Operator Learning

Operator learning is the machine-learning paradigm of approximating operators — maps between function spaces — from data, rather than approximating functions from vectors. Where standard supervised learning targets a function f: ℝᵈ → ℝᵏ with a fixed input dimension d, operator learning targets an operator G: 𝒜 → 𝒰 where 𝒜 and 𝒰 are Banach or Hilbert spaces of functions defined over bounded domains. The defining properties of an operator-learning model are (i) resolution- or discretization-invariance — the same trained model can be evaluated on inputs at any sampling of the underlying function — and (ii) generalization across the parameters of a problem family — one trained operator can solve new instances of the same PDE without retraining.

The paradigm's theoretical foundation is the universal approximation theorem for operators (Chen & Chen 1993, and refinements by Lu, Jin, and Karniadakis), which guarantees that sufficiently large neural networks can approximate any continuous operator between function spaces. Architectures in this family include the Fourier Neural Operator (FNO), DeepONet, Graph Neural Operator (GNO), wavelet neural operators, transformer-based neural operators, and various physics-informed variants. Together with Neural ODEs (continuous-depth networks) and convolutional PDE solvers they form the core of contemporary scientific machine learning — the subfield of ML targeting physical systems governed by partial differential equations.

Operator learning has become a foundational tool for applications where the task is to model an entire family of PDE solutions rather than a single instance: weather and climate modeling (FourCastNet, Pangu-Weather, GraphCast), industrial computational fluid dynamics, materials and protein design, and inverse problems in medical imaging and geophysics. The term is sometimes used interchangeably with "neural operator learning" but is broader — it pre-dates the specific neural-operator architectures (FNO, DeepONet, GNO) and was theorized before they existed.