The eikonal equation and why PINNs love it

Part 6 closed the FWI loop on the wave equation $u_{tt} = c^2 u_{xx}$. Part 7 turns to a different physical model: the EIKONAL EQUATION, which describes the propagation of WAVEFRONTS rather than wavefields. The eikonal equation is the high-frequency limit of the wave equation; it captures TRAVEL TIME from a source to every point of the domain, ignoring amplitude and waveform shape. In exchange, you get a PDE that PINNs solve cleanly, without the spectral-bias and cycle-skipping problems that plague wave-equation FWI.

Where the eikonal comes from

Plug a high-frequency ansatz $u(x, t) = A(x) e^{i \omega (T(x) - t)}$ into the wave equation $u_{tt} = c^2 \nabla^2 u$. At leading order in $\omega$ the result is

with boundary condition $T(x_{\mathrm{src}}) = 0$. This is the eikonal equation. Its solution $T(x)$ is the FIRST-ARRIVAL TRAVEL TIME from the source to every point $x$ of the medium. Wavefronts are the level sets ${x : T(x) = t_0}$. In a homogeneous medium they are concentric spheres centered on the source; in a heterogeneous medium they bend according to Snell's law of refraction.

Why first-order PDE? Because the $\omega^2 (T_t)^2 - c^2 \omega^2 |\nabla T|^2 = 0$ leading-order term in the wave-equation expansion is FIRST-ORDER in $T$. The amplitude $A(x)$ is governed by a SEPARATE TRANSPORT EQUATION at the next order. For travel-time inversion (§7.4), microseismic location (§7.5), and surface-wave analysis (§7.6) we only need $T$, so the eikonal alone is enough.

Why PINNs love the eikonal

Compare the eikonal solution to the wave-equation solutions of Part 4:

This is why every paper on PINN-for-eikonal converges in 1000-3000 epochs on toy problems and a few hours on Marmousi-class — vastly less than wave-equation PINN-FWI. EikoNet (§7.2) is the canonical demonstration; it has been deployed at Caltech for production microseismic monitoring.

Solving the eikonal: classical numerics

Two classical algorithms dominate eikonal solving:

• Fast Marching Method (FMM) (Sethian 1996). Treats the eikonal as a front-propagation problem; uses a heap-sorted priority queue to update grid cells in order of increasing T. Complexity $O(N \log N)$ where N is total cells. Gold standard for accuracy.
• Fast Sweeping Method (FSM) (Zhao 2005). Iteratively sweeps the grid in 4 alternating directions; at each cell, applies an upwind Godunov update of the eikonal equation. Converges in O(1) sweeps for most problems (~4-8 typical). Simpler to implement; comparable accuracy.

This widget uses FSM for its simplicity in JavaScript. The §7.2 EikoNet PINN will use the FSM solution as the reference for accuracy comparison.

Try it

The widget runs FSM on a 81 × 41 grid covering 2 km × 1 km. Pick a velocity model and a source position. Three panels render:

• Velocity c(x, z). The medium. Yellow source dot at the picked location.
• Travel time T(x, z) + isochrones. The eikonal solution, with white contours marking equally-spaced T levels. Wavefronts spread roughly as ellipses in homogeneous media; bend strongly at velocity contrasts.
• Eikonal residual |∇T|·c. Should be ≈ 1 everywhere (the eikonal equation re-arranged). Brighter cells = larger deviation from 1; expect non-1 values at the source neighborhood and at sharp velocity boundaries (numerical-discretisation artifacts).

The summary box reports the FSM wall-clock time and the average + max eikonal residual deviation excluding source-neighborhood cells. On the homogeneous model, residual $\sim 10^{-3}$; on the gradient and gas-pocket models, residual $\sim 10^{-2}$ (limited by grid resolution).

What the smoothness of T means for PINNs

The travel-time field $T(x)$ is smooth almost everywhere. It has ONE singularity, at the source: gradient $|\nabla T|$ is finite but the second derivative blows up like $1/r$ near the source. Away from this singularity (and away from caustics in heterogeneous media), $T(x)$ is a perfectly innocent function for an MLP to learn — no oscillations, no high-frequency content, just a smooth bowl rising from the source.

This is why §7.2's EikoNet works with a vanilla 1-32-32-32-1 Tanh MLP. No Fourier features, no SIREN, no multi-scale tricks. Just: parameterise T(x; θ) by an MLP, train against the eikonal residual $(|\nabla T_{\mathrm{NN}}|^2 - 1/c^2)^2$ at random collocation points, plus the source boundary condition $T(x_{\mathrm{src}}) = 0$. Adam converges in a few thousand epochs to a solution within $10^{-3}$ of the FSM reference.

What §7.2 will do

§7.2 builds the EikoNet PINN end-to-end: train a small MLP to satisfy the eikonal at random collocation points, race it against this widget's FSM, and quantify the cost / accuracy trade-off. The PINN is slower than FSM on a single source — but it has TWO advantages classical solvers do not: (1) it produces continuous T(x) anywhere in the domain (no grid interpolation), and (2) it can be conditioned on the source location, learning a function $T(x; x_{\mathrm{src}})$ that generalises to NEW sources without re-solving.

References

• Sethian, J.A. (1996). A fast marching level set method for monotonically advancing fronts. PNAS 93(4), 1591–1595. The FMM paper.
• Zhao, H. (2005). A fast sweeping method for eikonal equations. Math. Comp. 74(250), 603–627. The FSM paper used here.
• Smith, J.D., Azizzadenesheli, K., Ross, Z.E. (2021). EikoNet: Solving the eikonal equation with deep neural networks. IEEE Trans. Geosci. Remote Sens. 59(12), 10685–10696. The paper §7.2 builds on.
• bin Waheed, U., Haghighat, E., Alkhalifah, T., Song, C., Hao, Q. (2021). PINNeik: Eikonal solution using physics-informed neural networks. Computers & Geosciences 155, 104833. Independent contemporaneous PINN-eikonal work.
• Treister, E., Haber, E. (2016). A fast marching algorithm for the factored eikonal equation. J. Comput. Phys. 324, 210–225. The factored-eikonal trick used in §7.3.