Fix errors and modernize keras and jax_nn lectures

[jstac]

Summary

• Fix miscellaneous errors and modernize the keras and jax_nn neural network lectures

Changes

keras.md

• Fix validation bug: validation data was using the same seed as training data (seed=1234), so "validation" was evaluating on identical data. Changed to seed=5678
• Fix incorrect terminology: replaced "cross-validation" with "validation" throughout (the lecture does a train/validation split, not k-fold cross-validation)
• Modernize NumPy RNG: replaced legacy np.random.seed / np.random.randn with modern np.random.default_rng / rng.standard_normal API
• Fix heading level: "Regression model" was ## (h2) but should be ### (h3) under the "Models" h2, matching its sibling "Deep Network"
• Remove duplicate code: removed second identical definition of plot_results
• Fix typos: "adds a single layer the activation" → "with the", "train our models" → "train our models", "In a a follow-up" → "In a follow-up"
• Fix spelling: "Tensorflow" → "TensorFlow"
• Move Keras backend note: moved the detailed troubleshooting note about KERAS_BACKEND from jax_nn.md to here, where the backend is first introduced

jax_nn.md

• Fix incorrect description: parameters are stored as "named tuples" not "dictionaries" (two occurrences)
• Fix typos: "of a the network" → "of the network", "corresponding to" → "corresponding to", "since will not" → "since we will not"
• Fix capitalization: "ADAM" → "Adam" throughout (it's not an acronym — the original paper styles it "Adam")
• Modernize type hints: replaced deprecated jnp.ndarray with jax.Array throughout
• Remove unnecessary f-string: f"Training NN using Keras." → "Training NN using Keras."
• Extract reusable plotting function: added plot_jax_output to replace three identical scatter+line plot blocks; placed after f and loss_fn definitions so it follows its dependencies
• Simplify plot_keras_output: removed unused x, y parameters
• Clean up TODO comments: removed leftover development notes in summary table code
• Move Keras backend note: moved to keras.md where it belongs

Test plan

• Converted keras.md to Python via jupytext and ran successfully

🤖 Generated with Claude Code

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[jstac]

@mmcky please review and make live when ready. Thanks!

[mmcky]

Reviewed — all changes look correct. Fixed the CI error: plot_jax_output had curly/smart quotes (U+2018, U+2019) instead of ASCII quotes in Python string literals, causing the lexer failure. Pushed a fix.

[github-actions]

🚀 Deployed on https://69d5a4f496e388cf4a89bbde--incomparable-parfait-2417f8.netlify.app

[mmcky]

thanks @jstac. Merging this now.