Scientific Ml

DeepONet

DeepONet learns an operator by combining a branch network that encodes the input function with a trunk network that encodes the query point.

The branch-trunk design

DeepONet is built on a simple factorization. To approximate an operator G that maps an input function u to an output function G(u), it uses two subnetworks. The branch network reads samples of u at a fixed set of sensor locations and outputs a vector of coefficients. The trunk network reads a query coordinate y and outputs a vector of basis values. The predicted output at y is the dot product of these two vectors.

This structure mirrors a classical expansion: the trunk learns a set of basis functions of the output coordinate, and the branch learns the coefficients that combine them for a particular input. The universal approximation theorem for operators guarantees that such a form can approximate any continuous operator to arbitrary accuracy given enough width.

Kronos motion — operating point

Inputs and sensors

The input function is presented to the branch as its values at fixed sensor points. These sensors must be the same for every training and test example, though the output can be queried at any coordinate through the trunk. This makes DeepONet flexible in the output space while fixed in how it ingests the input.

A schematic

python

# G(u)(y) approx  sum_k b_k(u_sensors) * t_k(y)
b = branch(u_sensors)     # shape [batch, p]
t = trunk(y)              # shape [batch, p]
out = (b * t).sum(dim=-1) # operator evaluated at y

Strengths and trade-offs

Flexible output sampling: query the solution anywhere without a grid
Solid theoretical backing through the operator approximation theorem
Sensitive to sensor placement and to distribution shift in the inputs
Often paired with a physics-informed loss to reduce data requirements

Physics-informed variant

A physics-informed DeepONet adds the PDE residual of the predicted output to the loss, computed by automatic differentiation through the trunk coordinate. This lets the operator be trained with far fewer labeled solution pairs, because the equation itself supplies most of the supervision.