Deep Operator Network DeepONet
Definition
An operator-learning architecture with a branch net encoding the input function and a trunk net encoding the evaluation location; their inner product gives the solution value. It does not assume a regular grid and is the other mainstream neural operator design alongside FNO.
What it is
DeepONet has two subnetworks: a branch net that encodes the input function (sampled at fixed sensor locations) and a trunk net that encodes the query coordinate (x, y, t). Their outputs are combined by an inner product (plus a bias) to produce the solution value. This "function encoding x location encoding" factorization makes it naturally mesh-free.
Key points
- Mesh-free: the trunk takes arbitrary coordinates, so training and inference work directly on scattered points or unstructured meshes.
- Fixed sensors: the branch consumes function samples at fixed sensor locations, and sensor placement affects accuracy.
- Training tuples are
(input function, query point, solution value), more flexible than FNO's whole-field tensors. - Many variants exist; POD-DeepONet replaces the trunk with a POD basis and usually converges faster and more accurately.
Common pitfalls / boundaries
- It is not a "better than FNO" story: on regular grids with large fields FNO is usually more efficient, while scattered/unstructured data favors DeepONet.
- Sensor count and placement are the main tuning knobs; random placement loses accuracy badly in regions with steep gradients.