 solves steady heat conduction in a 10 mm silicon die with random rectangular power blocks, which is the simplest physics behind "faster chips that don't melt", using a finite-difference solver checked against an analytic solution to \(1.25\times10^{-5}\). Every parameter in it is a modelling choice made in this repository and is marked as such on that page. It is a first physics rather than the field: EUV mask diffraction is a Maxwell problem on nanometre features, drift-diffusion is a coupled nonlinear system, and neither has public neural operator data either.
The company launched on 25 August 2026, founded by Anima Anandkumar and Benedikt Jenik after declining a reported 35 percent stake, a salary package rising to 2 million dollars, and over 2 billion dollars of committed financing at Project Prometheus (company-episode.md:117-139). Funding and investors are undisclosed. The seed episode runs 26 minutes 56 seconds and was published on 4 September 2026.
The grades below are not a verdict on the company, which may well be doing exactly what it says. They are a statement of what a reader outside it can currently check, and the last column of the lab task turns each one into a specific experiment that would change the grade.
| Claim | Source | Public evidence that bears on it | Grade |
|---|---|---|---|
| Models operate in 3D plus time without flattening | company site | FNO-3D in the 2021 paper already convolves in space-time. This is standard practice rather than a differentiator | C |
| Up to 1 trillion parameters at training; 5 trillion tokens or context at inference; "5 trillion data points in a single prompt" in testing | company site; KuCoin, vktr.com, Anandkumar's own post | No benchmark and no independent verification by any outlet; a Tech Times headline reads "No Benchmark Proof Yet". Module 9 costs the claim at about 4.66 PiB of activation memory, roughly 65,000 accelerators for one forward pass, or about 8,000 under the most generous assumptions | C |
| Resolution invariance: any level of detail across training and inference | company site; both episodes | True for neural operators as a representational property (JMLR Theorem 8). Accuracy at an unseen resolution is a separate and conditional matter, and Lab 01 measured a 56 percent error rise at the first unseen grid, traced in Module 2 to a fixed Nyquist artefact | B for the property, C for the accuracy |
| Cross-physics uplift: one model beats equal-size specialists | company site; seed episode | The narrow version is supported by Poseidon, Subramanian et al. and DPOT with reproducible numbers. The broad version has documented counterexamples, chiefly MPP's shallow-water result and Subramanian's Helmholtz out-of-distribution result. No company data exists either way | B narrow, C broad |
| Self-improvement loop with physics laws as dense directional feedback | company site; seed episode 15:00 to 18:30 | Majumdar et al. 2023 reach 1.07× and 1.02× the supervised error on Burgers and Darcy with the physics loss as the only signal, which is the mechanism working on two small PDEs. Module 11 derives why it gets harder as the grid refines, at \(N^4\) in the condition number | B for the mechanism at small scale, C at the claimed scale |
| Not a transformer; built on neural operators | KuCoin, vktr.com | Consistent with the founders' published work, and unverifiable for the product. Module 9 notes that the usual argument for this position, the quadratic cost of attention, rules out one design and not the family | C |
| First markets: chip design, energy including geothermal, robotics, weather | analyticsinsight.net, vktr.com | Weather has the strongest public record of the four (Module 10). Chips have none, as the previous section documents | C |
Two items in the notes remain open and should stay visible. The Rappler founder profile and the body of the Tech Times article both returned HTTP 403 and were never read. And whether the MIT-affiliated Benedikt Jenik who works on self-driving perception is the same person as the co-founder is a plausible but unverified name match (company-episode.md:144, :176-183). An unverified identification is not a fact, and this course does not promote one by repetition.
LAB A reading task, one page, and the only lab in the course with no code in it.
Pick one row of the applications table and read the primary paper's experiment section and not its abstract. For every number in the row, record five things: what quantity was measured, what reference it was measured against, the hardware on both sides, whether offline cost is counted, and whether the number describes a field or a scalar. Where the paper gives \(T_{\text{off}}\), compute the break-even count \(n^*\) from the formula above and say how many evaluations the reported use case actually needs. Where it does not, say so, and say what the missing number would have to be for the claim to hold.
Then find the company or press statement that cites the same work, and grade the distance between the two. The deliverable is a table like the one above with one column added: "what I would need to see to raise the grade". That column is the useful output, and writing it is harder than it looks, because "more evidence" is not an answer and "a benchmark" is not either.
research_notes/lineage.md, with arXiv identifiers verified by abstract-page fetch; repository URLs found by search are marked there as reported and not verified (:331-336).research_notes/pdf-fno-jmlr.md:159-172. The break-even count of about 19,700 is computed on this page from those figures. Benchmark fairness caveats: research_notes/pdf-mgn-pdebench.md:331.research_notes/company-episode.md:102-183; seed transcript at ../transcript.txt.research_notes/foundation.md:156-159.../labs/01_burgers/. The activation-memory costing of the trillion-context claim: Module 9, from ../labs/03_scale/results_scale.json.You can now compute the break-even point of a surrogate, name the six things that must match before a speed-up ratio means anything, and write down for any claim the specific experiment that would change its grade.