An efficient and accurate operator learning for elliptic interface problems

In this paper, we propose an improved learning scheme–interface operator network based on extreme learning machines (ELM-IONet) to solve elliptic interface problems, where different coefficients, source terms, and other parameters are treated as input features. Here the extension-and-projection method is used in ELM-IONet to capture the discontinuities of functions. Meanwhile, the idea of the Gram-Schmidt orthogonalization is adopted to handle the output matrices of the trunk and branch networks of ELM-IONet, thereby improving its stability and generalization ability. Additionally, we integrate ELM-IONet with physics-informed neural networks, enabling it to remain effective even in the absence of paired input-output data within the computational domain. Numerous experiments are conducted to demonstrate that ELM-IONet significantly outperforms the traditional operator neural networks in terms of both accuracy and efficiency.