Picking a formulation: time vs frequency

Part 4 has been a tour: 1D and 2D acoustic wave (§4.1, §4.2), Helmholtz (§4.3), VTI/TTI (§4.4), free-surface BC (§4.5), Clayton-Engquist ABC (§4.6), PML (§4.7). Each was a piece of machinery; each had a specific niche. This closing section synthesises the menu into a decision tree.

The four questions

• What does the source look like? A broadband transient (Ricker wavelet) wants the time-domain. A narrowband CW (single dominant frequency) wants Helmholtz. A sequential-frequency FWI workflow wants Helmholtz + frequency continuation (§3.6).
• Is the medium isotropic or anisotropic? Isotropic is the default in §4.1–§4.3. VTI replaces a single $c^2$ with two coefficients $V_h^2$ and $V_v^2$ (§4.4). TTI rotates the symmetry axis — either rotate the PDE residual or add the tilt as a network input.
• What is the boundary configuration? Closed box: Dirichlet $u = 0$, soft enforcement is fine. Free surface at top: §4.5 hard-constraint reparameterisation. Open elsewhere: PML (§4.7) for oblique incidence; Clayton-Engquist (§4.6) only for near-normal.
• Forward or inverse? Forward problems fix all coefficients and compute $u$. Inverse problems promote the unknowns — typically $c(x, z)$ — to additional trainable parameters (or a second neural network), with a data-fit term added to the loss. This is the Part 6 workflow.

Try it

Three rules that override the decision tree

• Start in the time domain. Even when the decision tree says Helmholtz, a 5-second time-domain simulation is the fastest sanity check that your loss formulation is correct. Switch to frequency domain when you are confident the residuals are right.
• Validate against an analytic reference. Every section in Part 4 has used a known-solution test. For a real problem, build a simplified version (homogeneous velocity, idealised geometry) where an analytic answer or a high-fidelity FDTD reference exists. Run that first. The relative-L² against the known answer is the only metric that does not lie.
• Architecture is the second question. The first question is always: is the loss formulation correct? Are all required terms present? Are the residuals computed using the right derivatives? An expensive architecture will not save a wrong loss. Spectral-bias-resistant architectures (Fourier features, SIREN) are useful when the wavefield has many wavelengths across the domain — but only if the loss is right first.

Expertise checkpoint — end of Part 4

You should now be able to:

• Write down the PINN loss for a 1D, 2D, or 3D acoustic wave problem with any combination of free-surface, Dirichlet, ABC, and PML boundaries.
• Decide between time-domain and Helmholtz formulations based on the source character.
• Add VTI or TTI anisotropy to the PDE residual using the Thomsen 1986 / Alkhalifah 1998 parameterisation.
• Construct the hard-constraint reparameterisation $u = z \cdot \textrm{NN}$ for a free surface and recognise where it does and does not apply.
• Critique the wave-equation formulation in any current seismic-PINN paper using the language of this Part.

Pause-and-check. (1) Use the picker for a 2D acoustic forward problem on a flat-layered Earth with a Ricker wavelet source at 30 Hz, recording for 2 s. Does the recommendation match what you would have chosen? (2) Same problem but inverse: recover the velocity model from sparse surface data. What changes? (3) Pick a recent seismic-PINN paper from arXiv. Run its setup through the four questions and check whether its formulation matches what the picker would have recommended. If not, what extra information justifies the deviation?

Part 5 picks up the conversation by asking the harder question: are PINNs even the right tool for forward modelling? Spoiler: for pure forward modelling on a fixed velocity field, FDTD is faster and more accurate than any PINN. PINNs earn their place in the inverse setting (Parts 6–9), not the forward one.

References

• Rasht-Behesht, M., Huber, C., Shukla, K., Karniadakis, G.E. (2022). Physics-informed neural networks (PINNs) for wave propagation and full waveform inversions. JGR Solid Earth 127, e2021JB023120.
• Song, C., Alkhalifah, T., Waheed, U.B. (2023). A versatile framework to solve the Helmholtz equation using physics-informed neural networks. Geophys. J. Int. 232(3), 1750–1762.
• Alkhalifah, T., Song, C., Waheed, U.B., Hao, Q. (2021). Wavefield solutions from machine learned functions constrained by the Helmholtz equation. Artificial Intelligence in Geosciences 2, 11–19.
• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6), WCC1–WCC26.