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

arXiv: https://arxiv.org/abs/2503.15441 · HTML: https://mlsanity.com/paper/2503.15441/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 19 Mar 2025.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Entity Embeddings of Categorical Variables](https://mlsanity.com/paper/1604.06737/) (2016)
- [Solving inverse-PDE problems with physics-aware neural networks](https://mlsanity.com/paper/2001.03608/) (2020)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [Coercing Machine Learning to Output Physically Accurate Results](https://mlsanity.com/paper/1910.09671/) (2019)
- [A Shallow Ritz Method for Elliptic Problems with Singular Sources](https://mlsanity.com/paper/2107.12013/) (2021)

## Built on by (most cited first, 1 of 1)

- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)

## Abstract (arXiv)

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.