feat(domain): add autoregressive solver-loop curriculum

[andrinr]

State of this PR

This stacked follow-up turns the compact temporal-credit audit in #116 into a
paper-aligned autoregressive correction study. It keeps the existing
optimization/solver_in_loop experiment unchanged and adds the opt-in
optimization/solver_in_loop_curriculum experiment.

The new experiment separates three training protocols while holding the
solver, corrector architecture, data, optimiser schedule, and free-running
evaluation fixed:

• ONE: one-step correction from reference states;
• NOG: recurrent solver–corrector training with the solver outputs stopped
  in the backward pass;
• WIG: the identical recurrent forward trajectory with full temporal
  differentiation through the solver.

Protocol

• 32² candidate grid with correction after every native step (dt=0.02);
• 128² pseudo-spectral production target and gated 256² convergence audit;
• 32 training ICs and eight held-out ICs;
• 259,554-parameter, five-block periodic ResNet corrector;
• staged 1 → 2 → 4 → 8 → 16 → 32 look-ahead curriculum;
• 11,000 optimiser updates per training protocol and model seed;
• three paired model seeds;
• supervision at every corrected state, without the solver-terminal auxiliary
  used by feat(domain): benchmark all differentiable solvers in the loop #116;
• 300-step fully autoregressive held-out evaluation.

The output separates the gain from recurrent state exposure (NOG / ONE) from
the incremental gain from temporal solver gradients (WIG / NOG). It also
records curriculum checkpoint quality, a five-scale per-seed FD/AD sweep, update and wall-clock cost, correlation
horizon, energy/enstrophy/divergence diagnostics, final energy spectra, full
fields, and a five-column reference/solver/ONE/NOG/WIG rollout GIF.

Validation

• Ruff lint and format pass on all changed files.
• 530 tests pass and three are skipped.
• The curriculum has an end-to-end in-process solver smoke covering staged
  optimisation, changed learning rates, ONE/NOG/WIG rollout arrays,
  correlations, checkpoint shapes, and rendering.

Offline results

Production results are pending. All numerical experiments will run offline on
Kander through the same Slurm/Pyxis runner and solver images used by #116. No
hosted benchmark label will be used.

Literature correspondence

The terminology follows the controlled ONE/NOG/WIG comparison in
Differentiability in Unrolled Training of Neural Physics Simulators
and the recurrent SOL_n protocol in
Solver-in-the-Loop.

[PasteurBot]

📊 View the full benchmark results

No benchmarks ran for this PR, so there is no status report.

---

[andrinr]

Folded into #118. The full solver-in-loop curriculum implementation is now on that draft PR; the ongoing offline Kander runs and their aggregated plots/GIFs will be reported there.