$φ-$DeepONet: A Discontinuity Capturing Neural Operator

2026-04-09

AI Summary

This work addresses the challenge that conventional neural operators struggle to model strong or weak discontinuities in inputs or outputs arising from material interfaces. To overcome this limitation, the authors propose φ-DeepONet, which for the first time integrates one-hot encoded domain decomposition into a neural operator architecture. The method employs a multi-branch network to handle discontinuous inputs and fuses interface-aware one-hot encodings with spatial coordinates in the trunk network to construct a nonlinear latent embedding capable of capturing output discontinuities. A joint loss function incorporating physical laws and interface information is also introduced. By relaxing the continuity assumptions inherent in classical neural operators, φ-DeepONet achieves accurate and stable predictions across multiple one- and two-dimensional benchmark problems, demonstrating robust performance even in the presence of strong interfacial discontinuities.

Abstract

We present $φ-$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 $φ$-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 $φ$-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.

Problem

• discontinuities
• neural operators
• function spaces
• material interfaces
• non-smooth behavior

Innovation

• neural operator
• discontinuity capturing
• physics-informed learning
• domain decomposition
• interface embedding