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}

[oroikono]

I’ve addressed the reported path and output-directory issues, rebased on main, and pre-commit is passing. Full NVIDIA CI appears to require external-contributor approval. Could a maintainer confirm whether this example is in scope and whether you would prefer the generated artifacts removed or the PR split into smaller pieces? I’m happy to reduce the review surface.