The 2D acoustic wave equation

The 2D acoustic wave equation,

is the underlying physics of every 2D seismic shot record. The PINN has three inputs ($x, z, t$) instead of two, and the PDE residual involves three Hessian-diagonal entries ($u_{xx}, u_{zz}, u_{tt}$). The IC and BC bookkeeping is otherwise identical to §4.1.

Setup

To get a clean analytic solution we use the 2D analogue of §4.1's separable mode:

• Domain $(x, z) \in [-1, 1]^2,\ t \in [0, 1]$.
• IC: $u(x, z, 0) = \sin(\pi x) \sin(\pi z), \quad u_t(x, z, 0) = 0$.
• BC: $u = 0$ on all four edges.

Because $\sin(\pi x) \sin(\pi z)$ is an eigenmode of the Laplacian with eigenvalue $-2 \pi^2$, the exact solution is

which oscillates uniformly across the domain at angular frequency $c \pi \sqrt{2}$. The temporal frequency is $\sqrt{2}$ times higher than the 1D case at the same $c$.

The compute budget

Going from 1D to 2D triples the per-point AD cost (3 input dimensions instead of 2). To stay tractable in-browser we use a smaller collocation grid: 8 × 8 × 6 = 384 PDE points, 36 IC points, ~96 BC points. With 1500 epochs of 4-term NTK-balanced training, the run takes ~2–3 minutes. Production seismic-PINN papers (Rasht-Behesht 2022, Moseley 2020) use 10⁴–10⁵ collocation points and run on a GPU for hours; the recipe scales but the wall-clock cost grows roughly linearly with $N_c \times \textrm{epochs}$.

Try it

What you should observe

• For $c = 1.0$ the PINN reaches relative-L² of ~10–15% in the browser-feasible compute budget. The IC and BC terms are tight ($\sim 10^{-4}$); the PDE residual is what bottlenecks the spacetime accuracy. This is the 2D-vs-1D lesson: the same recipe scales but at quadratic compute cost, and the in-browser budget is not enough to drive PDE residual to the $10^{-2}$ level §4.1 reaches in 1D.
• For $c = 1.5$ the temporal frequency reaches $1.5 \pi \sqrt{2} \approx 6.66$, putting ~1.06 cycles inside $[0, 1]$. Spectral-bias error grows; relative-L² climbs into the 20–30% range. Production codes (Rasht-Behesht 2022) use Fourier features (§2.2) and 10× more colloc points to handle higher temporal frequencies.
• The IC-velocity term $u_t(x, z, 0) = 0$ is the slowest to converge initially, because it depends on $\partial_t u_\theta$ at a 2D boundary surface. NTK rebalancing is what eventually drives it down.
• The PINN and analytic spacetime fields differ visibly only at the late-time crests where the temporal frequency stresses the network width. Time-zero and the first peak match cleanly.

From the eigenmode toy to the real shot record

The eigenmode test is pedagogical, not realistic. A real 2D seismic forward model has:

• A localised source term (Ricker wavelet) on the right-hand side of the wave equation, not just an initial condition. The PINN PDE residual gains a source term: $u_{tt} - c^2 (u_{xx} + u_{zz}) - f(x, z, t) = 0$.
• A spatially-varying velocity field $c(x, z)$ — the medium being imaged. For the forward problem this is a fixed input to the PDE; for the inverse problem (Part 6) it becomes another trainable network.
• Boundary conditions tailored to a half-space: a free surface at $z = 0$ (§4.5) and absorbing/PML boundaries on the other three sides (§4.6, §4.7).

All of these change the residuals in the PINN loss, not the structure of the loss itself. The 4-term NTK-balanced recipe is robust to these changes; it is the same recipe Moseley et al. (2020) used for the original wave-equation PINN paper.

References

• Moseley, B., Markham, A., Nissen-Meyer, T. (2020). Solving the wave equation with physics-informed deep learning. arXiv:2006.11894.
• 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.