Add wave equation PINN example

[gpartin]

Summary

Adds a Jupyter notebook example demonstrating how to solve the 1D wave equation using TorchPhysics:

feature/add-wave-equation-example\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}feature/add-wave-equation-example

The wave equation is fundamental in physics and engineering (acoustics, electromagnetics, seismology), but the current examples don't include one. This fills that gap.

What's included

• examples/pinn/wave-equation.ipynb: Complete notebook with:
  • Problem definition (fixed-end string with sinusoidal IC)
  • PDE residual using tp.utils.grad for second derivatives
  • Dirichlet boundary conditions (fixed ends)
  • Initial displacement and velocity conditions
  • Training with pytorch-lightning
  • Animation of the learned solution
  • Quantitative comparison with analytical solution

Problem setup

• Domain: x in [0, 1], t in [0, 1], wave speed c = 1
• BCs: u(0,t) = u(1,t) = 0 (fixed ends)
• ICs: u(x,0) = sin(pi*x), u_t(x,0) = 0
• Analytical solution: u(x,t) = sin(pix) cos(pit)

Technical notes

• Follows the same structure as the heat-equation example
• Uses NormalizationLayer for stable training
• Four PINNConditions: PDE residual, boundary, initial displacement, initial velocity
• The wave equation requires enforcing the initial velocity separately since it's a second-order PDE in time

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