A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems

In this paper, we propose a categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems. The architecture comprises three hidden layers: a discontinuity-capturing layer, which maps domain segments to disconnected sets in a higher-dimensional space; a categorical embedding layer, which reduces the high-dimensional information into low-dimensional features; and a fully connected layer, which models the continuous mapping. This design enables a single neural network to approximate piecewise smooth functions with high accuracy, even when the number of discontinuous pieces ranges from tens to hundreds. By automatically learning discontinuity embeddings, the proposed categorical embedding technique avoids the need for explicit domain labeling, providing a scalable, efficient, and mesh-free framework for approximating piecewise continuous solutions. To demonstrate its effectiveness, we apply the proposed method to solve anisotropic elliptic interface problems, training by minimizing the mean squared error loss of the governing system. Numerical experiments demonstrate that, despite its shallow and simple structure, the proposed method achieves accuracy and efficiency comparable to traditional grid-based numerical methods.