Frequency-domain (Helmholtz) formulation

Take the wave equation $u_{tt} = c^2 \nabla^2 u$ and Fourier-transform in time. With $u(x, z, t) = \textrm{Re}[U(x, z; \omega) e^{-i \omega t}]$ the result is the Helmholtz equation

where $k = \omega / c$ is the wavenumber and $f$ is the (frequency-domain) source. The Helmholtz equation has no time variable — it is a steady-state problem at one frequency. The PINN ansatz becomes $U_\theta(x, z)$ (2 inputs); the loss is just PDE + BC.

Why the frequency domain matters in seismic

Industrial seismic FWI codes (Plessix 2006 RAM3D, Operto et al. 2007, Pratt 1999) overwhelmingly use the frequency domain for inversion. Reasons:

• One frequency at a time means a smaller linear system per iteration.
• Frequency continuation (§3.6) is the natural fit: invert at low frequency first, warm-start to higher.
• Cycle-skipping is easier to diagnose by frequency.

The PINN literature followed: Song, Alkhalifah & Waheed 2023 GJI is the modern reference for Helmholtz PINNs in seismic.

The widget setup

To verify against an exact answer we choose a forced source so $U(x, z) = \sin(\pi x) \sin(\pi z)$ is the analytic solution for any $k$:

For $k = 2 \pi$ (one wavelength across the unit-square domain) the source amplitude $k^2 - 2 \pi^2 = 2 \pi^2 \approx 19.7$. As $k \to \sqrt{2} \pi \approx 4.44$ the source amplitude vanishes — that is the (1, 1) resonant eigenmode of the closed box, where the Helmholtz operator becomes singular. Stay safely above or below.

What you should observe

• For low $k$ (1–5) the PINN converges in a few thousand epochs to relative-L² of a few percent.
• At $k \approx 4.44$ (the (1, 1) resonance) the PDE is singular — the PINN may produce arbitrarily large outputs. Step the slider over and around it to see the failure.
• For high $k$ (8–12) the wavefield has multiple wavelengths across the domain. Spectral bias of the vanilla Tanh MLP starts to bite; relative-L² grows. Fourier features (§2.2) or SIREN (§2.3) would fix this — modern Helmholtz PINNs use them by default.
• The compute is much cheaper than time-domain (no time dimension, no IC) — typically 30 seconds for 2000 epochs.

References

• Pratt, R.G. (1999). Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model. Geophysics 64(3).
• Plessix, R.-E. (2006). A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int. 167.
• Operto, S., et al. (2007). 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver. Geophysics 72(5).
• Song, C., Alkhalifah, T., Waheed, U.B. (2023). A versatile framework to solve the Helmholtz equation using physics-informed neural networks. Geophys. J. Int.