Add 1D wave equation example

[gpartin]

Summary

Add a 1D wave equation example to the PINNs-TF2 examples collection.

Problem

The wave equation u_tt = c^2 * u_xx is one of the fundamental PDEs in physics and engineering, but there is currently no wave equation example in the repository.

Solution

This PR adds a complete wave equation example following the existing example structure:

• examples/wave_equation/train.py — Defines the PDE residual, initial conditions (displacement and velocity), boundary conditions, and generates analytical reference data
• examples/wave_equation/configs/config.yaml — Hydra configuration with training parameters
• examples/wave_equation/README.md — Problem description and setup table

Problem Setup

The example solves:

u_tt = c^2 * u_xx,  x ∈ [0,1], t ∈ [0,1], c = 1
u(0, x) = sin(πx)           (initial displacement)
u_t(0, x) = 0               (initial velocity)
u(t, 0) = u(t, 1) = 0       (Dirichlet BCs)

Exact solution: u(x,t) = sin(πx) cos(πt)

Design Choices

• Uses analytical data generation (root_dir: null) instead of loading from .mat files, keeping the example self-contained with no external data dependencies
• Enforces both displacement (u) and velocity (v = u_t) initial conditions for the second-order-in-time PDE
• Network architecture: [2, 50, 50, 50, 50, 50, 1] (5 hidden layers of 50 neurons)
• 20,000 training epochs with MSE loss

This mirrors the wave equation example in PINNs-Torch, adapted for the TensorFlow 2 API.

[Copilot]

Pull request overview

Adds a new self-contained 1D wave equation example to the examples/ collection, following the repo’s existing Hydra + pinnstf2.train() example structure and using analytically generated reference data.

Changes:

• Introduces examples/wave_equation/train.py with PDE residual, initial/boundary conditions wiring, and analytical reference solution generation.
• Adds examples/wave_equation/configs/config.yaml defining domains, samplers, network architecture, and training settings.
• Adds examples/wave_equation/README.md documenting the PDE setup and training configuration table.

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 2 comments.

File	Description
examples/wave_equation/train.py	Implements wave-equation residual and analytical solution generation for mesh-based training/validation.
examples/wave_equation/configs/config.yaml	Provides Hydra config for mesh, samplers (collocation/IC/BC), model, and trainer settings.
examples/wave_equation/README.md	Documents the PDE, exact solution, and problem setup table for the new example.

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[Copilot]

config.yaml requests initial-condition supervision for both u and v, but read_data_fn() only returns {"u": exact_u}. This will raise a KeyError when Mesh.on_initial_boundary() tries to slice self.solution["v"] for the initial condition sampler. Include an analytical v solution array in the returned dict (e.g., v = ∂u/∂t) or adjust the config to avoid requesting v from the mesh solution.

[Copilot]

The markdown table is not formatted consistently with other example READMEs: the rows start with || (double pipe), which renders as an extra empty column in many Markdown parsers. Use a standard table format (single leading | per row) like the other examples under examples/*/README.md.

Suggested change

	| 1D Wave Equation | |
	|------------------------------|---|
	| Quantity | Value |
	|------------------------------|-------------------------------|