refactor(notebooks): use internal adam for autograd27 smatrix

[marcorudolphflex]

Summary

This updates Autograd27Smatrix.ipynb as a standalone extraction from #465.

• Replaces the direct optax optimizer setup and manual update loop with Tidy3D's internal adam and optimize helpers.
• Uses learning_rate = 0.04, which avoids the large objective spike seen with the earlier 0.05 rerun.
• Removes the notebook-level optax dependency.
• Sanitizes local output metadata and local path warning outputs from the rerun.

Optimization Result

Lower-LR internal-Adam rerun, final saved step:

• Step: 25
• Objective J: 2.6270e-02 (vs 1.0909e-02 before)
• Gradient norm: 1.5076e-02
  before

after

Validation

• uvx ruff format --check --diff Autograd27Smatrix.ipynb
• uvx ruff check Autograd27Smatrix.ipynb
• uv run --no-project python scripts/sync_metadata_tags.py --check
• uv run --no-project python scripts/validate_notebook_metadata.py Autograd27Smatrix.ipynb
• uv run --no-project python scripts/check_notebook_private_paths.py Autograd27Smatrix.ipynb
• python3 misc/check_misc_references.py
• uv run spellcheck.py Autograd27Smatrix.ipynb

---

Note

Low Risk
Notebook-only refactor with no production code paths; main review note is that the rerun yields a higher final objective than before, which is expected from the hyperparameter change rather than a functional regression.

Overview
Autograd27Smatrix.ipynb now follows the same optimization pattern as other autograd examples: the inverse-design loop uses tidy3d.plugins.autograd adam and optimize instead of a third-party optax setup and hand-written update steps.

The optimization cell wires record_step as a optimize callback (density plots plus step logs for J, beta, and gradient norm) and keeps parameters in [0, 1] via bounds. learning_rate is set to 0.04 for the rerun (replacing 0.05), which the author reports avoids a large objective spike. Committed outputs were refreshed for the new run (e.g. final step 25, J ≈ 2.63e-02).

This is documentation/example-only: no library or simulation API changes.

^{Reviewed by Cursor Bugbot for commit fa22e8c. Bugbot is set up for automated code reviews on this repo. Configure here.}

[github-actions]

Spell check passed successfully for 1 notebook(s).
Generated by GitHub Action run: https://github.com/flexcompute/tidy3d-notebooks/actions/runs/27758889360

[groberts-flex]

looks good thanks @marcorudolphflex