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Measured in Grid Points
PINN, DeepONet and FNO against the finite-difference solver they all have to beat, on one equation with an exact solution
Part 3 of a series on scientific machine learning. Part 1: Beyond PINNs: A Map of Modern Neural Network Paradigms. Part 2: PINNs vs Neural Operators: Solving a PDE Is Not the Same as Learning the Solver.
The previous article argued that a physics-informed neural network and a neural operator are not two answers to the same question. A PINN learns one solution field, x ↦ u(x), for one fixed coefficient. DeepONet and the Fourier neural operator learn the solution operator, a ↦ u, over a whole family of coefficients. That article ended with the design of an experiment to put numbers on the distinction, and with a caveat that belonged in the design rather than the discussion: in one dimension a classical solver is so cheap that the usual case for operator learning, amortising a training cost over many queries, has nothing to amortise.
This article runs the experiment. The code is public (github.com/DiogoRibeiro7/pde-operator-experiment). Every number below comes from the result files the code writes, directly or by simple arithmetic on them such as ratios and unit conversions. The main run takes 64 minutes on a two-core CPU, and the follow-up scripts under twenty minutes more.
One rule shaped the design more than any other: the classical solver is a competitor, not only the source of training…
