Stop arguing about which neural operator is better. We do not have to guess any more.

Two papers posted 11 hours apart to arXiv this week move this entire field past seven years of conflicting benchmark claims and into actual engineering. One proves there is no universal best spectral basis, and gives an exact predictive rule for which one you should use. The other delivers the first production grade CFD surrogate that reliably resolves both global flow patterns and near wall boundary layers.

This is not incremental progress. This is the point where scientific ML stops being a collection of clever tricks and starts being something you can reliably specify for a production job.

The control experiment no one ran for 7 years

Since Fourier Neural Operators were published in 2020, every single paper proposing an alternative operator has run benchmarks on a small set of hand picked PDEs, then declared their architecture universally superior. Every single rebuttal paper has done exactly the same thing with a different set of test problems.

No one ever ran a fair comparison. Every comparison published to date compared operators with different parameter counts, different activation functions, different learning rates, different normalisation. None of them controlled for all variables except the spectral basis.

This changed with the Hartley Neural Operator paper. The authors did one very simple, very powerful thing: they constructed an operator that is identical to FNO in every single respect except the transform used.

HNO replaces the complex FFT with the purely real Discrete Hartley Transform. Because the Hartley spectrum does not have conjugate symmetry, HNO retains twice as many frequency modes as FNO for the same truncation level. It then uses one real weight per mode, where FNO uses one complex weight per mode. At equal model width the two operators have exactly identical parameter count, exactly identical memory footprint, and exactly identical forward pass wall time.

They differ in exactly one property: the spectral basis.

Green's function alignment is everything

The central result of the paper is not that one operator beats the other. It is that performance is almost entirely determined by alignment between the basis symmetry and the symmetry of the underlying PDE operator.

All spectral neural operators work by approximating the kernel of the PDE solution operator, also called the Green's function. They do this by applying a diagonal multiplier in the transformed domain. This approximation is exact if and only if the Green's function shares the symmetry of the basis functions.

For self adjoint elliptic operators, Green's functions are real, symmetric, and have zero phase. The Hartley basis diagonalises these operators exactly. The complex Fourier basis does not, and wastes half its parameter budget representing conjugate pairs that will never be used.

For time dependent operators with transport or oscillation, Green's functions carry non zero phase. A real diagonal multiplier cannot represent phase shift. In this regime the Fourier basis is strictly more expressive, and all of the Hartley advantage reverses completely.

This is not an empirical observation. The authors prove this result analytically first, then verify it across 11 different PDE families, 3 boundary condition types, and 7 initial condition distributions.

The basis decision rule

This is the table you can take to work tomorrow. Every value is measured relative error between iso-parametric HNO and FNO trained under identical protocol. Values below 1 mean HNO is better. Values above 1 mean FNO is better.

Operator class	Example PDEs	Recommended basis	HNO / FNO relative error
Self-adjoint elliptic	Poisson, Biharmonic, Linear Elasticity	Hartley (HNO)	0.42
Diffusive phaseless	Heat equation	Either	0.97
Weakly advective low Re	Shallow water, porous flow	Either	1.12
Strongly advective turbulent	Burgers, Navier-Stokes	Fourier (FNO)	1.71
Oscillatory time dependent	Wave, Helmholtz, Acoustics	Fourier (FNO)	2.39

This result explains every contradictory benchmark result published in the last 7 years. Every paper that claimed HNO was superior ran tests on elliptic problems. Every paper that claimed FNO was superior ran tests on advective problems. None of them were lying. All of them were measuring basis alignment, not general architecture quality.

There is no universal winner

The most important conclusion of the paper is negative. There will never be a single best neural operator. There is no architecture that will dominate across all PDE classes. Anyone that tells you otherwise is selling something, or only ever tests on one type of problem.

This is good news. It means we can stop running endless benchmark bakeoffs. We can stop arguing on Twitter. We can instead measure the symmetry properties of the operator we need to learn, pick the correct basis, and move on to solving actual problems.

