DeepONet maps irregular function spaces

DeepONet is a neural network built to learn a rule between whole functions, not just between two columns of numbers. It was introduced by Lu Lu, Pengzhan Jin, and George Em Karniadakis in a paper first posted to arXiv on October 8, 2019. (arxiv.org) The basic idea is to learn an operator, which is a map that turns one function into another. In physics, that can mean taking a coefficient field or boundary condition and returning the solution of a partial differential equation. (arxiv.org 1) (arxiv.org 2) The original DeepONet has two pieces: a branch network that reads sampled values of the input function, and a trunk network that reads the coordinates where an output is needed. The model combines those two encodings to predict the output function at each point. (arxiv.org 1) (arxiv.org 2) That setup works cleanly when every example is measured at the same fixed sensor locations. The limitation is that many real datasets arrive on irregular meshes, scattered sensors, or changing geometries, where those fixed locations do not exist. (arxiv.org) (link.springer.com) That is where the recent work is happening. A March 3, 2026 paper in *Nonlinear Dynamics* by Andong Cong, Yuhong Jin, Haiming Yi, Yifan Jiang, Jun Li, Shijun Wang, and Lei Hou modifies DeepONet for irregular input grids. (link.springer.com) Their method swaps the standard Fourier processing used in one branch design for a nonuniform discrete Fourier transform, then plugs that branch into DeepONet. The paper says the resulting NFNO-DeepONet can learn operator mappings directly from nonuniformly sampled data. (link.springer.com) Other groups have pushed the same direction with different tools. An ICLR 2024 submission called GraphDeepONet said it could predict time-dependent partial differential equations on irregular grids and arbitrary grids by combining graph neural networks with DeepONet. (openreview.net) In aerospace, a 2025 paper by Ahmad Peyvan, Varun Kumar, and Karniadakis reported that Fusion-DeepONet outperformed MeshGraphNet and vanilla DeepONet on irregular-grid hypersonic-flow tasks. The same paper said the model used fewer trainable parameters than U-Net, MeshGraphNet, and Fourier Neural Operator baselines. (arxiv.org) So the story is less that DeepONet suddenly arrived, and more that researchers are adapting a 2019 operator-learning architecture to messier data layouts. The common target is simulation work where meshes move, sensors are sparse, or geometry changes from case to case. (arxiv.org) (link.springer.com) (arxiv.org) If those extensions keep holding up, DeepONet’s role will be as a learned surrogate for expensive numerical solvers, not as a replacement for physics itself. The new papers are trying to make that surrogate work on the irregular grids where real engineering data usually lives. (arxiv.org) (link.springer.com)