1D layered medium: PINN vs analytic

Part 5 — Forward modelling and where PINNs fall short

Learning objectives

Solve the 1D wave equation in a smoothly varying two-layer medium with a PINN
Measure reflection and transmission coefficients R, T from PINN output
Compare to the Fresnel formulas R = (c₂-c₁)/(c₁+c₂), T = 2c₂/(c₁+c₂)
Establish a quantitative baseline for the rest of Part 5

Part 5 is the textbook's honest reckoning: pure forward modelling is not where PINN earns its keep. To make the case rigorously we need quantitative benchmarks against problems where we know the right answer. This section starts with the simplest case: the 1D acoustic wave equation in a smooth two-layer medium.

The setup

Solve

\frac{\partial^2 u}{\partial t^2} = c(z)^2 \frac{\partial^2 u}{\partial z^2}

on $z \in [0, 1], t \in [0, 0.8]$ with smoothly-varying velocity

c(z) = 1 + 0.5 \, \sigma(40(z - 0.5)) ,\qquad \sigma(\cdot) = \textrm{sigmoid}(\cdot) ,

so $c(0) = 1$ near the surface and $c(1) = 1.5$ deep below, with a smooth transition of width $\sim 1/40 = 0.025$ centred at $z = 0.5$ . A Gaussian downgoing pulse centred at $z_0 = 0.2$ propagates downward, hits the velocity contrast near $z = 0.5$ , and partially reflects + partially transmits. The 1.5× contrast (rather than 2× or 3×) is chosen so the PINN can converge in the browser-feasible compute budget; the same recipe extends to any contrast.

The Fresnel coefficients

For a sharp acoustic interface with constant density $\rho$ (so impedance $Z = \rho c \propto c$ ) the pressure-wave reflection and transmission coefficients are

R = \frac{Z_2 - Z_1}{Z_1 + Z_2} = \frac{c_2 - c_1}{c_1 + c_2} = 0.2 ,\qquad T = \frac{2 Z_2}{Z_1 + Z_2} = \frac{2 c_2}{c_1 + c_2} = 1.2 .

Note that $T > 1$ : pressure amplitude in the lower-impedance medium increases on transmission. Energy is conserved because the energy flux for a plane wave is $\propto Z |A|^2$ :

|R|^2 + \frac{Z_1}{Z_2} |T|^2 = 0.04 + \frac{1}{1.5} \cdot 1.44 = 0.04 + 0.96 = 1 . \checkmark

The PINN setup

4-term NTK-balanced loss (PDE + IC pos + IC vel + BC) on a 3-48-48-1 Tanh network. Same recipe as §4.1 but with the spatially-varying $c(z)^2$ baked into the PDE residual $r = u_{tt} - c(z)^2 , u_{zz}$ .

Try it

What you should observe

The wavefield snapshot at $t = 0.35$ shows the pulse just hitting the interface — a small reflected lobe is forming above $z = 0.5$ and a transmitted pulse is emerging below.
At $t = 0.70$ the reflected pulse has travelled back to about $z = 0.25$ (above interface, speed $c_1 = 1$ ), and the transmitted pulse has travelled to about $z = 0.80$ (below interface, speed $c_2 = 1.5$ ).
The widget extracts peak amplitudes from the time traces at $z = 0.25$ and $z = 0.85$ and computes measured $R, T$ . Expected at this PINN budget: $R \approx 0.10$ – $0.20$ (correct sign, magnitude within ~30% of Fresnel) and $T \approx 0.85$ – $1.20$ . The smoothed-step velocity slightly blurs the sharp-interface formula, and the in-browser PINN budget caps spacetime accuracy.
The honest framing: even a tame 1D layered problem is borderline for a vanilla PINN at browser-feasible compute. This is itself a Part 5 message — the FDTD reference in §5.2 will recover $R, T$ to 1–2% in milliseconds.

References

Aki, K., Richards, P.G. (1980, 2002). Quantitative Seismology. Standard reference for Fresnel coefficients and reflection/transmission at acoustic interfaces.
Sheriff, R.E., Geldart, L.P. (1995). Exploration Seismology. Cambridge.