PINNs vs Neural Operators

Solving a PDE Is Not the Same as Learning the Solver

A mathematical comparison of Physics-Informed Neural Networks, DeepONet, and Fourier Neural Operators using the same differential equation.

The question usually arrives in a compact form: I have a PDE — should I use a PINN, a DeepONet, or a Fourier Neural Operator?

Phrased that way it sounds like a question about architectures, and it invites answers about layer counts, parameter budgets, and benchmark tables. That answer is premature. The three methods are not three implementations of one task; they are three different mathematical learning problems that share a vocabulary. Choosing between them is mostly a matter of deciding which problem you have.

What follows works through the distinction on a single equation, kept fixed throughout, so that the comparison never depends on which example flatters which method.

One equation, used throughout

Consider the one-dimensional variable-coefficient elliptic problem

−d/dx ( a(x) du/dx ) = f(x), x ∈ (0, 1), u(0) = u(1) = 0.

The coefficient a(x) is bounded and strictly positive, a(x) ≥ a_min > 0, which keeps the problem well posed. Physically, a is what varies from one specimen to another: thermal conductivity in a composite bar, hydraulic permeability in a soil column, diffusivity in a layered medium. The forcing f is held fixed; the…