DeepONet (Deep Operator Network)

DeepONet (Deep Operator Network) is a neural operator architecture introduced by Lu, Jin, Mengu, and Bahri in 2019 (arXiv 1910.03193, Nature Methods 2021) that approximates an operator G: G(a)(y) ≈ ∑ᵢ bᵢ(a) · tᵢ(y), where the branch network b: ℝʷ → ℝᵖ encodes the input function a at w sensor locations and the trunk network t: ℝᵈ → ℝᵖ encodes the output location y. The output of G at y is the dot product of the two evaluations, giving a factorization of the operator rather than a discretized pointwise map.

The branch-plus-trunk architecture is a learning-realization of the universal approximation theorem for operators proved by Chen and Chen (1993) — any continuous operator between Banach spaces can be approximated to arbitrary accuracy by such a factored network. In DeepONet, the branch is evaluated at sensor measurements of the input function, the trunk is evaluated at the desired output coordinates, and their inner product gives the predicted operator output at those coordinates. This factorization separates input-function representation (the branch) from output-location mapping (the trunk) and makes DeepONet both efficient and a natural fit for irregular geometries and mixed boundary conditions.

DeepONet complements the Fourier Neural Operator: where FNO parameterizes the learned kernel in Fourier space and works well on structured grids, DeepONet handles irregular input functions and output coordinates without a Fourier ansatz, and is the workhorse for problems in fluid dynamics, materials science, and parametric PDEs where the domain is non-grid-aligned or the sensor placement is irregular. Variants include PI-DeepONet (physics-informed), S-DeepONet (stacked / multi-output), and DeepONet-based foundation models for parametric operator learning.