Neural Operators

Neural operators learn mappings between function spaces, so one trained model solves a family of PDEs at any resolution.

Learning operators, not functions

A standard neural surrogate learns a map from a fixed input vector to a fixed output vector. A neural operator learns a map between whole functions - for example from an initial condition or coefficient field to the solution of a partial differential equation. Once trained, it approximates the solution operator of the PDE for a range of inputs, not a single instance.

Discretization invariance

Kronos motion — family decades

The defining property of a neural operator is that it is discretization-invariant: trained on data at one mesh resolution, it can be queried at a finer or coarser resolution without retraining. This distinguishes it from a convolutional surrogate tied to a fixed grid, and it means the operator represents a continuous mapping rather than a matrix of a fixed size.

Main architectures

Fourier neural operators apply learnable filters in the Fourier domain, capturing global structure efficiently
Graph neural operators pass messages on a graph to handle irregular geometries
DeepONet uses a branch network for the input function and a trunk network for the query location
Wavelet and spectral variants target multiscale or non-periodic problems

The Fourier neural operator

The Fourier neural operator transforms the input to the frequency domain, multiplies by learned weights on the low frequency modes, transforms back, and adds a local linear term, repeating over several layers. Truncating to low modes gives efficiency and a smoothing prior well suited to many physical fields.

Strengths and cautions

Neural operators can emulate PDE solvers orders of magnitude faster and generalize across inputs and resolutions. They still require substantial training data, can struggle with sharp fronts and shocks, and extrapolate poorly beyond the training distribution. Uncertainty quantification for operators is an active research area.

Relevance

For fusion design, neural operators are a candidate for emulating field solvers whose inputs are spatial profiles rather than a few scalars. In Kronos research they are evaluated as fast emulators of electromagnetic and transport fields for the machines, subject to the same validation discipline as any surrogate.