  Scientific computing has been an indispensable tool in applied sciences and
engineering, where traditional numerical methods are often employed due to
their superior accuracy guarantees. However, these methods often encounter
challenges when dealing with problems involving complex geometries. Machine
learning-based methods, on the other hand, are mesh-free, thus providing a
promising alternative. In particular, operator learning methods have been
proposed to learn the mapping from the input space to the solution space,
enabling rapid inference of solutions to partial differential equations (PDEs)
once trained. In this work, we address the parametric elliptic interface
problem. Building upon the deep operator network (DeepONet), we propose an
extended interface deep operator network (XI-DeepONet). XI-DeepONet exhibits
three unique features: (1) The interface geometry is incorporated into the
neural network as an additional input, enabling the network to infer solutions
for new interface geometries once trained; (2) The level set function
associated with the interface geometry is treated as the input, on which the
solution mapping is continuous and can be effectively approximated by the deep
operator network; (3) The network can be trained without any input-output data
pairs, thus completely avoiding the need for meshes of any kind, directly or
indirectly. We conduct a comprehensive series of numerical experiments to
demonstrate the accuracy and robustness of the proposed method.