This is how engineering works. You do not debate whether a hammer is better than a screwdriver. You pick the right tool for the job.

The multiscale CFD surrogate problem

While one paper was resolving the fundamental basis question, the second paper was solving the largest practical failure mode of existing neural operators.

All pure spectral operators fail catastrophically at local structure. FNO will correctly predict the gross shape of a cylinder wake 1000 timesteps ahead. It will completely garbage the 1mm thick boundary layer right at the cylinder surface. That boundary layer is the part that determines drag, lift, separation, and every other quantity anyone actually cares about for engineering.

This failure mode is inherent to pure Fourier bases. Global spectral transforms cannot represent sharp local gradients efficiently. They will always bleed information across the domain.

Fourier-wavelet transformer architecture

The proposed architecture fixes this by splitting the input field into two parallel encoding branches, and using physics signals directly inside the attention mechanism.

There are three non obvious design choices that make this work:

1. The Fourier branch handles long range pressure propagation. The wavelet branch handles near wall gradients and shear layers. No single basis has to do everything.
2. Attention scores are biased directly by the local PDE residual calculated on the previous forward pass. The model automatically pays more attention to regions where it is currently wrong.
3. Masked pretraining does not mask random pixels. It masks the regions with highest residual error from the last evaluation pass.

That last point is the single most effective improvement to physics informed training published in the last three years. Instead of forcing the model to reconstruct arbitrary regions, you force it to practice fixing exactly the mistakes it is already making.

Benchmark performance

All tests were run against standard public benchmarks with fixed train/test splits, and compared against every major baseline from the last five years. No cherry picking.

On cylinder wake flow the model achieved an all channel Pearson correlation of 0.9702, compared to 0.9388 for standard FNO. Scale separated error analysis shows almost all of the gain comes from the highest frequency modes corresponding to near wall shear layers. The global low frequency error is almost identical across all models.

What still does not work

This is not a silver bullet. The paper is very clear about limitations, which is refreshing for this field.

The model will still not extrapolate more than 12 timesteps outside the training distribution. It will still fail catastrophically at flow separation points above Reynolds number 2e6. It runs 7x faster than an optimised RANS solver, not 1000x faster like you will see in press releases next month. It still requires 120GB of VRAM to train at industrial resolution.

None of these are fatal. They are known, documented tradeoffs. That is exactly what we need.

Engineering takeaways for production

These are the actionable conclusions you can use right now:

1. Stop defaulting to FNO for every problem. If you are solving elliptic boundary value problems for structural mechanics, heat transfer, or electrostatics, HNO will give you 2.3x lower error for exactly zero extra cost. Just swap the transform. Nothing else changes.
2. If you are doing any time dependent flow, stick with FNO. All of the Hartley gains reverse completely.
3. For any CFD surrogate work, stop using pure spectral operators. You need a wavelet branch. There is no way around this.
4. Masked pretraining works. But mask by residual, not uniform random. This one change will give you more improvement than any architectural modification you can make.
5. Stop running cross PDE architecture comparisons. They do not measure anything useful.

Open remaining questions

We still do not know what the correct basis is for non linear operators. We still do not have a good computable measure for phase content of an arbitrary PDE. No one has extended this analysis to spherical harmonics, unstructured meshes, or graph operators. We still have no good theory for how basis alignment interacts with non linear activation functions.

None of that stops you from using these results today.

Closing

For a very long time scientific ML has operated like alchemy. People would try 20 different architectures, pick the one that happened to work on their problem, and write a paper declaring it the one true approach.

That era is over. We now have testable rules. We have known tradeoffs. We can pick the correct tool before we start training. We can predict performance before we burn ten thousand dollars on GPU time.

That is not incremental improvement. That is the point where a field stops being research and starts being engineering. These two papers are the most important work published in this space in the last three years. Go read them.