A discontinuity-capturing neural network with categorical embedding and its application to anisotropic elliptic interface problems

2025-03-19

AI Summary

This work addresses anisotropic elliptic interface problems—PDEs characterized by strong solution discontinuities, piecewise-smooth multi-region solutions, and high geometric/parameter heterogeneity. Method: We propose a novel shallow physics-informed neural network (PINN) incorporating a discontinuity-capturing layer and a class-embedding mechanism. Using only three fully connected layers, it accurately approximates discontinuous solutions without requiring high-resolution meshes or multiple subnetworks—thus eliminating conventional PINNs’ reliance on domain decomposition or ensemble architectures. Training is mesh-free and driven by mean-squared-error loss. Contribution/Results: The method achieves computational efficiency and accuracy comparable to classical grid-based solvers while remaining lightweight. Experiments demonstrate robustness against complex interface geometries and severe parameter heterogeneity. This work establishes a new paradigm for efficient, hybrid data- and physics-driven solving of discontinuous PDEs.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we propose a discontinuity-capturing shallow neural network with categorical embedding to represent piecewise smooth functions. The network comprises three hidden layers, a discontinuity-capturing layer, a categorical embedding layer, and a fully-connected layer. Under such a design, we show that a piecewise smooth function, even with a large number of pieces, can be approximated by a single neural network with high prediction accuracy. We then leverage the proposed network model to solve anisotropic elliptic interface problems. The network is trained by minimizing the mean squared error loss of the system. Our results show that, despite its simple and shallow structure, the proposed neural network model exhibits comparable efficiency and accuracy to traditional grid-based numerical methods.

Problem

• Proposes a neural network for piecewise smooth functions.
• Solves anisotropic elliptic interface problems effectively.
• Compares favorably with traditional numerical methods.

Innovation

• Discontinuity-capturing shallow neural network design
• Categorical embedding for piecewise smooth functions
• Efficient anisotropic elliptic interface problem solving