The cost-vs-accuracy front

§5.2 raced one PINN against one FDTD. To make the comparison rigorous we need to vary the compute budget on both sides and trace the achievable Pareto front. This section does that on a clean 1D problem (the §4.1 standing-wave reference, where we have an analytic solution).

The setup

• Same problem as §4.1: $u_{tt} = u_{xx}$ on $x \in [-1, 1], t \in [0, 1]$, IC $u(x, 0) = \sin(\pi x), u_t(x, 0) = 0$, BC $u(\pm 1, t) = 0$. Analytic $u(x, t) = \sin(\pi x) \cos(\pi t)$.
• FDTD swept across $N_x \in {20, 40, 80, 160}$ (4 budget points). Time step set by CFL.
• PINN swept across epochs $\in {300, 800, 1500}$ (3 budget points).

For each run we record (wall-clock time, relative-L²-vs-analytic) and plot the seven points in log-log.

Try it

What you should observe

• FDTD points: even the cheapest (Nx = 20, ~few ms) reaches few-percent relative-L². Increasing $N_x$ drops the error like $O(\Delta x^2)$ — quadratic convergence.
• PINN points: 300 epochs runs in ~10 s with relative-L² near 30%. 1500 epochs runs in ~50 s with relative-L² near 1%.
• The two clouds are orders of magnitude apart. FDTD's entire curve sits below-left of PINN's entire curve.
• The structural gap: FDTD's convergence rate is $O(\Delta x^2)$; PINN's effective convergence rate from gradient-descent training is much slower and stalls due to spectral bias.

What this means in practice

A seismic processing pipeline that needs 100 forward simulations per FWI iteration and 100 iterations to converge will run $10^4$ forward solves total. With FDTD at $N_x = 80$ that takes $\sim 100,$s; with PINN at 800 epochs that takes $\sim 7,$hours. For pure forward modelling, the choice is unambiguous.

The §5.5 niches matter precisely because they break this 1:1 forward-solve assumption. If you can train PINN once and evaluate at $10^4$ (parameter, source) configurations cheaply, the trade-off changes. But for the standard "run the wave equation on this velocity field" task, FDTD wins.

References

• Trefethen, L.N. (1996). Finite difference and spectral methods for ordinary and partial differential equations. Online textbook. Standard reference for FDTD convergence theory.
• Karniadakis, G.E., et al. (2021). Physics-informed machine learning. Nature Reviews Physics 3, 422–440. Survey that includes a fair PINN-vs-classical comparison.