Two algorithms, same PDE, same starting gun. The finite-difference solver (left, teal) crawls timestep by timestep, the way numerical codes have for decades. The operator surrogate (right, coral) jumps straight from initial condition to any time t in one shot. Same answer to within a couple percent, watch which one finishes. This is the story neural operators (FNO, DeepONet) tell, and why they're one of the hottest directions in scientific ML.
Both sides solve the 1D viscous Burgers equation, the canonical nonlinear PDE benchmark mixing advection and diffusion:
Classical explicit finite differences. The CFL stability condition forces tiny timesteps, for ν = 0.02 at Nx = 256 that's thousands of sequential steps to reach t = 0.5, and you can't compute step n before step n−1 exists. The animation deliberately shows the time-marching.
Burgers admits the Cole-Hopf transformation: substitute u = −2ν·φₓ/φ and φ obeys the linear heat equation, which is diagonal in Fourier space, every mode just decays exponentially. So the "operator" maps the initial condition to its spectral coefficients once, and evaluating the solution at any time t is a single O(Nx·K) pass. No time marching at all.
A neural operator like FNO (Li et al., 2021) does conceptually the same thing, it learns a map from initial conditions to solutions, parameterized in frequency space, for problems where no analytic transform exists (Navier-Stokes, Darcy flow, weather). Trained once on solver outputs, then deployed at constant inference cost. Published FNO speedups reach 1000×+; what you measure here is the same structural advantage in its purest form.