Tailored Finite Point Operator Networks for Interface problems

TL;DR

This paper introduces TFPONets, a novel deep learning approach combining DeepONets and the Tailored Finite Point method, to effectively solve complex parameterized interface problems with discontinuities and singularities.

Contribution

The paper presents TFPONets, a new method that improves accuracy and generalization in solving interface problems without complex equation manipulation.

Findings

TFPONets outperform existing methods like DeepONet and IONet.

Enhanced accuracy in 1D and 2D interface problems.

Superior learning capabilities with limited data.

Abstract

Interface problems pose significant challenges due to the discontinuity of their solutions, particularly when they involve singular perturbations or high-contrast coefficients, resulting in intricate singularities that complicate resolution. The increasing adoption of deep learning techniques for solving partial differential equations has spurred our exploration of these methods for addressing interface problems. In this study, we introduce Tailored Finite Point Operator Networks (TFPONets) as a novel approach for tackling parameterized interface problems. Leveraging DeepONets and integrating the Tailored Finite Point method (TFPM), TFPONets offer enhanced accuracy in reconstructing solutions without the need for intricate equation manipulation. Experimental analyses conducted in both one- and two-dimensional scenarios reveal that, in comparison to existing methods such as DeepONet and IONet, TFPONets demonstrate superior learning and generalization capabilities even with limited locations.