Perfectly matched layers (PML)

Part 4 — Wave equations in PINN form

Learning objectives

Recognise PML as the absorbing-strip alternative to Clayton-Engquist
Derive the time-domain PML wave equation u_tt + σ(x) u_t = c² u_xx in a damping strip
Train a PINN with and without PML and watch energy absorption
Connect to Bérenger 1994's stretched-coordinate construction

Bérenger (1994 JCP) introduced the perfectly matched layer (PML) as an absorbing strip that — unlike Clayton-Engquist — absorbs waves at all incidence angles. The construction is mathematically beautiful: a coordinate stretching

\partial_x \to \frac{1}{1 + i \sigma(x) / \omega} \partial_x

in the frequency domain converts propagating waves to exponentially-decaying ones inside the PML strip. The damping function $\sigma(x)$ is zero in the interior and ramps smoothly to a maximum $\sigma_{\max}$ at the outer wall. Crucially the impedance match between PML and interior is perfect at every frequency and incidence angle — "perfectly matched".

Time-domain PML

The time-domain version — suitable for explicit FDTD or PINN solvers — is, in the simplest split-field-free form,

\frac{\partial^2 u}{\partial t^2} + \sigma(x) \frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2} ,

where $\sigma(x)$ is the damping profile. For $\sigma = 0$ this is the ordinary wave equation. For $\sigma > 0$ it is the wave equation with velocity-proportional damping — the analogue of friction. The pulse loses amplitude exponentially as it propagates through the PML strip, and (with a smooth ramp) does not reflect off the inner edge of the strip. Komatitsch & Tromp (2003 GJI) standardised the seismic implementation.

Same 1D rightgoing pulse as §4.6. We add a PML strip $x \in [1.6, 2.0]$ with quadratic damping ramp $\sigma(x) = \sigma_{\max} \big( (x - 1.6) / 0.4 \big)^2$ , $\sigma_{\max} = 30$ . Both walls are Dirichlet $u = 0$ ; the PML strip absorbs the pulse before it reaches the wall.

What you should observe

NO-PML: the pulse hits the right wall, reflects, and moves leftward at $t = 1.4$ . Same as §4.6 reflecting case.
WITH-PML: the pulse enters the PML strip (shaded region), loses amplitude, and effectively vanishes by the time it would have reached the wall. The $t = 1.4$ snapshot is essentially zero everywhere.
Residual energy decays exponentially in the PML strip; the energy curve drops faster than the §4.6 absorbing-BC case because the PML absorbs across a thick strip rather than at a single boundary point.

PML in modern seismic-PINN codes

Modern seismic-PINN papers (Rasht-Behesht 2022, Alkhalifah-Song-Waheed 2022) use PML on all open boundaries plus the §4.5 hard-constraint free surface at the top. The PML strip is typically 5–10 dominant wavelengths thick, with $\sigma_{\max}$ tuned so the round-trip transmission through the strip is < $10^{-3}$ (Komatitsch & Tromp 2003 give the formula). The PINN handles the spatially-varying $\sigma(x, z)$ exactly as it handles the spatially-varying velocity $c(x, z)$ — just additional inputs to the PDE residual.

References

Bérenger, J.-P. (1994). A perfectly matched layer for the absorption of electromagnetic waves. JCP 114(2), 185–200.
Komatitsch, D., Tromp, J. (2003). A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation. Geophys. J. Int. 154(1).
Festa, G., Vilotte, J.-P. (2005). The Newmark scheme as velocity-stress time-staggering: An efficient PML implementation for spectral element simulations of elastodynamics. Geophys. J. Int.