Add gray-box term-discovery example

[oroikono]

Add gray-box term-discovery example

Closes #1834

Description

Adds examples/graybox_discovery/, an example that recovers an unknown term in a
PDE (a reaction/closure term) rather than an unknown scalar coefficient.

A 1D reaction-diffusion problem u_t = D u_xx + R(u) is set up with R unknown.
A field network learns u(x, t) and a second network learns R(u), trained
jointly from point observations. The residual is declared in SymPy with R left
as a Function; a PhysicsInformer evaluates it each step (autodiff for the
spatial derivative, a manual torch.autograd.grad for the time derivative since
autodiff spatial gradients only cover x/y/z), so no change to the
residual-assembly code is needed.

The generated data uses a non-polynomial R(u) = 3 sin(pi u) and only observes
u <= 0.75, which leaves R under-determined near u = 1. Adding the known
equilibria R(0) = R(1) = 0 as a direct penalty on the reaction network pins
the closure in the unobserved range.

What's included

• examples/graybox_discovery/generate_data.py — synthetic data
• examples/graybox_discovery/graybox_reaction_diffusion.py — training script
• examples/graybox_discovery/conf/config.yaml — config
• examples/graybox_discovery/README.md — description, run instructions, and
  the recovered-vs-true R(u) plot
• examples/graybox_discovery/graybox_reference.py — numpy-only reproduction
  of the mechanism (no PhysicsNeMo/GPU required)
• examples/graybox_discovery/test_graybox_verifiers.py — checks the prior
  reduces the spread of the recovered closure and improves recovery in the
  unobserved region (per CONTRIBUTING.md, example tests live under
  examples/, not the top-level test/)

No changes to core APIs.

Testing

pytest examples/graybox_discovery/test_graybox_verifiers.py runs the
reference-based checks with no GPU/torch requirement (one additional guarded
test is skipped until the physicsnemo/sym/eq/verifiers.py follow-up PR
lands). The example itself was run end to end in a real PhysicsNeMo
environment via python generate_data.py && python graybox_reaction_diffusion.py; training converges (loss ~0.12 -> ~5e-4 over
20k steps) and the recovered R(u) tracks the true closure well past the
observed range (u_obs = 0.75), which is the effect the physics prior is
meant to produce -- see the plot in the example README.

Checklist

• Commits signed off (DCO)
• Apache-2.0 headers on new files
• Docstrings on new public functions/classes
• Ran pytest examples/graybox_discovery/test_graybox_verifiers.py
• Ran ruff check / ruff format --check (clean)
• Ran test/ci_tests/header_check.py and
  test/ci_tests/check_docstring_coverage.py locally (clean)
• Ran markdownlint-cli on changed Markdown (clean)
• Updated CHANGELOG.md

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[greptile-apps]

Greptile Summary

This PR adds a gray-box term-discovery example for a reaction-diffusion PDE. The main changes are:

• New synthetic data generation and committed example artifacts.
• New Hydra training script with field and reaction networks.
• New NumPy/SymPy reference implementation and example tests.
• New README and changelog entry.

Important Files Changed

Filename	Overview
examples/graybox_discovery/graybox_reaction_diffusion.py	Adds the main training script; the default data load depends on the caller's working directory.
examples/graybox_discovery/generate_data.py	Adds deterministic data generation; the output file is written to the caller's working directory.
examples/graybox_discovery/graybox_reference.py	Adds a deterministic NumPy/SymPy reference implementation for the example tests.
examples/graybox_discovery/test_graybox_verifiers.py	Adds tests for the reference mechanism and a guarded verifier-helper import check.
examples/graybox_discovery/conf/config.yaml	Adds training, scheduler, batch, loss, and data path configuration.
examples/graybox_discovery/README.md	Documents the example workflow and expected recovered reaction plot.
CHANGELOG.md	Adds a changelog entry for the new gray-box discovery example.

_{Reviews (1): Last reviewed commit: "Add gray-box term-discovery example" | Re-trigger Greptile}

[greptile-apps]

Data Path Depends On CWD

When the training script is launched from the repo root, np.load("graybox_data.npz") looks in the repo root instead of examples/graybox_discovery/, so it misses the bundled data file and can fail before training starts. Resolve the default relative path from the example directory, while still allowing absolute override paths.

[greptile-apps]

Generated Data Leaves Example Directory

When this script is run as python examples/graybox_discovery/generate_data.py, it writes graybox_data.npz into the caller's current directory instead of next to the example. A later training run that expects the example-local data can then read stale bundled data or fail to find the newly generated file.