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.

翻译：科学计算一直是应用科学与工程领域不可或缺的工具，其中传统数值方法因其优越的精度保证而被广泛采用。然而，在处理涉及复杂几何结构的问题时，这些方法常常面临挑战。相比之下，基于机器学习的方法是无网格的，从而提供了一种有前景的替代方案。特别是，算子学习方法被提出来学习从输入空间到解空间的映射，使得一旦训练完成，便能快速推断偏微分方程（PDEs）的解。在本工作中，我们针对参数化椭圆界面问题展开研究。基于深度算子网络（DeepONet），我们提出了一种扩展界面深度算子网络（XI-DeepONet）。XI-DeepONet展现出三个独特特征：（1）界面几何结构作为额外输入被整合到神经网络中，使得网络在训练后能够推断新界面几何结构的解；（2）与界面几何相关的水平集函数被视为输入，解映射在其上是连续的，并能被深度算子网络有效逼近；（3）网络可以在没有任何输入-输出数据对的情况下进行训练，从而完全避免了直接或间接使用任何类型的网格。我们进行了一系列全面的数值实验，以验证所提方法的准确性与鲁棒性。