“Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,” by Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis, was published in Nature Machine Intelligence in March 2021 (volume 3, pages 218-229), with an earlier arXiv version in 2019. It is one of the two foundational architectures, alongside the Fourier Neural Operator, for the field now called operator learning.
The paper builds on a less famous result than the usual universal approximation theorem: a neural network can approximate not just any continuous function, but any continuous operator, meaning a map that takes a whole function as input and returns another function. DeepONet realizes this with two coordinated networks. A branch network encodes the input function sampled at fixed points, and a trunk network encodes the location where you want the output evaluated. Their combination produces the value of the output function at that location.
This design lets a single trained DeepONet stand in for a differential-equation solver, an integral operator, or other complex transformations, and it generalizes to inputs it never saw during training. The authors demonstrated it on deterministic and stochastic differential equations and on operators such as integration and the fractional Laplacian.
For a general reader, DeepONet and the neural operators that followed represent a shift in what machine learning is asked to learn: not a single answer, but the rule that turns one function into another. That is closer to how physics and engineering describe the world, and it is why operator learning has become a core tool of AI for science.