feat(domain): add 2D solver-in-the-loop corrector benchmark

[andrinr]

Summary

Add a solver-in-the-loop neural-correction benchmark to 2D Navier–Stokes
alongside the existing drag-optimization task.

The benchmark owns the train/test split, references, Equinox corrector,
initialization, optimizer, recurrent objective, paired stop-gradient control,
and evaluation. Candidate cells vary only in the differentiable solver
transition and its VJP.

The central result is a two-stage distinction:

1. all six differentiable solvers pass the same analytic forward control;
2. only PICT and Warp-NS currently preserve enough state across the canonical
   Tesseract boundary to pass a 100-call recurrence test.

No solver is hidden to make the comparison agree. The four failed cells remain
visible as adapter diagnostics, and the complete intrinsic six-solver VJP
ranking is withheld until their recurrent state is preserved.

Type of change

• New solver backend
• Solver tuning / improvement
• New benchmark domain
• Harness / infrastructure change
• Documentation
• Bug fix

Corrector and paired training

The corrector is an Equinox PeriodicResidualCNN, not a U-Net:

• circularly padded 5×5 convolutions, channels 2 → 32 → 64 → 2;
• 56,098 identical parameters for every solver;
• zero-initialized residual head, so update zero is exactly solver-only;
• additive correction after each solver interval;
• zero-mean, divergence-free spectral projection of each correction.

Every model seed is trained twice:

• full VJP: temporal credit propagates through the differentiable solver;
• stop-gradient: the identical forward solver recurrence and local
  supervision are retained, but temporal credit through the solver is cut.

The reported solver-VJP lift is the paired
log(error_stop-gradient / error_full-VJP) over the same seeds and held-out
ICs. Each cell's loss is normalized by its own uncorrected recurrent baseline,
so raw solver accuracy does not change the optimizer scale.

Fair all-six forward gate

The analytic control is the smooth N=64 Taylor–Green case from the established
forward benchmark (nu=0.05, dt=0.01). Admission thresholds are declared as
1% relative error for the first interval, 1% for an uninterrupted t=1
trajectory, and 1% for repeated/native closure after 100 canonical calls.

XLB receives four internal substeps per canonical interval. This is the
smallest tested Mach-safe budget that makes its uninterrupted forward solution
pass the same 1% gate; two substeps left 1.79% long-forward error.

Solver	First interval	Uninterrupted t=1	100-call closure	Ranking eligible
JAX-CFD	0.243%	0.441%	21.203%	no
INS.jl	0.241%	0.233%	21.230%	no
PhiFlow	0.241%	0.238%	21.230%	no
PICT	0.0056%	0.0122%	0.000%	yes
Warp-NS	0.0090%	0.0098%	0.000%	yes
XLB	0.677%	0.804%	10.351%	no

This resolves the apparent contradiction in the earlier plots: the native
forward trajectories are accurate, while repeated velocity-only calls are not
the same state transition for four adapters.

• JAX-CFD, INS.jl, and PhiFlow discard native staggered face velocity. Repeated
  center↔face conversions attenuate the state.
• XLB discards density and non-equilibrium population moments, then rebuilds
  equilibrium populations from velocity on every call.
• PICT and Warp-NS have velocity-complete periodic transitions and close at
  numerical precision.

These are interface-state failures, not evidence that the underlying native
solver VJPs are intrinsically poor.

Nonlinear shared-reference task

The learning stress task uses a benchmark-owned 64² dealiased pseudo-spectral
vorticity RK4 reference, spectrally restricted to the shared 32² grid. It is
not an upscaled candidate solver.

• multimode k0=6, amplitude 0.5, nu=0.001;
• four native steps of dt=0.02 per correction interval;
• 16 training ICs (seeds 0–15) and eight disjoint held-out ICs (100–107);
• three paired model seeds and 200 optimizer updates;
• recurrent training to t=1.92, held-out rollout to t=2.88.

The flow changes materially over the rollout and uses doubly periodic
[0,2π]² boundaries, appropriate for homogeneous decaying turbulence and the
translation-equivariant corrector. The benchmark does not claim wall or inlet
generalization; the retained drag task covers that separate regime.

All six cells run and render. PICT and Warp-NS are the conditional intrinsic
comparison because they pass recurrence admission and begin at matched
first-interval p95 error (6.32% and 6.28%).

Quantity	PICT	Warp-NS
Geometric correction gain	2.091×	1.608×
Final held-out error	0.182	0.207
Incremental solver-VJP lift	1.08%	3.34%
Paired bootstrap 95% CI	−0.69% to 2.40%	2.09% to 4.44%
Median full-VJP update	2.234 s	0.846 s

Warp-NS's incremental VJP lift is distinguishably larger: its lift ratio over
PICT is 1.022× (95% CI 1.008–1.036×). PICT nevertheless learns the stronger
overall correction and lower final error. Warp-NS is about 2.6× faster per
full-VJP update.

So the differentiable solver VJP does help, but modestly. In this nonlinear
task the evidence favors Warp-NS for temporal credit assignment and efficiency,
while PICT is more correctable in absolute terms.

Solver-specific refined-reference control

optimization/solver_in_loop_self_reference asks the upscaling question:

For a recurrence-admitted solver, does differentiating through its coarse
transition help imitate that solver's own higher-resolution, smaller-step
trajectory?

Each solver supplies its own 64², half-step reference, restricted to the 32²
coarse grid. Absolute errors are not compared across solvers because their
targets differ; the paired within-solver VJP lift is comparable.

Quantity	PICT	Warp-NS
Geometric correction gain	1.923×	1.527×
Incremental solver-VJP lift	3.56%	4.05%
Paired bootstrap 95% CI	1.72% to 5.27%	2.32% to 5.83%
Median full-VJP update	2.214 s	0.810 s

Both VJPs help at 95% confidence, but their lift difference is not significant.
Warp-NS remains about 2.7× faster.

Analytic TGV null control

The all-six TGV experiment is intentionally smooth and near the error floor.
Among the admitted adapters, PICT and Warp-NS obtain incremental VJP lifts of
2.19% and 0.69%; total correction gains are 1.421× and 1.055×. This confirms
the paired machinery but is not a substitute for the nonlinear learning task.

Supporting JSON, complete plot inputs, physics diagnostics, and job provenance:
local artifact index.

Other fixes included

• Select the reference per forward-agreement run. Multimode fields now use
  cross-solver consensus instead of being compared with an unrelated analytic
  TGV field.
• Repair the PICT projection path and build its runnable container through the
  daemon-free Slurm/Pyxis workflow.
• Preserve the existing drag-optimization registrations.
• Keep all Slurm/build helper tooling outside this repository.

Validation

All experiments and validation ran through Slurm on Kander.

• nonlinear production: JAX-CFD 1642897, INS.jl 1642898,
  PhiFlow 1642899, PICT 1642900, Warp-NS 1642901, XLB 1642902;
  merge/render 1644051;
• self-reference production: PICT 1642930, Warp-NS 1642931;
  merge/render 1644052;
• final all-six TGV: JAX-CFD 1644252, INS.jl 1644253,
  PhiFlow 1644254, PICT 1644255, Warp-NS 1644256, XLB 1644258;
  merge/render 1644264;
• final Ruff 1644419, format 1644422, full pytest 1644423
  (499 passed, three skipped), problem configs 1644424, Tesseract configs
  1644425, git-aware pre-commit 1644426, artifact integrity 1644825.

Generated benchmark results live only on the artifact branch. This PR remains
a draft with no benchmark label while the comparison and native-state follow-up
are reviewed.