嘗試用擴散模型取代 CFD：查表法勝出，並助方法奏效

The idea

Designing aerodynamic shapes for EVs or delivery drones means running CFD (computational fluid dynamics) for every shape change, and serious runs can take days. My pitch was:

Train a diffusion model constrained by the Navier-Stokes equations to instantly generate and optimize surface geometries in real time, based on live wind-tunnel or flight telemetry.

The constraints: cost $0, start building now, and base every design decision on a set of 18 research papers.

Code, data and every result: github.com/Flotrpy/airfoil-diffusion

Scoping it down so it was buildable

Full Navier-Stokes CFD at zero cost isn't realistic, so the first plan shrank the problem:

• Shapes: 2-D wing sections (airfoils) for a small delivery drone, described by 11 numbers (a CST/Bernstein parameterisation, following the Airbus diffusion paper).
• Solver: NeuralFoil, a fast neural stand-in for XFOIL. It is not Navier-Stokes, and I say so throughout.
• Regime: Reynolds number 2×10⁵ to 3.5×10⁵, angle of attack 0–10°.
• The task: given a flight condition and a target lift and drag, propose an airfoil. A hit means NeuralFoil confirms lift within 1% and drag within 5% (the tolerances from Geng et al.).
• Compute: everything in plain NumPy (no PyTorch) on a free 2-CPU cloud sandbox.

The dataset is 12,000 random airfoils × 44 flight conditions ≈ 350k labelled examples, split by airfoil, so the test shapes were never seen in training.

Iteration 1: diffusion loses to a lookup table

I trained a conditional DDPM (the Airbus architecture) and compared it against a boring baseline: find the nearest airfoil in the training set and return it.

More data helped both, but diffusion never caught up. My own Phase 1 gate failed.

Lesson: always run the dumb baseline. A lookup table over a dense dataset is a strong inverse-design method.

Iterations 2–4: stop generating from scratch

Instead of throwing away the lookup, I built on top of it with retrieve → refine → verify:

1. Retrieve the nearest training airfoils.
2. Refine them by gradient descent on a small MLP surrogate of the solver (optionally after a light diffusion "warm start").
3. Verify every candidate with NeuralFoil; each check counts against the budget.

The first version won at 10 and 100 tries but lost on the first try (57% vs 75%). It kept "fixing" airfoils that were already correct.

Adding a gate ("leave it alone if the surrogate says it's already a hit") only got to a tie. The surrogate's error was about as big as the 1% tolerance, so it couldn't tell hits from misses.

The fix that worked was almost embarrassingly simple. The targets sit on the same (Re, angle) grid as the training data, and NeuralFoil had already scored every training airfoil at those exact conditions. So the gate reads the stored result instead of guessing: zero error, a free bias correction for the surrogate, and no extra solver calls.

The gate passed: 83% / 97% / 99% vs the lookup's 75% / 91% / 98%.

Then came the uncomfortable result: with diffusion switched off, the same pipeline scored 93% on the first try. A strength sweep on validation data agreed: no diffusion beat every diffusion setting.

Iteration 5: the realistic test, between grid points

Real telemetry doesn't arrive on a neat grid. Between the grid points, the lookup falls apart and the stored-result trick has nothing to read.

• Lookup: 1% on the first try, 16% within 10.
• Retrieve-refine-verify: ~50% on the first try, 98% within 10.
• Iteration 7 interpolates each retrieved airfoil's stored results from the 4 surrounding grid points. That estimate is ~3.5× more accurate than the surrogate and lifts the first try to 58%.

Iteration 6: making diffusion physics-aware

Back to the original idea: a constrained diffusion model. I fine-tuned it with a PIDM-style loss (the surrogate's lift/drag of each denoised shape must match the request), balanced against the normal denoising loss with ConFIG gradient combining. All of it is hand-written backprop in NumPy, checked against finite differences.

The physics error of generated shapes fell ~30%. Diffusion alone improved modestly, most clearly off-grid (10% → 18% on one sample, 58% → 70% within 10). It still trails direct surrogate polishing.

What I'd tell myself at the start

1. Beat the baseline before celebrating the model. The lookup table was the most important experiment in the project.
2. Your stand-in solver must be more accurate than your tolerance, or every gate and filter built on it is noise.
3. Reuse what you've already paid for. Stored solver results gave an exact gate and a free bias correction.
4. Test the realistic case. On-grid numbers looked great; off-grid is where the method actually earns its keep.
5. Report the ablation that hurts. Diffusion turned out to be optional here, and that's useful to know.

Honest limits

• NeuralFoil both labels the data and judges the answers. Results need re-checking with XFOIL or a RANS/Navier-Stokes solver before any real wing.
• 2-D sections only: no 3-D wing, no structure, no manufacturing constraints.
• 100 targets per test set, so differences of a few points are noise.

Try it

git clone https://github.com/Flotrpy/airfoil-diffusion
cd airfoil-diffusion
pip install -r requirements.txt
python scripts/01_make_dataset.py --n-airfoils 12000
python scripts/03_train_diffusion.py --steps 200000 --width 128
python scripts/07_refine_eval.py

All the numbers in this post are in docs/blog/results.csv and results/*.json. Feedback is very welcome, especially from anyone with access to a real CFD solver.