Neural Operator

Neural operators are a class of neural network architectures introduced by Lu, Jin, Mengu, and Bahri at MIT (2020; arXiv 2003.03485) that learn maps between infinite-dimensional function spaces rather than between finite-dimensional vectors. Where a standard neural network approximates a function f: ℝᵈ → ℝᵏ, a neural operator approximates an operator G: 𝒜 → 𝒰, where 𝒜 and 𝒰 are function spaces defined over bounded domains. This makes the architecture mesh-invariant: a neural operator trained on a coarse discretization can be evaluated on a finer discretization (or vice versa) without retraining, because the underlying objects it acts on are functions, not grids.

The defining property of a neural operator is that it replaces the input-to-output mapping of a standard neural network with a sequence of (i) a lifting layer that maps input functions to higher-dimensional representations, (ii) a stack of iterative kernel-integration blocks operating on those representations — typically parameterised by learned integral kernels rather than pointwise nonlinearities — and (iii) a projection layer that maps the result back to the output function space. By learning the integral kernel, the model gains resolution-independence and can transfer across geometries or samplings of the same underlying physical problem. Architectures in this family include the Graph Neural Operator (GNO) for irregular domains, the Fourier Neural Operator (FNO), which parameterizes the kernel in Fourier space for grid-aligned problems, and DeepONet, which uses a branch-plus-trunk factorization of the operator.

Neural operators have become a foundational primitive for scientific machine learning, weather and climate modeling, fluid dynamics, materials design, and inverse problems, where the goal is to learn a parametric family of PDE solutions rather than a single instance. They are distinct from Neural ODEs (continuous-depth networks) and from convolutional PDE solvers: they learn the operator, not the trajectory, and the kernel-integration step is what gives them cross-resolution generalization.