A neural operator is a neural network designed to learn an operator, which is a map from one function to another, rather than the more familiar map from a vector of numbers to another vector. The motivating example is a differential equation: the inputs and outputs of a physics simulation are really functions, such as a velocity field over space and time, and a neural operator learns the rule that turns the input field into the solution field.
Two properties distinguish neural operators from ordinary networks. First, they are discretization-invariant: because they approximate an operation on functions, a model trained on data sampled at one resolution can be queried at a finer resolution without retraining. Second, a single trained operator generalizes across an entire family of problems, for example many different initial conditions or material parameters of the same equation, instead of being tied to one instance the way a classical solver is.
The two best-known realizations are the Fourier Neural Operator, which parameterizes the operator in frequency space, and DeepONet, which splits the work between a branch network that reads the input function and a trunk network that reads the query location. Both arrived around 2020 to 2021 and now underpin fast emulators in weather, fluid dynamics, and materials.
For a business or general reader, the neural operator is the engine behind the AI surrogate model: when an exact simulation is too slow to run repeatedly, a neural operator can learn it once and then approximate it almost instantly, which is what makes machine-learned weather forecasts and engineering design loops practical.