Free-surface boundary conditions

The acoustic free-surface boundary condition models the air-rock interface where the pressure perturbation must vanish: $u(x, z = 0, t) = 0$. (The elastic version requires zero traction, $\sigma_{zz} = \sigma_{xz} = 0$, which is more involved.) For the acoustic case it is a clean Dirichlet BC.

This section combines two threads we have already met:

• The §3.2 loss-balance crisis: BC penalty terms with naive $\lambda = 1$ get drowned out by the PDE-residual gradient.
• The §1.5 / §2.4 hard-constraint trick: reparameterise the network output so the BC is identically satisfied, eliminating the BC term from the loss entirely.

The reparameterisation

For the free-surface BC $u(x, z = 0, t) = 0$, write

At $z = 0$ the wrap factor $z$ is zero, so $u_\theta(x, 0, t) = 0$ identically — the free-surface BC is satisfied by construction. The PDE residual involves derivatives of $u_\theta$; the chain rule gives

For both BCs at $z = 0$ AND $z = 1$ we use $u_\theta(x, z, t) = z(1 - z) \cdot \textrm{NN}$. The $z(1 - z)$ wrap satisfies both Dirichlet BCs identically; the BC term drops out of the loss entirely.

Try it: 1D depth-only problem

For a clean comparison we use a 1D depth problem:

• Domain $z \in [0, 1],\ t \in [0, 1]$.
• IC: $u(z, 0) = \sin(\pi z),\ u_t(z, 0) = 0$.
• BC: $u(0, t) = 0$ (free surface), $u(1, t) = 0$ (closed bottom).
• Analytic: $u(z, t) = \sin(\pi z) \cos(\pi t)$.

Three modes race: SOFT (both BCs in loss); HARD-FS (only free surface hard, bottom soft); HARD-BOTH (both hard).

What you should observe

• SOFT often gets the IC and PDE right but leaves a small residual at the BC. The relative-L² is typically a few percent.
• HARD-FS shows mild improvement — one less BC term to balance.
• HARD-BOTH eliminates both BC terms from the loss entirely. The optimiser only sees PDE + IC, and the BC is satisfied by construction. Convergence is faster and final accuracy higher.

For 2D and 3D wave problems

The same idea extends. In 2D, the reparam $u_\theta(x, z, t) = z \cdot \textrm{NN}$ enforces the free surface along the entire $z = 0$ line. The other three boundaries (left, right, bottom) typically need ABC or PML (§4.6, §4.7), which are NOT Dirichlet and so cannot be hard-enforced. The hybrid — hard free surface, soft/PML elsewhere — is the de-facto standard in modern seismic-PINN papers.

References

• Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G. (2021). Physics-informed neural networks with hard constraints for inverse design. SIAM J. Sci. Comput. 43(6).
• Sukumar, N., Srivastava, A. (2022). Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. CMAME.