When forward PINN earns its keep

§5.1-§5.4 painted the case for FDTD on a single forward run. But "forward modelling" covers more than that. Real seismic workflows have shapes that the bare FDTD comparison misses. This closing section is a scenario gallery, eight forward-modelling setups, with the verdict for each: FDTD wins, comparable, or PINN wins (with the reason).

The four genuine forward-PINN niches

• Parameterised forward / surrogate modelling. If you need wavefields for many configurations of a parameter (source location, velocity-model parameter, layer geometry), train a network conditioned on those parameters as extra inputs. Once trained, evaluation at any new parameter is a single forward pass, milliseconds. FDTD has to re-run from scratch each time. (Wang & Perdikaris 2021; Lu et al. 2021 DeepONet; Karniadakis 2021 review.)
• Irregular geometry. Body-fitted FDTD grids, mortar elements, and staircase boundaries are all expensive to set up. PINN residuals are mesh-free, just sample collocation points anywhere in the domain. Spectral-element FDTD is the alternative but the engineering effort dwarfs PINN setup.
• Sparse-data assimilation. When you need a wavefield consistent with both the PDE and a few sparse direct observations, PINN's natural multi-term loss handles it. FDTD has no analogous data-fit term in the forward solver.
• Train-once-evaluate-many. After training, the PINN produces $u$ at any $(x, z, t)$ in $O(P)$ network-forward time. FDTD output is on a fixed grid; arbitrary-point queries need interpolation. For one-off plotting this is moot, but for downstream uses with $10^4$+ random queries the network is more elegant.

The flip side: everything else is FDTD-wins territory. A single 2D Marmousi forward run, a high-frequency simulation on a structured grid, a production seismic processing chain with thousands of forward solves, all are FDTD problems. PINN doesn't come close.

Try it: scenario gallery

The Part-6 transition

Part 5 closes the door cleanly: for forward modelling, PINN is rarely the right tool. Part 6 opens a different door. In the inverse problem, given sparse seismic recordings on the surface, recover the velocity field $c(x, z)$ that produced them, FDTD has nothing useful to say. The unknown is the velocity field itself; FDTD can compute the forward simulation for any candidate velocity but cannot tell you which candidate to try. PINN-FWI repurposes the entire forward-PINN machinery as the inner loop of the inverse solve, and adds $c(x, z)$ as a second neural network trained jointly with the wavefield network. This is where PINN's claim to fame in seismology lives.

Expertise checkpoint, end of Part 5

You should now be able to:

• Decide whether a forward-modelling task is best served by FDTD or PINN by walking through the §5.5 scenario tree.
• Defend the FDTD-wins verdict for any vanilla single-run forward problem on first principles (cost, convergence rate, spectral bias).
• Recognise the four niches where forward PINN earns its keep (parameterised surrogate, irregular geometry, sparse-data assimilation, transfer/warm-start) and explain each in one sentence.
• Be ready to read Part 6 with the right mindset: PINN is about to demonstrate dramatic value in the inverse setting, building on every piece of forward-PINN machinery from Parts 1-4.

Pause-and-check. (1) You need to compute the wavefield in a 2D anisotropic-VTI medium with a curved free surface and complex internal layering. Should you reach for PINN or FDTD? (2) You need 1000 forward simulations across a parameter sweep of layer geometry. Same question. (3) You are doing FWI and need 50 forward solves per outer iteration on a fixed velocity. Same question.

References

• Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., Mahoney, M.W. (2021). Characterizing possible failure modes in physics-informed neural networks. NeurIPS.
• 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.
• bin Waheed, U., Haghighat, E., Alkhalifah, T., Song, C., Hao, Q. (2021). PINNeik: Eikonal solution using physics-informed neural networks. Comput. Geosci. 155, 104833.
• Moseley, B., Markham, A., Nissen-Meyer, T. (2020). Solving the wave equation with physics-informed deep learning. arXiv:2006.11894.
• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6), WCC1-WCC26.