$\phi-$DeepONet: A Discontinuity Capturing Neural Operator

TL;DR

$DeepONet is a neural operator that effectively captures discontinuities in functions by using multiple branch networks and a nonlinear embedding, enabling accurate predictions in complex scientific problems.

Contribution

It introduces a novel neural operator architecture that handles discontinuities via multiple branches and a nonlinear interface embedding, improving over classical methods.

Findings

Achieves accurate predictions on benchmark problems with discontinuities.

Demonstrates stability and robustness in complex interface-driven scenarios.

Outperforms traditional neural operators in discontinuous settings.

Abstract

We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In $\phi$-DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a {\it one-hot} representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics- and interface-informed loss function. We evaluate $\phi$-DeepONet on several one- and two-dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface-driven discontinuities.